A Simplified LPTN Model for a Fault-Tolerant Permanent Magnet Motor under Inter-Turn Short-Circuit Faults

Abstract

This paper proposes a simplified lumped parameter thermal network (LPTN) model for thermal analysis under the inter-turn short-circuit (SC) faults of a five-phase fault-tolerant permanent magnet (PM) motor for electric vehicles. The proposed model can consider circumferential heat transfer, variable copper loss, and uneven loss distribution. Firstly, an analytical calculation formula of inter-turn SC current is proposed to separate the copper loss of SC turns from electromagnetic simulation, and the accuracy of the formula is verified by finite element analysis (FEA). Secondly, a simplified LPTN model is proposed, the calculation formulas of the thermal resistance are given, and the simplified principle is explained. By comparing the results of different simplifications, it is found that the error between the simplified model and the original model is small. Finally, the proposed model is verified by experiments. The results show that the simplified model can achieve not only a high calculation speed but also a high accuracy. Moreover, the proposed model is applicable to all cases of asymmetrical temperature distribution.

1. Introduction

In electric vehicles, it is required that the main drive motor can still run smoothly in the case of motor failure; otherwise, it will cause significant damage to lives and properties. Therefore, the motor is required to have strong fault tolerance performance to minimize the impact of the fault part on the remaining normal part. At the same time, using fault-tolerant control algorithms can further enhance the fault-tolerant performance of the motor and ensure the safety of users [1,2,3].

The most important requirement of a fault-tolerant permanent magnet (PM) motor is the ability to manage and mitigate faults. In the operation of a PM motor, winding is one of the most fault-prone components [4]. In addition, 15–21% of winding problems are caused by high temperatures and insulation damage. During the operation of the motor, the insulation layer of the winding bears certain mechanical stress, electrical stress, and thermal stress. Additionally, when these stresses exceed a certain limit, the insulation layer of the winding will be damaged, leading to the occurrence of inter-turn short-circuit (SC) faults. Inter-turn SC faults play a key role in the faults of windings [5,6], especially in the case of a few SC turns. The SC current can reach the value that is higher than dozens of times the rated current, and the large amount of heat generated in the winding can cause a very high temperature in a short time. The high temperature will aggravate the damage of the insulation layer in the winding and lead to the motor eventually burning down. Therefore, an accurate thermal analysis taking all effects into account is particularly important during the design phase for predicting temperature distributions and hot spot temperatures under an inter-turn SC fault. Quantifying the rate of temperature rise and the maximum allowed time duration is of great significance. Furthermore, faults should be discovered in time, and appropriate actions need to be taken to prevent further damage. A thermal analysis of fault-tolerant motors under inter-turn SC faults can provide guiding value for fault-tolerant control and maintenance.

There is extensive research on the thermal analysis of PM motors under healthy conditions based on three methods [7,8,9,10,11,12,13]: lumped parameter thermal network (LPTN), finite element analysis (FEA), and computational fluid dynamics (CFD). Few papers consider the temperature rise in PM motors under inter-turn SC faults. A LPTN model was built with Motor-CAD in [14] to calculate the temperature rise in SC turns, and the results showed that the temperature of SC turns is much higher than that of healthy turns. However, Motor-CAD is only suitable for symmetric models; when inter-turn faults occur, the copper loss distribution is no longer uniform, and it is necessary to consider the circumferential heat transfer of the PM motor. A FEA model was built in [15], which considered the circumferential heat transfer of the PM motor under inter-turn SC faults. However, Ref. [15] only changed the input copper loss without modeling the SC turns, which may have a significant impact on the final temperature distribution. A 3D LPTN model and 3D FEA model were proposed in [16,17]; these papers used the method of electromagnetic–thermal (EM-thermal) simulation to analyze the transient temperature rise in SC turns under the inter-turn SC fault. First, the SC current is calculated in the electromagnetic simulation; then, the SC current is used to calculate the temperature rise; and then, the transient temperature of the PM motor is updated to the electromagnetic field to calculate the SC current again. The transient temperature rise in the PM motor is obtained by iterating continuously. The results show that the temperature obtained by EM-thermal simulation is more accurate and much higher than that obtained by thermal simulation. However, the time constants of electromagnetic simulation and thermal simulation are different, and it takes a long time to accurately simulate the effective value of SC current. As a result, solving the 3D FEA model by EM-thermal simulation is time-consuming and requires a well-configured computer. The 3D LPTN model in [16] has a high discretization level, however, this model has hundreds of nodes and the cost of modeling and computing is much higher than that of a typical LPTN model.

This paper proposes a simplified LPTN model for thermal analysis under inter-turn SC faults of a five-phase fault-tolerant PM motor. Circumferential heat transfer, variable copper loss, and uneven loss distribution can be considered in this model. Moreover, to separate the copper loss of SC turns from the electromagnetic simulation, an analytical calculation formula of inter-turn SC currents is derived. This formula can accurately calculate the copper loss with a different number of SC turns at different temperatures. The simplified LPTN model can achieve not only a high calculation speed but also a high accuracy. The remaining sections are organized as follows. In Section 2, the five-phase fault-tolerant PM motor is introduced. In Section 3, the loss under an inter-turn SC fault is calculated by the analytical calculation formula. In Section 4, the proposed simplified LPTN model is presented. In Section 5, the transient temperature rise is calculated for the healthy condition and the inter-turn SC fault condition. In addition, the calculation results of different simplified models are compared. In Section 6, the proposed model is validated by experiments. Finally, Section 7 concludes this paper.

2. Five-Phase Fault-Tolerant PM Motor

The motor studied in this paper is the rotor flux-switching PM motor proposed in [18]. Figure 1 shows the five-phase rotor flux-switching PM motor. Meanwhile, Table 1 presents the main geometry parameters of the motor. As depicted in Figure 2, the excitation directions of the main PMs (MPMs) are the same, which is different from the conventional motor type. In addition, the red line indicates the main flux path and the blue line indicates the auxiliary flux path. For the position of the stator and rotor in Figure 2a, the flux is firstly generated by the PM and then flux through the air gap into the stator tooth, through the winding, and finally back to the PM. Additionally, when the stator and rotor are in the position presented in Figure 2b, the flux generated by the PM first passes through the air gap into the winding and finally passes through the stator tooth back to the PM. Therefore, the magnitude and polarity of the flux in the windings change accordingly to the rotor position. It has the following advantages:

(1)

Multiple working harmonics. The armature pole pair number of the motor is 9, the modulation teeth (stator teeth) is 20, and the stator pole pair number of the motor is 11, which meets the design criteria of the magnetic field modulation motor. Therefore, the motor can not only use the fundamental magnetic field but also use the harmonic magnetic field modulated to generate torque.

(2)

Strong rotor robustness. The PM on the rotor is arranged in an inverted T shape. The lower part of PM is similar to the Halbach structure. On the one hand, it has a magnetic shielding function to reduce magnetic leakage at the bottom; on the other hand, the adjacent PM can assist in increasing the main magnetic field. At the same time, the integrated and hollow structure of the rotor not only enhances the robustness of the rotor but also reduces the moment of inertia of the rotor, which is conducive to the rapid response of the servo system.

(3)

Strong fault tolerance. The single-layer fractional slot concentrated winding and fault-tolerant teeth are used in the stator, which ensures the physical and magnetic circuit isolation between the windings and significantly reduces the coupling between the windings. Figure 3a shows phase coupling of the fault-tolerant PM motor. Compared with the self-inductance of windings, the mutual inductance between windings can be negligible, as shown in Figure 3b. At the same time, the number of harmonics in single-layer fractional winding is significantly higher than that of other winding structures, which is beneficial to restrain the short-circuit current of the motor. In addition, the five-phase winding makes it capable of continuous operation even in single-phase or multi-phase failure [19].

Table 1. Main parameters of the fault-tolerant PM motor.

Parameters	Value
Stator diameter	145 mm
Rotor diameter	91.7 mm
Axial stack length	60 mm
Airgap thickness	0.5 mm
Number of turns in one slot	35
Slot depth	20.5 mm
Rated current	6 A
Rated speed	1500 r/min

Table 1. Main parameters of the fault-tolerant PM motor.

Parameters	Value
Stator diameter	145 mm
Rotor diameter	91.7 mm
Axial stack length	60 mm
Airgap thickness	0.5 mm
Number of turns in one slot	35
Slot depth	20.5 mm
Rated current	6 A
Rated speed	1500 r/min

Figure 1. Topological structure of the fault-tolerant flux-switching PM motor.

Figure 1. Topological structure of the fault-tolerant flux-switching PM motor.

Figure 2. Flux analysis of the FTPM motor. (a) A1 is positive. (b) A1 is negative.

Figure 2. Flux analysis of the FTPM motor. (a) A1 is positive. (b) A1 is negative.

Figure 3. Phase coupling of the fault-tolerant PM motor. (a) Single-phase excitation magnetic field lines. (b) Self-inductance and mutual-inductance between phases.

Figure 3. Phase coupling of the fault-tolerant PM motor. (a) Single-phase excitation magnetic field lines. (b) Self-inductance and mutual-inductance between phases.

3. Analysis of Loss under Inter-Turn SC

Inter-turn SC fault is a very complex fault. Since the insulation layer between SC turns is not completely destroyed at the beginning, there is a contact resistance that makes the SC current smaller between the SC turns. The degree of inter-turn SC fault deepens as the temperature of SC turns increases, and it eventually becomes complete SC. Additionally, at this time, the insulation layer between SC turns is completely destroyed and the contact resistance tends to be infinitesimal. If the number of SC turns is only a few, the SC current may reach the value that is higher than dozens of times the rated current. Moreover, the value of the contact resistance is unpredictable. Therefore, in order to study the most serious temperature rise in the SC turns, the inter-turn SC fault studied in this paper is a complete SC.

The number of SC turns and the position of SC turns in the slot affect the inter-turn SC current. It can be known from Ref. [5] that the SC current reaches its maximum when SC turns are close to the slot opening. An equivalent processing of the windings should be carried out to distinguish the wires at different positions before analyzing the inter-turn SC current. As shown in Figure 4 and Figure 5, it is assumed that each wire has the same thickness and is tiled in the slot in turn. Inter-turn SC fault occurs in slot A1 and slot A2, the position of SC turns is close to the slot opening.

Since the phase windings of the fault-tolerant PM motor are magnetically well-isolated from each other, the mutual coupling between the phases can be neglected. Thus, the inter-turn SC current can be predicted based on the analysis of a single-phase winding. The illustration of inter-turn SC is given in Figure 6, in which E_f is the electromotive force of one SC turn, R_f is the resistance of one SC turn, I_f is the SC current, ω is the angular velocity of the motor, L_f is the inductance of one SC turn, and N_f is the number of SC turns. 1, 2, 3, …, N_f−1 indicates the node number.

Since an inter-turn SC current is much larger than a current in healthy turns and the healthy turns are far away from the SC turns, the mutual inductance between healthy turns and SC turns can be ignored. The calculation of SC current usually ignores the coupling effect between SC turns [20]. In order to consider the coupling effect between SC turns, the coupling coefficient k is introduced. Assuming that the coupling coefficient k between any two SC turns is equal, the SC current and the copper loss of SC turns can be estimated in Equations (1) and (2). The relationship between resistance and temperature is shown in Equation (3).

$$I_f=\frac{E_f}{\sqrt{R_f^2+{\left[\omega L_f+k(N_f-1)\omega L_f\right]}^2}}$$

(1)

$$P_f=I_f^2{N}_f{R}_f=\frac{E_f^2{N}_f{R}_f}{R_f^2+{\left[\omega L_f+k(N_f-1)\omega L_f\right]}^2}$$

(2)

$$R_f=R_0\left[1+\alpha \left(T_f-T_0\right)\right]$$

(3)

where R₀ is the resistance of one SC turn at room temperature, α is the temperature coefficient of copper resistivity, the value of α is approximately 0.393%/°C, and T_f is the real-time temperature of SC turns.

It can be seen from Equation (1) that, when other conditions remain unchanged, the larger the number of SC turns N_f is, the smaller the SC current in each turn will be. In the case of a one-turn SC, the SC current reaches the maximum, in which the values of E_f, L_f, and k are unknown, and the values of R_f, N_f, and ω are known. Through simulation calculations, the corresponding I_f can be obtained when N_f is 1, 2, and 3. The simultaneous equations can be used to solve the values of L_f, E_f, and k.

Table 2. Calculation of inter-turn SC current.

Number of SC Turns	Analytical (A)	FEA (A)	Error (%)
1	135.04	135.04	0
2	121.06	121.06	0
3	107.19	107.19	0
4	94.83	94.86	0.032
5	84.26	84.29	0.035
10	52.00	52.18	0.345
15	36.91	37.40	1.310
20	28.48	29.14	2.265
25	23.14	24.03	3.703
30	19.48	20.43	4.650
35	16.81	17.84	5.773

shows the error between the SC current calculated by the analytical formula and the SC current obtained by the FEA method.

It can be seen from Table 2 that with an increasing number of SC turns, the error tends to increase. This is due to the flux leakage not being the same everywhere in the slot, and the influence caused by the flux leakage variation in the slot not being considered in the analytical formula. However, when the number of SC turns is small, the error of the analytical formula is small, and the analytical formula can be considered accurate.

Figure 7 shows the relationship between the loss of SC turns and the number of SC turns in one slot. It can be seen from the figure that the loss of SC turns is far greater than the loss of healthy turns, when the number of SC turns is 4, the copper loss in the slot reaches the maximum. Although SC current reaches its maximum under one-turn SC, the copper loss of four turns SC is almost twice that of one-turn SC. Therefore, the transient temperature rise in SC turns under four-turn SCs is studied in this paper.

Table 3. Calculation of the inter-turn SC current under different temperatures.

Temperature (°C)	Analytical (A)	FEA (A)	Error (%)
20	94.83	94.86	0.032
30	93.20	93.05	0.161
40	91.60	91.30	0.329
50	90.02	89.59	0.480
100	82.63	81.75	1.076
150	75.86	74.98	1.174
200	70.28	69.11	1.693
250	65.17	64.02	1.796
300	60.67	59.57	1.847
350	56.68	55.66	1.833

shows the error between the SC current calculated by the analytical formula and the SC current obtained by the FEA method at different temperatures. With the increase in temperature, the resistance of SC turns increases, leading to the decrease in SC current. The results show that although there are errors, the accuracy is still very high at different temperatures. When using FEA method, hundreds of instantaneous currents are needed to obtain an effective value of SC current. When the temperature of the SC turns keeps rising, new SC currents need to be constantly calculated. Considering the increase in calculating speed, the analytical formula has more advantages than the FEA method.

In addition to copper loss, there are also iron losses, PM eddy current losses, and friction losses during the operation of PM motors. Iron loss and PM eddy current loss are calculated via FEA. Because the temperature coefficient of ferromagnetic material is far less than that of copper, iron loss and PM eddy current loss are less affected by temperature and can be considered as constants independent of temperature [21]. Since the research target of this paper is a low-speed PM motor, the friction loss accounts for less than one percent of the total loss, which can be negligible [22].

4. Simplified LPTN Model

The advantage of the LPTN method is that the calculation speed is very fast and it does not need the help of FEA. However, the LPTN method can only calculate the average temperature of each node. Moreover, the more nodes there are, the more complex the modelling will be and the higher the accuracy will be.

Due to the uneven distribution of loss, the temperature of the motor is not symmetrical in the circumferential direction. For example, the temperature of each stator tooth and winding is different. Therefore, it is not reasonable to take only one tooth and one slot to establish the LPTN model. According to the characteristics of LPTN method, the simplified idea of this paper is as follows. A few nodes are set up in the area with a small temperature gradient to obtain the average temperature, and a large number of nodes are set up in the area with a large temperature gradient to obtain the temperature distribution.

Before using the LPTN method to analyze the temperature rise in the fault-tolerant PM motor, the following assumptions should be made during this analysis:

(1)

All the loss of the fault-tolerant PM motor is converted into heat.

(2)

Only heat conduction and convection are considered, and the effect of heat radiation is ignored.

(3)

Except for copper, the effect of temperature on material properties is not considered.

(4)

Skin effect and proximity effect of winding are not considered.

(5)

Only copper loss, iron loss, and PM eddy current loss are considered.

(6)

Because of the structure of fractional slot concentrated winding, the length of the end winding is very short and the heat transfer between the end winding and the air at the end is not considered.

When inter-turn SC fault occurs, the SC turn generates a lot of heat. Part of the heat is transferred to the surrounding teeth and slot through the stator teeth, part of the heat is transferred to the housing along the radial direction through the stator yoke, and the rest of the heat is transferred to the rotor through the airgap. The heat on the rotor is transferred to the bearings and endcap along the axial direction. Finally, the heat generated by the fault-tolerant PM motor is emitted to the external air through the endcap and housing.

Since the loss on the rotor is small and the rotor will rotate continuously during normal operation and evenly absorb heat from the stator, and the difference in the thermophysical parameters of the rotor core, PM and shaft is not great, and their temperature is relatively close and can be equivalent to a node. As shown in Figure 8, the rotor (PM, rotor core and shaft) is viewed as equivalent to a uniform temperature distribution. The thermophysical parameters of equivalent nodes are weighted according to the volume (area) of each material and are the same in the radial and axial directions. The only difference is that the influence of the insulation layer between laminations needs to be considered axially [15,16,17].

The physical parameters of the equivalent can be calculated by the area weight method, and the calculation formula is shown in Equations (4)–(7).

$$C_{eq1}=m_{pm}{c}_{pm}+m_{ro}{c}_{ro}+m_{sh}{c}_{sh}$$

(4)

$${\lambda }_{eq1-rad}=\frac{A_{pm}}{A_{eq1}}{\lambda }_{pm}+\frac{A_{ro}}{A_{eq1}}{\lambda }_{ro}+\frac{A_{sh}}{A_{eq1}}{\lambda }_{sh}$$

(5)

$${\lambda }_{eq1-axi}=\frac{A_{pm}}{A_{eq1}}{\lambda }_{pm}+\frac{A_{ro}}{A_{eq1}}\frac{1}{\frac{k_{\epsilon }}{{\lambda }_{ro}}+\frac{1-k_{\epsilon }}{{\lambda }_{in}}}+\frac{A_{sh}}{A_{eq1}}{\lambda }_{sh}$$

(6)

$$A_{eq1}=A_{pm}+A_{ro}+A_{sh}$$

(7)

where Ceq₁ is the thermal capacitance of the equivalent; m_pm, m_ro, and m_sh are the mass of the PM, rotor core, and shaft, respectively; c_pm, c_ro, and c_sh are the specific heat capacities of the PM, rotor core, and shaft, respectively; λ_eq1-rad and λ_eq1-axi are the radial and axial thermal conductivity of the equivalent, respectively; A_pm, A_ro, and Ash are the area of the PM, rotor core, and shaft, respectively; λ_pm, λ_ro, and λ_sh are the thermal conductivity of the PM, rotor core, and shaft, respectively; and k_ε is the lamination factor of the rotor core.

The heat from the rotor is eventually dissipated into the air through the bearings and end caps, as shown in Figure 9. The thermal resistance in Figure 9 can be calculated by Equations (8)–(11).

$$R_{ra}=\frac{1}{k_r{S}_r}$$

(8)

$$R_{ea}=\frac{1}{k_e{S}_e}$$

(9)

$$R_{rb}=\frac{\ln \left(r_{bo}/r_{bi}\right)}{4\pi {\lambda }_b{l}_b}$$

(10)

$$R_{be}=\frac{\ln \left(r_{bo}/r_{bi}\right)}{4\pi {\lambda }_b{l}_b}+\frac{\ln \left(r_{eo}/r_{ei}\right)}{4\pi {\lambda }_e{l}_e}$$

(11)

where k_r and k_e are heat transfer coefficients between the rotor and air, and between the endcap and air, respectively; S_r and S_e are contact areas between the rotor and air, and between the endcap and air, respectively; r_bo and r_bi are outside diameter and inside diameter of the bearing, respectively; r_eo and r_ei are outside diameter and inside diameter of the endcap, respectively; λ_b and λ_e are thermal conductivity of bearing and endcap, respectively; and l_b and l_e are the thicknesses of the bearing and endcap, respectively.

When inter-turn SC occurs, SC turns will generate a lot of heat, and due to the low thermal conductivity of the windings, there will be a large temperature gradient in the SC slot, so more nodes should be allocated to calculating the temperature distribution. Figure 10 shows the LPTN model of the healthy slot and SC slot, and Figure 11 shows the LPTN model of the internal air.

The motor studied in this paper has concentrated winding. The end windings are short in length and fully in contact with the stator core, so the temperature of the end winding is very close to the temperature of the stator winding, so the end winding is not considered.

An accurate calculation of the thermal conductivity of winding is the key to the LPTN method. Since the winding is composed of multiple copper conductors and wire insulation, and there is a large amount of air in the slot and their positions are randomly distributed, it is necessary to consider the winding as an equivalent model, taking the equivalent model as a mixture of copper, insulation, and air [23]; the thermal conductivity of the equivalent model is shown in Equation (12).

$${\lambda }_{eq}=\left({\displaystyle \sum _{i=1}^n{\delta }_i}\right)/\left[{\displaystyle \sum _{i=1}^n\left({\delta }_i/{\lambda }_i\right)}\right]$$

(12)

where λ_eq is the thermal conductivity of the equivalent model, δ_i is the thickness of different materials, and λ_i is the thermal conductivity of different materials. The value of the thermal conductivity of the equivalent winding refers to Motor-CAD and is set to 1.831 W/(m∙k) in this paper.

All the thermal resistances can be derived using the governing principle of the heat conduction [24,25], and some important thermal resistances are given in (13) to (20).

$$R_{ty}=\frac{H_z}{2W_z{\lambda }_{ro}l}+{\theta }_1\frac{Q\ln \left(r_o/r_i\right)}{4\pi {\lambda }_{ro}l}$$

(13)

$$R_{th}=\frac{W_z}{2H_h{\lambda }_{ro}l}+\frac{l_{in}}{k_{in}{H}_{h}l}+\frac{W_h}{2H_h{\lambda }_{eq}l}$$

(14)

$$R_{st}=\frac{W_z}{2H_c{\lambda }_{ro}l}+\frac{l_{in}}{k_{in}{H}_{c}l}+\frac{W_c}{2H_c{\lambda }_{eq}l}$$

(15)

$$R_{hy}=\frac{H_h}{2W_h{\lambda }_{eq}l}+\frac{l_{in}}{k_{in}{W}_{h}l}+{\theta }_2\frac{Q\ln \left(r_o/r_i\right)}{4\pi {\lambda }_{ro}l}$$

(16)

$$R_{sh}=\frac{H_h+H_c}{2W_h{\lambda }_{eq}l}$$

(17)

$$R_{rt}=\frac{H_z}{2W_z{\lambda }_{ro}l}+\frac{Q}{{\alpha }_s{S}_s}+\frac{Q}{{\alpha }_r{S}_r}+\frac{Q\ln \left(r_{ro}/r_{ri}\right)}{2\pi l{\lambda }_{eq1-rad}}$$

(18)

$${\alpha }_s=28\left[1+{\left(k_{s}v\right)}^{0.5}\right]$$

(19)

$${\alpha }_r=28\left[1+{\left(k_{r}v\right)}^{0.5}\right]$$

(20)

where H_z and W_z are the height and width of the stator teeth, respectively; H_h and W_h are the height and width of the healthy turns, respectively; H_c and W_c are the height and width of the SC turns, respectively; l is the length of the axial stack; r_o and r_i are the outside diameter and inside diameter of the stator yoke, respectively; k_in is the thermal conductivity of the slot insulation; l_in is the length of the slot insulation; θ₁ is the correction coefficient of the contact area between the stator teeth and stator yoke; θ₂ is the correction coefficient of the contact area between the winding and stator yoke; Q is the number of stator teeth; S_s and S_r are the area of the stator inner surface and rotor outer surface, respectively; r_ro and r_ri are the outside diameter and inside diameter of the rotor core, respectively; v is the tangential velocity of the rotor surface; α_s and α_r are heat transfer coefficients of the stator inner surface and rotor outer surface, respectively; the value of k_s ranges from 0.7 to 0.8; and the value of k_r ranges from 0.1 to 0.2. The coefficient of convective heat transfer with the external air is generally 5–20 W/(m²∙k), The value of the coefficient of convective heat transfer with the external air refers to Motor-CAD and is set to 6.648 W/(m²∙k) in this paper.

Since the temperature distribution is mirrored along the axis of symmetry of the SC slot A1 and A2, only half of the fault-tolerant PM motor needs to be modeled to save computational costs, as shown in Figure 12. Thick black lines represent adiabatic surfaces on which there is no circumferential heat transfer due to the symmetrical temperature distribution. Because the thermal resistance between teeth and slots is much greater than that of the stator yoke, the heat flow path emitted by the SC slot is shown in Figure 13. The heat generated by the SC turns is difficult to transfer to distant slots. Additionally, the temperature of distant slots rises because of the rising temperature of the stator yoke. Because the temperature gradient of the stator yoke far away from the SC turns is smaller, the temperature of slots far away from the SC turns is similar. Therefore, slots and teeth far away from the SC turns may be regarded as equivalent nodes, as shown in Figure 12. The circumferential thermal resistance of this equivalent can be thought of as a series of R_th. The thermal resistance between the equivalent and the rotor can be considered in parallel with R_rt. The thermal resistance between the equivalent and the stator yoke can be considered in parallel with R_ty and R_hy.

The matrix model of the LPTN method is given in Equation (21).

$$\left[C\right]\left[\overline{T}\right]+\left[G\right]\left[T\right]=\left[Q\right]$$

(21)

where [C] is the heat capacity matrix of each node, $\left[\overline{T}\right]$ is the temperature rise matrix of each node, [T] is the temperature matrix of each node, [G] is the thermal conductivity matrix of each node, and [Q] is the heat source matrix for each node. The transient temperature of each node can be obtained by solving $\left[\overline{T}\right]$. The $\left[\overline{T}\right]$ obtained is the [T] in the next iteration, and the transient temperature curve can be obtained by iterative calculation. Moreover, due to the change in heat capacity caused by temperature change having no significant effect on the calculation results, the heat capacity is set as a constant [26].

5. Transient Temperature Rise in Fault-Tolerant PM Motor

5.1. Transient Temperature Rise in the Winding under Healthy Conditions

When the fault-tolerant PM motor runs under the healthy rated condition, there is no SC-turn node. Since the heat source is symmetrically distributed, there is no circumferential heat transfer. The temperature of each winding is the same, so only one tooth and one slot can be used to build the model. The speed of the motor is 1500 r/min, and the rated current amplitude is 6 A. The iron loss, copper loss, and eddy current loss obtained by simulation are input into the stator yoke, stator teeth, winding, and rotor node. Due to the low speed and the small size of the fault-tolerant PM motor, the friction loss on the rotor and bearing can be ignored. Finally, the transient temperature rise in the winding is obtained as shown in Figure 14. Since there is no additional cooling device, the winding reaches a steady state temperature of 103 °C.

When the fault-tolerant PM motor runs under healthy conditions, the effective value of the current in the winding is constant and the resistance of the wire does not change much with the temperature, so the change in copper loss caused by the change in the resistance of the wire can be ignored. In this case, all losses are considered constant.

Table 4. Temperature under healthy condition.

Component	Temperature (°C)	Component	Temperature (°C)
Winding	103.7	Rotor	98.7
Stator teeth	102.0	Bearing	95.0
Stator yoke	99.9	Endcap	91.9
Housing	94.3

shows the temperature rise in the fault-tolerant PM motor under healthy conditions. Since the iron loss on the stator is not very different from the copper loss on the winding, the final temperature of the stator teeth is similar to that of the winding, which is the premise of equivalent treatment of stator teeth and winding. If the temperature gradient between the stator teeth and winding is large under healthy conditions, the equivalent method proposed above may cause a large error, and other equivalent methods must be adopted.

5.2. Transient Temperature Rise in the Winding under Inter-Turn SC Fault

Since the copper loss in the SC slot is the largest under four-turn SC, as shown in Figure 7, this paper mainly studies the transient temperature rise in the winding under four turns. Since a large amount of heat is generated in SC turns, the temperature rise in the winding under inter-turn SC fault is much greater than that under healthy conditions and the change in winding resistance cannot be ignored. It should be noted that the change in winding resistance will cause the change in SC current. The change in SC current and winding resistance will lead to the change in copper loss in SC slot, and there are three possibilities as the temperature rises:

(1)

The resistance of the wire is much greater than the inductive reactance (R_f >> ωL_f), which is common in low-speed motors or motors with small wire diameters. In this case, the rise in resistance will lead to a significant decrease in SC current, and the loss of SC turns calculated at a high temperature will be less than the loss of SC turns calculated at room temperature.

(2)

The resistance of the wire is far less than the inductive reactance (R_f << ωL_f), which is common in high-speed motors or motors with large wire diameter. In this case, the rise in resistance has little effect on the SC current, and the loss of SC turns calculated at a high temperature will be greater than the loss of SC turns calculated at room temperature.

(3)

The third case is between the above two cases, in which the loss of SC turns calculated at a high temperature may be greater than, less than, or approximate to the loss of SC turns at room temperature.

In order to calculate the accurate temperature rise in the fault-tolerant PM motor under inter-turn SC fault, it is necessary to consider the change in the resistance with the temperature.

Figure 15 shows the process of this analysis, with the help of MATLAB R2018b developed by MathWorks. Two termination conditions are set in the process: one for reaching the maximum motor running time and one for temperature convergence. When the termination condition is not met, the temperature at this second will be recorded, and the resistance, inter-turn SC current, and copper loss will be updated to continue the calculation. The two termination conditions can be customized. In this paper, the maximum motor running time is set to 3 h, and the temperature convergence condition is that the absolute value of that second temperature minus the previous second temperature is less than 10⁻⁶ °C.

Figure 16a,b show the variation in SC current and the loss of SC turns with temperature under four-turn SC, respectively. Figure 16c shows the transient temperature rise in the SC turns. As shown in Figure 16, the SC current decreases obviously with the increase in temperature, but the final loss of SC turns shows a slight decrease due to the increase in resistance of SC turns.

As can be seen in Figure 16c, when inter-turn SC faults occurs, the heat capacity of SC turns is small due to the number of SC turns is 4. Before a large temperature difference is formed, the heat produced cannot be transferred out in time. Additionally, there will be a large temperature rise at the beginning of the fault, as shown in Figure 16d. The temperature rise in the first 50 s is about 150 °C, while the total temperature rise is about 300 °C. The research in this paper can provide some reference for fault-tolerant control of inter-turn SC faults, faults should be detected within the specified time, and an appropriate mitigation action is taken before causing further damage.

In Figure 16c,d, “Thermal only” indicates that the curve of copper loss with temperature is not considered, while “EM-thermal” indicates that the curve of copper loss with temperature is considered. Two temperatures of the fault-tolerant PM motor in Figure 16c are similar, but this is only a very coincidental situation and does not mean that the consideration of resistance changes is not important. When the number of SC turns, speed, or rated current change, the size relationship between R_f and ωL_f may change, and there may be a large difference in the temperature rise under the two calculation methods. Therefore, in order to accurately calculate the transient temperature rise under inter-turn SC fault, it is necessary to consider the change in resistance with temperature.

Table 5. Temperature under inter-turn SC faults.

Component	Temperature (°C)	Component	Temperature (°C)
SC turns	282.9	equivalent	151.6
Healthy turns in SC slot	221.2	Housing	131.3
Healthy turns in H slot 1	171.8	Rotor	149.8
Stator teeth near SC turns	185.4	Bearing	139.0
Stator yoke	148.1	Endcap	127.8

shows the temperature rise in the fault-tolerant PM motor under inter-turn SC faults. As can be seen from Table 5, due to the large heat flow and large thermal resistance, the temperature gradient at the SC turns is very large. The temperature gradient is much smaller elsewhere away from the SC turns. The temperature difference between the equivalent node and the stator yoke node is 3.5 °C under inter-turn SC faults, and the temperature difference is 3.8 °C under healthy condition. The temperature difference under the two conditions is very similar, which indicates that the temperature rise at the equivalent node is due to the increase in the temperature of the stator yoke rather than the heat flow from the SC turns. This proves the correctness of the simplification idea proposed in this paper.

5.3. Temperature Comparison under Different Simplifications

Since the above results are based on the equivalent of eight slots and teeth into a node, in order to prevent the error caused by too high of a degree of simplification, the steady-state temperature under different degrees of simplification is calculated as shown in

Table 6. Temperature under different degree of simplification.

Degree of Simplification	Temperature of SC Turns (°C)
9 slots and teeth	277.5
8 slots and teeth	282.9
7 slots and teeth	284.1
6 slots and teeth	284.7
5 slots and teeth	284.9

.

The rules can be drawn from Table 6 are as follows: As the degree of simplification decreases, the results become more and more accurate but the increase in accuracy becomes smaller and smaller, and by the end, it hardly changes at all, which conforms to the simplification theory proposed in this paper. Therefore, an appropriate degree of simplification should be selected according to the structure and loss of the motor. Generally, for small and medium-sized motors with low speeds, one or two slots can be left as transition slots. For high-speed motors with large inter-turn SC current, a few more slots can be left as transition slots.

6. Experimental Verification

In order to verify the accuracy of the simplified LPTN model, a prototype has been built, the motor is mounted on the test platform, as shown in Figure 17. PT100 temperature sensors are placed in the SC slot, health slot, and stator teeth to measure temperatures in these positions. An intelligent digital display controller is connected externally, and it shows the real-time temperature of temperature sensors. As shown in Figure 17, all wires of one phase are connected to the outside of the prototype by additional wires, and any wire can be short-circuited artificially outside the prototype.

The prototype is first tested under healthy conditions with a speed of 900 r/min and a load current set at 6 A for all phases. The temperature of ambient air is 13 °C. The temperature on the temperature sensor is recorded at certain intervals, and the temperature rise in winding is shown in Figure 18. The temperatures obtained by the two methods differ by 3.3 °C. Therefore, this error is acceptable and the LPTN model can be considered to be accurate.

Since there is no additional cooling equipment, in order to ensure the safe operation of the prototype, the prototype is tested under an inter-turn SC fault at the speed of 300 r/min. The load current in all phases is set to 3 A, and the number of SC turns is 5. The temperature rise in SC turns is shown in Figure 19.

Since the fault-tolerant PM motor has a low speed and a small loss, which leads to a low temperature rise, the change in temperature has little influence on the loss of SC turns. It can be seen from the results that the result obtained by considering the resistance change with temperature is more consistent with the experimental result, and the error is small, so the results obtained by this method can be considered to be accurate.

For a period of time after the inter-turn SC fault occurs, the measured temperature is lower than that calculated by the simplified LPTN method. This is because the PT100 temperature sensor has a certain heat capacity, so it cannot respond to the rapid temperature rise in time in the early stage, and it has a certain lag. After a period of time, this lag will disappear.

Figure 20 shows the transient temperature rise in each part of the FTPM motor, and

Table 7. Error between LPTN model and experimental results.

Component	LPTN (°C)	Measured (°C)	Error (%)
SC turns	45.9	44.3	3.6
Healthy turns in SC slot	41.2	39.7	3.8
Healthy turns in H slot 1	36.8	35.1	4.8
Stator teeth near SC turns	37.5	34.3	9.3

shows the error between LPTN model and experimental results. The error at the stator teeth is relatively large, probably because the temperature sensor is attached to the stator teeth and the contact surface between the two is small. The temperature sensor cannot reflect the true temperature of the stator teeth well.

7. Conclusions

In this paper, a simplified LPTN model for thermal analysis under inter-turn SC faults of a five-phase fault-tolerant PM motor has been proposed. An analytical calculation method of inter-turn SC current has also been proposed to separate the copper loss of SC turns from electromagnetic simulation. The LPTN method is used to analyze the temperature field of the motor under healthy conditions and inter-turn SC faults, and the experimental verification is carried out. The experimental results are consistent with the simulation results. Therefore, it can be concluded that the model has good accuracy under different heat sources, that is, it has good accuracy under different operating conditions.

In addition, a computer is used to compare the computational speed of the proposed simplified LPTN model and CFD. The CPU of the computer used is i7-12700, and the total running memory is 32 G. The results show that CFD takes about 4 h from the start of the calculation to convergence, while the proposed simplified LPTN model takes only 5 s.

It is shown that the simplified LPTN model can satisfy both calculation speed and accuracy. Furthermore, it can be applicated to all cases of asymmetrical temperature distribution.