Fault diagnosis of power transistors in a power converter of SRM drive based on a state inverse solution

Switched reluctance motor drives are a common technology used in traction motor drives. Herein, an online method is proposed for the fault diagnosis of power transistors in the popular asymmetric half ‐ bridge power converter of the SRM drive. Based on the rearrangement of three current sensors, each phase current was first calculated by solving equations associated with their detected values and the drive signals of the transistors. The faults in the transistors were then preliminarily detected by monitoring the error between the calculated sum of currents and the sum of actual phase currents detected by a current sensor. Once a fault was identified, the actual states of all transistors of the power converter were solved for inversely using a mathematical model and the necessary rule for a trade ‐ off. Then, the faulty transistors and the fault types were identified by comparing the actual states with the drive signals. Compared with prevalent methods, the diagnostic range of the proposed scheme was wider, and its control modes and the number of motor phases were not limited. A higher accuracy than currently available methods was its prominent advantage. The effectiveness of the proposed solution was also validated on a three ‐ phase 6/4 SRM drive.


| INTRODUCTION
The switched reluctance motor (SRM), as a new type of motor, has the advantages of a simple structure, high system efficiency [1,2], good robustness, and fault tolerance [3,4]. The SRM drive system has good economic indicators and operating characteristics in electric/hybrid automobiles, aerospace applications, and home appliances. It is a powerful competitor in the field of electric drive [5][6][7].
Because the SRM drive system often works in harsh environments, featuring high temperature and large vibrations, a variety of faults may occur [8,9]. For safety-critical systems, minor failures may also result in significant loss of personnel and property if not handled in a timely manner [10]. Therefore, to improve system reliability, the control system should have fault-tolerant control ability to enable the motor to operate normally even under some fault conditions. To achieve this goal, the system should first have a powerful fault diagnosis scheme to accurately identify and locate faults [11].
Power converters are the reliability-dominant part of SRM drive system [12]. The power transistors of a converter are prone to failure owing to excessive current, voltage, and heating caused by high frequency of operation. Therefore, research on fault diagnosis in power converters is important to improve the reliability of SRM drive systems. Methods of fault diagnosis for power converters can be mainly divided into spectrum analysis [13,14], current analysis [15][16][17], the drive signal assistant [12,[18][19][20], and the current reconfiguration method with a fault diagnosis function [21,22]. In the study by Gan et al. [13], the Blackman window function was used in the fast Fourier transform of the current of the bus and open-circuit faults were diagnosed based on the spectral distribution of the fault current. In the study by Gan et al. [14], the energy dispersion of wavelet packet nodes of the phase current were used as fault feature to determine the type of fault. However, the scheme cannot distinguish between the short-circuiting of the double-transistor and that of the upper transistor. In the study by Ro et al. [15], the coordinates of the four-phase current were transformed and various faults were diagnosed by the d-q axis current distribution pattern. In the study by Marques et al. [16], the phase currents were normalized by the reference current and averaged in the turn-on region. The fault type was accurately identified by the difference between the normalized currents. However, the scheme has poor real-time performance in terms of locating fault devices. In the studies by Chen et al. [12,19], extra current sensors were installed to identify different characteristics of the detected current in different fault states, and then cooperated with the drive signal to locate the faults. In the study by Shin et al. [18], the difference S erro in the drive signal between the upper and lower transistors was used as fault feature, and the fault was located when the sample value T cnt exceeded a discriminant threshold. In the study by Han et al. [20], three current sensors were rearranged, distinct fault features were extracted, and driving signals were used to formulate a mathematical model of the fault characteristics and create a fault dictionary. A method of configuration of the current sensors was proposed in the study by Peng et al. [21]. By changing the period of application period of PWM and the timing of A/D conversion in the overlapping area of the phase current, only two sensors were needed to control and diagnose the four-phase SRM. However, this can be used only for PWM control. In the study by Han et al. [22], a virtual current sensor was used to detect three-phase current in case of a short-circuit fault, but was easy to misjudge in case of sudden changes in load and different speeds because the slops of the current under excitation was considered the fault feature.
According to previous reports, on the premise of ensuring that the diagnostic method has good universality and a wide range of diagnosis, accuracy and cost are often incompatible. Therefore, it is advisable to appropriately increase the cost of the system to avoid a negative impact on its reliability due to the implementation of the diagnostic method. Therefore, to improve the accuracy of the fault diagnosis of the power transistor of the asymmetric half-bridge power converter (AHPC), a simple and efficient method is proposed. Compared with prevalent methods, it has a wider diagnostic range and higher diagnostic accuracy regardless of the control strategy used (including the VPWM, CCC, and APC modes). The general applicability and accuracy of the proposed method are our contributions to research in the area. In addition, the core innovation of the proposed solution is that it can directly obtain the actual states of power transistors in the power converter online. Therefore, the fault diagnosis of power transistors in power electronic circuits that require current feedback also has a certain reference significance here. In short, the method is more scalable. To visually compare the proposed solution with currently available diagnostic methods, a comparison of the methods of fault diagnosis is provided in Table 1.
The study is structured as follows: Section 2 introduces the working modes of the AHPC and summarizes the failures modes of a transistor. The proposed methods of fault diagnosis based on an inverse state solution are detailed in Section 3. The results of experiments on a 6/4 SRM drive are provided in Section 4, and Section 5 contains the conclusions of this paper.

| Topology of AHPC
The topology of the three-phase AHPC is shown in Figure 1, where A, B, and C denote the winding of phases A, B, and C, respectively, S 1 -S 6 denote power transistors, D 1 -D 6 denote freewheeling diodes, and LEM A -LEM C denote current sensors. The current sensors measure the winding current directly, and the three phases operate independently of one another. Two transistors in each phase constitute four states, corresponding to four working states in each phase. Taking phase A as an example, four paths of the current are shown in Figure 2a-d: excitation, lower freewheeling, upper freewheeling and demagnetization. For convenience, the four states are called ST1, ST2, ST3, and ST4, respectively.
To suppress torque ripple, and reduce switching loss and iron loss, each phase in the CCC and VPWM methods uses the soft chopping strategy in the turn-on region. Taking phase A as an example, in the turn-on region, the lower transistor S 2 is controlled by the position signal to stay ON, and the upper transistor S 1 is controlled by the chopping signal. The winding thus constantly switches between ST1 and ST2. In the turn-off region, both transistors are kept OFF and the winding state is ST4. Therefore, in normal operation, phase A has only three working states, ST1, ST2, and ST4, and their corresponding equations of circuit balance are, respectively, given as where U S denotes the power supply voltage, i, R, and L denote the current, resistance, and inductance of the phase winding, respectively, θ denotes the angle of the position of the rotor, ω denotes its angular speed, and iω ⋅ ∂L= ∂ θdenotes the rotating electromotive force in a direction opposite to that of U S in the rising region of phase inductance, and in the same direction as U S in the falling region of inductance.

| Short-circuit fault in the transistor
A 550-W three-phase 6/4 SRM prototype, the specifications of which are listed in Table 2, was used as simulation object, and the torque-current-rotor position (T − i − θ) and flux-currentrotor position (ψ − i − θ) data of the body of the motor were obtained through the finite element analysis software JMAG. The data were then imported into MATLAB/Simulink to build a nonlinear simulation model of the SRM control system, in which the power converter was built by the simpowersystem module and the step signal was used to cooperate with the control signal to simulate the fault. Taking phase A as an example, the failure modes of the power transistor in CCC mode are analysed. When S 1 is short, it is not controlled by the chopper signal, and phase A remains in ST1 in the turn-on region. However, the fault current has different characteristics at different speeds: When running at low speed, the waveform of the current is shown in Figure 3a. Figure 3b is an enlargement of part of Figure 3a at the fault moment, where S S2 * denotes the drive signal of S 2 , � denotes the moment of failure, i a denotes the current of phase A, and i ref denotes the reference current. Once the fault has occurred, as shown in region I of Figure 3b, i a in the turn-on region rises continuously under the action of the power supply voltage, and gets rid of the limitation of. In region II, the rotor is in the falling region of the inductance of phase A, and the short-circuiting of S 1 causes ST4 to be replaced by ST3. According to (2), the rate of change in the current at this time is Because the rotating electromotive force is negative, the current rate is positive and i a increases, which produces braking torque and reduces system efficiency. At high speed, the fault current is shown in Figure 3c. In region I, even if S 1 is short, the fault current and phase current under normal conditions are identical within the turn-on region because S 1 in the normal state is always ON throughout the turn-on region. There is thus no fault feature at this time. However, in the turn-off region, because ST4 is shielded by ST3, phase A cannot be demagnetized, and i a rises under the action of the rotating electromotive force, resulting in a braking torque.
The results of the simulation of the short-circuiting of S 2 are shown in Figure 4a, and Figure 4b is a local amplification of  Figure 4a at the moment of failure. The fault feature was similar to that of S 1 when a short-circuit occurs at high speed. This was because S 2 was always on in the turn-on region under normal conditions, and the fault current under the short-circuiting of S 2 did not change immediately in the turn-on region regardless of the speed. Once phase A was OFF, the short-circuit of S 2 helped maintain phase A in zero-voltage freewheeling state ST2, and i a did not decrease rapidly. When the falling region of inductance is reached, i a rises under the action of the rotating electromotive force as indicated by Eqauation (4). When S 1 and S 2 are both short, only ST1 works in phase A. Within the turn-on region, phase A is not controlled by the chopping signal, and i a rises freely under the action of the power supply voltage. The fault feature is identical to that of the separate S 1 short. In the falling area of inductance in the turn-off region, by (1), the rate of i a is Because the rotating electromotive force is negative at this time, i a rises rapidly under the combined action of the power supply voltage and the rotating electromotive force, resulting in a larger braking current that significantly affects the torque balance of the motor and the efficiency of the system.

| Open-circuit fault of transistor
When S 1 or S 2 is open separately, the changes are as shown in Figure 5a. In the turn-on region, phase A immediately enters the zero-voltage freewheeling state after fault occurrence, and i a decreases slowly under the rotating electromotive force.
Within the turn-off region, the winding enters ST4 and i a drops rapidly to zero under the reverse voltage. Because the winding cannot be excited again, i a cannot remain zero. The difference is that if S 1 is open, the zero-voltage freewheeling path consists of S 2 and D 1 ; if S 2 is open, the freewheeling path consists of S 1 and D 2 . Figure 5b shows the results of the simulation of S 2 and S 1 as open circuits at the same time. The winding enters ST4 immediately after fault occurrence, where i a drops rapidly and remains zero under the dual action of the reverse power supply voltage and the rotating electromotive force within the turn-on region. In case of an open circuit of transistors, the fault phase cannot be excited again and the SRM is in the phase-loss condition. The output of the torque is insufficient and unbalanced, which may damage the load equipment in extreme cases.
In summary, the failure of the transistor has a significantly bad impact on the operation of the SRM system. It is thus necessary to accurately diagnose different fault types and locate the fault transistor to improve the reliability of the SRM system.

| Phase current detection method
The accurate detection of the phase current is a prerequisite for closed-loop control of an SRM system. The traditional location of current sensors is shown in Figure 1. In traditional location detection, different faults often exhibit the same feature, which is not conducive to extracting the feature and location of faults.
According to the analysis in Section 2: when S 1 is turned off, the current flowing through D 1 is equal to i a ; when S 2 is turned off, the current flowing through D 2 is equal to i a ; when S 1 and S 2 are both turned on, no current flows in either diode. The current flowing through the diode reacts differently to different ON-OFF states of the transistors. Therefore, to implement the proposed method of fault diagnosis on the premise of accurately detecting phase current without affecting the normal operation of the system, the current sensors are rearranged: The positions of installation of LEM 1 ∼LEM 3 are shown in Figure 6a, and that of LEM 4 is shown in Figure 6b. The relationship between the sensors and each instance of current is represented in Figure 6c, where i fup and i fdn denote the upper and lower freewheeling bus currents, respectively, P and N denote the positive and negative directions, respectively, and n:2:1 denotes the ratio of turns. If it is assumed that the positive direction of all current sensors is from inside to outside, the detected values of the current sensors are where n denotes the number of times the winding wires pass through LEM 1 ∼LEM 3 in Figure 6a ∼ c, and i 1 ∼ i 4 denote the values of LEM 1 ∼LEM 4 , respectively.

| Current calculation method
The actual ON-OFF states of the transistors are defined as S Sk | k¼1;2;::: where S Sk | k¼1;2;:::;6 represents the actual ON-OFF states of S 1 ∼ S 6 .
According to the topology of the AHPC, the upper and lower freewheeling bus currents can be rewritten as whererepresents the operation of logic NOT. By entering (11) and (12) into (6), (7), and (8), we get 8 < : where m 1 , m 2, and m 3 denote the defined state coefficients, and are expressed as 8 < : It is clear from Equation (14) that the state coefficients m 1 , m 2 , and m 3 are completely determined by the ON-OFF states of the transistors, and the corresponding relationship is shown in Table 3.
Transforming Equation (12) into matrix form yields Regardless of the working condition of the motor, to ensure that the phase current can be solved for, Equation (15) must have a unique solution. The coefficient matrix must thus satisfy where rank [ ] denotes the rank of the matrix. After simulation calculation, when n = 1, Equation (16) can be satisfied in any case. Then, Equation (15) can be solved, and the expression for the solved value of phase currents i as , i bs , and i cs are shown in Equation (17).
However, in physical systems, the ON-OFF state of each transistor of the power converter cannot be obtained directly online. Given that when there is no fault in the transistor, the actual ON-OFF state is controlled by the drive signal: When the drive signal is high, the transistor is ON, and is OFF otherwise. The drive signal S Sk � is defined as S Sk * | k ¼ 1;2;::: where S Sk * denotes the drive signal of S k . The control coefficients of the power converter are defined as m 1 � , m 2 � , and m 3 � , and are determined by the drive signal of the transistors as follows: where m 1 � , m 2 � , and m 3 � denote the defined control coefficients. At this time, when there is no fault in the converter, the actual ON-OFF state and the drive signal are equal, because of which the control coefficients and the state coefficients are identical. In this case, the control coefficients of each transistor can be substituted as state coefficients into Equation (17) to calculate the phase current. Figure 7 shows the results of the simulated calculation of the phase current TA B L E 3 The relationship between the actual states of transistors and the state coefficients

| Fault location method
To correctly calculate the phase current, the necessary and sufficient condition is that the control coefficients and state coefficients of the power converter be the same. In short, the correct calculation of the phase current depends on all the transistors being in a healthy condition. For example, if S 2 is open when phase A is separately excited, the actual state coefficients (m 1 , m 2 , m 3 ) of the power converter are (-1,1,1), and the control state coefficients (m 1 � , m 2 � , m 3 � ) are (0,1,1). Then, the correctness of current calculation cannot be guaranteed if the control coefficient is used instead of the state coefficient to calculate the phase current.
After fault occurrence, if the state coefficients can be obtained and compared with the control coefficients, the fault type can be determined and the faulty device located. For example, if the state coefficient m 1 ¼ 1 is obtained at a certain time, it can be known from Table 3 that both S 1 and S 2 are OFF at this time. If the control coefficient m 1 � is one, this indicates that S 1 and S 2 are controlled to be ON. In short, id S 1 and S 2 are still OFF under a high-level drive signal, both transistors can be considered open. For convenience of locating faults, the relationships among the state coefficients, control coefficients, and fault types are shown in Table 4. In addition, when and only when there is no fault in the power converter is the calculated current equal to the actual current. Thus, the following is true: where i sum denotes the sum of the values of the calculated current, and i 4 denotes the detected value of LEM 4 , as shown in Equation (9). If the difference between i sum and i 4 exceeds a certain threshold, the solved phase current is not correct, where one or more faults in the transistors has led to the incorrect result. As shown in Table 4, to locate the fault position, the control coefficients and state coefficients of each phase need to be obtained. The former can be derived directly from the controller and the latter need to be further calculated. Even if the transistor fails, the correctly calculated phase currents can be obtained as long as the drive signal is consistent with the actual ON-OFF state. Therefore, if the difference between i sum and i 4 does not exceed a certain threshold, the calculated currents are considered correct, and the control coefficients may be the actual state coefficients of the fault phase. The mathematical model that can be used to inversely solve for the state coefficients can be shown as where m * k denotes the possible state coefficient. The threshold for the difference between i sum and i 4 is set to 0.05 A. When the difference between i sum and i 4 is greater than 2 A, the calculated current is considered incorrect. At this time, the system inputs the digital signal of the current sensors, i 1 ∼i 4 , into Equation (21) to obtain the possible state coefficients in case of fault. Figure 8 shows the simulated waveforms of i sum and i 4 at the instant that S 2 short circuits, where � indicates the fault moment. To avoid the current spike caused by the instantaneous switching of the transistors, a filter circuit is used. In case of a fault, when the difference between i sum and i 4 is greater than 2 A, the control coefficients (1,1,0), are collected. And the data collected from the current-sampling modules are 9.4383, 6.287, 7.0236, and 3.8861 A, respectively. Substituting them into Equation  Table 4 to determine that S 2 is short. Therefore, the rule of the trade-off for the state solution is that if the coefficients of a phase in all possible solutions are equal, it is assigned to the state solution; otherwise, the control coefficient of the phase is assigned to m k . The mathematical expression for this rule is where m k * denotes the control coefficient, m k 1 ; m k 2 ; :::; m k n denote all possible state coefficients obtained through Equation (21), and m k denotes the actual one.
To ensure the correctness of the solution for the current in case of a fault and render the ON-OFF state of the transistor consistent with its drive signal, the system keep the drive signal of the faulty transistors consistent with their actual ON-OFF states after locating the fault. For example, after diagnosing the S 2 short, its drive signal S S2 * is maintained at a high level. A flowchart of diagnosis using the proposed fault diagnosis method is shown in Figure 9.

| Experimental platform
A three-phase 6/4 switched reluctance prototype was used to verify the proposed method. Its key parameters are listed in Table 2. Figure 10a shows a block diagram of the experimental setup, and a photograph of the experimental setup used for the three-phase SRMs is given in Figure 10b. Three Hall-effect current sensors with a rated current of 50 A, LA-50P, were installed at specified locations according to the proposed current detection scheme. The current was sampled with a 16-bit A/D conversion chip, AD7606. For easy observation, the TLC5615 was used to convert the output signals, such as current to D/A. The modules of A/D and D/A were integrated in the control board. An incremental rotary encoder E6B2-CWZ6C with a resolution of 1000 pulses/rotation was used to calculate the position of the rotor. The control system used TMS320-F28335 as the core, supplemented by the necessary high-speed logic circuit, and had a drive circuit to achieve Fuzzy PI speed closed-loop control as well as a fault diagnosis function based on CCC. The FQA160N08 metal-oxide-semiconductor field-effect tube was used as the main transistor in the AHPC and they are installed on the heat sink on the back of the converter. Fault in the transistor was provoked by controlling the drive signals through external relays. A block diagram of the control system is shown in Figure 11. To prevent the drive signal from being used as both the output and the input of the closed-loop system, which leads to an inaccurate solution, the system used a delay link to delay the current by one sampling period to ensure the stability of the system. The control conditions are set as follows: the turn-on angle θ on ¼ 0°, turn-off angle θ off ¼ 27°, reference speed ω ref ¼ 1000r=min, and load torque Under the proposed diagnostic method, the drive signal of the faulty transistor was always consistent with its actual ON-OFF state: That is, it remained at a high or low level in case of a conditions. Therefore, the fault-related information of the power converter was obtained by observing the drive signal of each transistor. Figure 12 shows the calculated results of the phase currents under the fault-free condition, where i as , i bs , and i cs denote the calculated phase currents, i sum denotes the sum of the calculated currents, i 4 denotes the output of LEM 4 , and i err denotes the absolute value of the difference between i sum and i 4 . Figure 12a shows the result of the calculation of current, which verifies the correctness of the current detection method. In Figure 12b, the maximum value of i err is 0.21 A, considerably lower than the threshold. This shows that the calculated phase currents could be used not only in the feedback control of the motor, but also in the proposed power converter fault diagnosis scheme in the experimental system. in i err . When i sum increased above 2 A and remained above it for several sampling cycles, m 1 * jumped from zero to two, and m 2 * and m 3 * remained constant; S S1 * changed from a high to a constant, low level, which indicated that S 1 was open. Because S S1 * was always low, m 1 * only changed between one and two in case of a fault. The results showed that i err triggered the operation of the diagnostic program; i 1 ∼i 4 were thus entered at that time into Equation (21). All possible combinations of state coefficients and the control coefficients m 1 * , m 2 * , and m 3 * were then entered into the trade-off rule (22). The final status of the power converter (2, 1, 1) was solved for. Finally, the calculated state coefficients (2, 1, 1) and control coefficients (0,1,1) were entered into Table 4, where the open-circuit fault of S 1 was identified, and S S1 * was kept at low level. Figure 14 shows the results of diagnosis under the condition of short-circuiting of. As shown in Figure 14a, in the turnoff region of phase B, due to the inconsistency between the actual state of S 4 and its drive signal, an error occurred in the calculated current, and i sum began to deviate from i 4 at the falling edge of S S4 * .The fault diagnosis program was triggered when the threshold of 2 A was reached and exceeded by i err . The control coefficients of the power converter were (1,1,1) and the state coefficient of the output of the diagnostic program were (1,2,1). S 4 was determined to be short from the lookup table (Table 4). As shown in Figure 14b, the drive signal S S4 * was maintained at high level; thus, m 2 * was either zero or two under the fault condition. At the same time, the experimental results show that m 2 and m 2 * were consistent within the turn-on region of the phase B after short-circuit occurred. i err thus did not rise when fault occurred. In the turn-off region of phase B, m 2 * is 0 and m 2 is 1, which makes i err rise and fault feature is presented. The experimental results were consistent with the theoretical analysis.

| Faults in multiple power transistors
When a short-circuit fault occurs in a transistor, the winding generates an overcurrent under the action of the forward rotating electromotive force in the region of a drop in the winding inductance. At the same time, another transistor in the same phase is prone to short due to the overcurrent. As S 4 was already short, the experimental waveforms under an S 3 short fault are shown in Figure 15. When the short-circuit fault of S 4 was located by the system, i sum followed i 4 and i err did not exceed the trigger diagnostic threshold. m 2 * was either zero or two, indicating that S S4 * remained at a high level in case of a fault in S 4 , thus ensuring the consistency of m 2 and m 2 * . When S 3 was short as well as S 4 , as shown in Figure 15a, i err rapidly rose, and the diagnostic program was triggered. when the control coefficient of the power converter was (0,2,1) and the state coefficient of the output of the diagnostic program was (0,0,1). From Table 4, it can be concluded that the fault was a short circuit in S 3 , in which case S S3 * was maintained at high level. Because only ST1 was in phase B, m 2 * was zero throughout the fault condition. Figure 16 shows the results of the diagnosis of S 1 and S 2 , when both were open and S 4 was short. Before the opencircuit fault, as shown in Figure 16b, m 2 * was either zero or two, and i sum in Figure 16a followed i 4 , indicating that S S4 * was maintained at a high level. When S 1 and S 2 were both open and S 4 was short, the peak current of i sum occurred, which caused i err to rise rapidly and exceeded the 2 A threshold. After several current sampling cycles, i sum continued to follow i 4 and i err dropped rapidly to zero. In addition, the drive signals S S1 * and S S2 * were maintained at a low level, indicating that the system had diagnosed the fault. The experimental results also prove that when the threshold was reached, the fault diagnosis program was triggered and the data of i 1 ∼ i 4 were entered into (21). All possible states m 1 k ∼ m 3 k and control coefficients m 1 * ∼ m 3 * were then entered into Equation (22) to obtain the actual state coefficient (1,2,1). By entering the calculated state  Table 4, the faults in open circuits and could be diagnosed. Finally, both S S1 * and S S2 * were maintained at high level at the same time to ensure the correctness of the calculation of the current.

| CONCLUSION
Research on fault diagnosis in power converters is important for improving the reliability of the SRM drive system. Here, a method is proposed to diagnose faults in power transistors in the widely used AHPC. By comparing the control coefficient and the state coefficient of the power converter, faults can be located quickly and accurately. The proposed method can diagnose multiple faults, and it is not limited by the number of motor phases and type of control strategy. Moreover, it does not impose a heavy computational burden on the control system, and is suitable for online implementation. The results of experiments verified the effectiveness of the proposed method. The work here can help enhance the competitiveness of SRM drive systems in safety-critical applications. The focus here was on the topology of the AHPC, and our future work in the area will consider extending the method to a greater variety of topologies of power converters with fault tolerance.