The Effect of the Number of Parallel Winding Paths on the Fault Tolerance of a Switched Reluctance Motor

Abstract

Achieving increased fault tolerance in an electric motor requires decisions to be made about the type and specifications of the motor machine and its appropriate design. Depending on the type of motor, there are generally three ways to achieve an increased resistance of the drive system to tolerate resulting faults. The simplest way is to select the right motor and design it appropriately. Switched reluctance motors (SRMs) have a high tolerance for internal faults (in the motor windings). Failure tolerance can be improved by using parallel paths. The SRM 24/16 solution has been proposed, which allows for operation with four parallel paths. In this paper, a mathematical model designed to analyse the problem under consideration is provided. Based on numerical calculations, the influence of typical faults (open and partial short circuit in one of the paths) on the electromagnetic torque generated as well as its ripple and (source and phase) currents were determined. The higher harmonics of the source current (diagnostic signal) were determined. Laboratory tests were performed to verify the various configurations for the symmetric and emergency operating states. The feasibility of SRM correct operation monitoring was determined from an FFT analysis of the source current.

1. Introduction

Motors used in a fault-tolerant drive system are required to be able to continue operating after a fault has occurred [1,2,3,4,5,6]. In the case of variable-speed systems, a fault can occur on the controller side [2,3,4] or on the motor itself [1,4,5,6]. The failure of a transistor (open circuit) in a power converter results in symptoms that are identical to those that occur in an open phase in a motor [1,2,6]. Depending on the type of fault, the resulting failures may have different statuses. In the case of critical faults, further operation of the motor can be limited in time (e.g., until the motor is completely destroyed). This is extremely important for drive systems used in the aerospace [7,8,9,10] and automotive industries [11,12] in both electric [13], and hybrid driving systems [14,15].

Achieving increased fault tolerance in an electric motor requires decisions to be made about the type and specifications of the motor machine and its appropriate design. Depending on the type of motor, there are generally three ways to achieve an increased resistance of the drive system to tolerate resulting faults. The first way uses a physical duplication of the drive systems. This approach is often used in the aerospace industry, for example [16]. Where there are dimensional constraints (no space for duplicating drive systems), multichannel solutions are used [1,3,6,17]. In this case, each motor channel has its own power converter. Such systems can be classified as partial redundancy systems. Methods one and two are very expensive solutions (particularly for systems with full redundancy). The third approach is based on achieving increased fault tolerance through appropriate motor selection and design. Brushless permanent magnet motors (PMSM or BLDCM) or squirrel cage induction motors are very popular nowadays. They are very good machines, as evidenced by their use in industrial applications, for example. They are used in full-redundancy systems [18,19,20,21]. However, they do have one significant disadvantage. They tolerate the condition of partial winding short circuiting very poorly. This is particularly true for permanent magnet motors [22,23]. A motor that tolerates a partially short-circuited winding condition well is the switched reluctance motor (SRM) [1,4,5,6]. An SRM is suitable if a solution with increased fault resistance is required. The number of SRM phases significantly affects fault tolerance. This increases naturally with the number of phases. However, this has consequences. For example, the cost of the power converter increases. According to the authors, the use of a three-phase SRM allows the required level of fault tolerance resistance to be achieved. However, this requires that the SRM be designed appropriately.

For drive systems with a higher fault tolerance, monitoring is important. The task of the monitoring system is to determine the condition of the drive system. The monitoring system acquires this information from diagnostic output signals. Typical diagnostic signals used in practical applications include electrical signals (e.g., current or voltage) and non-electrical signals (vibration or temperature) [23]. In general, there are no universal diagnostic methods based on electrical signals. Each motor has its own peculiarities that require the use of dedicated diagnostic methods, e.g., the MCSA method for IM rotor fault detection [24,25]. In the case of SRMs, due to their specifics (electric independence of the phases), most known diagnostic methods are not applicable [26,27,28,29,30]. Some methods have been presented [31,32] for the detection of short-circuited SRMs operating with current regulation. Phase currents were used for fault detection. This is both an advantage and a major disadvantage of both methods. For motors with multiple phases, independent monitoring of each phase is required. A slightly different approach was taken in [33]. In this case, the Signal Injection (SI) technique was used for short circuit detection. The SI technique can only be used in current-free winding states. This is a very important limitation of the proposed method. As with the methods in [31,32], independent control of each motor phase is required for monitoring. In practice, the number of diagnostic signals should be limited. Monitoring and diagnostic capabilities should be sought on the basis of a minimum number of signals. According to the authors, the source current can be used for monitoring and diagnostics [34].

This work was carried out to determine the effect of changing the number of parallel winding paths on fault tolerance. The effect of the number of parallel winding paths on the SRM monitoring and diagnostic capabilities was also determined. The effects of two typical fault conditions, i.e., a discontinuity or open circuit (OC) and a short circuit (SC) in one of the motor winding paths, were analysed.

The monitoring and diagnostics were based on a Fast Fourier Transform (FFT) analysis. The FFT analysis was performed only for the source current. The source current of an SRM can be used as a signal for monitoring and diagnostics [34]. When all poles of each phase are connected in series, a good diagnostic signal can be provided. When analysing a source current with a larger number of parallel winding paths, there can be problems with correctly interpreting the resulting fault (SC or OC). In this work, the diagnostic signal was analysed as a function of the number of parallel winding paths or the motor control method. Recommendations for achieving resistance to failure are identified.

2. Model and Winding Configuration

A prototype switched reluctance motor (SRM) was designed as a multipole motor. Figure 1 shows a cross section (Figure 1a) and a physical model of the SRM (Figure 1b). The different phases of the SRM are marked in different colours (Figure 1a).

Table 1. Selected parameters of the SRM.

Lp.	Parameter	Value
1	Number of phases	3
2	Number of stator poles	24
3	Number of rotor poles	16
4	Stator diameter	140 mm
5	Rotor diameter	82 mm
6	Shaft diameter	25 mm
7	Active length	140 mm
8	Air-gap length	0.56 mm 1
9	Magnetic material	M230-35A
10	Number of turns per pole	105
11	Nominal current per path	3.5A
12	Nominal voltage	270 VDC

¹ Designed 0.3 mm.

provides selected geometrical dimensions and parameters of the prototype SRM.

Each SRM phase has eight stator poles. For the actual model, the connections of all the stator poles were provided. This is shown in Figure 1b. This allows for a high degree of flexibility with regard to stator pole configuration options. When all eight poles are connected in series, the classic configuration is obtained (a = 1, Figure 2a). With four stator poles per phase, two parallel winding paths are obtained (a = 2, Figure 2b). Typically, the minimum number of stator poles per phase is two. This provides four paths (a = 4, Figure 2c).

When configuring the SRM windings, the way each pole is supplied with power is also important. The authors assumed that each adjacent stator pole will have an inverted magnetic field direction. This would ensure that all magnetic couplings have the same sign. The polarity of the individual poles for conventional phase 1 is shown in Figure 2.

In Figure 2, three keys are introduced: S1, S2, and S3. Their configuration allows the cases of symmetry (SYM), partial short-circuit (SC), open circuit in one path (OC) and open circuit in one winding path with simultaneous partial short-circuit (OC/SC) to be analysed.

Table 2. Key configuration.

Working Condition	S1	S2	S3
SYM	on	on	off
SC	on	on	on
OC	off	off	off
OC/SC	off	on	on

provides the key configuration for the operating states analysed.

The OC/SC state is a combination of an open circuit (OC) with a short circuit (SC). Its practical importance is, according to the authors, comparable to that of an OC. For the sake of clarity, the following has been omitted. The configuration of the three keys still makes it possible to analyse the case of pole 1 being deenergised (S1 = on, S2 = off, S3 = on). However, this case was not analysed due to its very low probability of occurrence. Figure 3 shows a schematic of a typical power converter used to power the SRM.

3. Maths Models of SRM with Different Winding Paths in a Phase

3.1. SRM Model for Classic Configuration with One Winding Path in a Phase

The mathematical models of the three-phase SRM machines are presented. In these machines with nonlinearity of the magnetic circuit, the phase flux-linkages are ${\psi }_{\mathrm{i}}={\psi }_{\mathrm{i}}(\theta ,i_1,i_2,i_3)$ and the stator phase number is $i\in (1,2,3)$, depending on the electrical rotor position angle θ and the phase currents $i_1,\hspace{0.33em}{i}_2,\hspace{0.33em}{i}_3$. The model was formulated assuming there was a short-circuit in the winding coils in phase 1 with current $i_{\mathrm{sc}}$, flux ${\psi }_{\mathrm{sc}}$ and resistance $R_{\mathrm{sc}}$. The general structure of the mathematical model for a three-phase SRM with a classical configuration with one winding path per phase (a = 1, Figure 2a) can be written in the form:

$$\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ 0\end{array}\right]=\left[\begin{array}{cccc}{R}_1& 0& 0& 0\\ 0& R_2& 0& 0\\ 0& 0& R_3& 0\\ 0& 0& 0& R_{\mathrm{sc}}\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\\ i_{\mathrm{sc}}\end{array}\right]+{\displaystyle \frac{\mathrm{d}}{\mathrm{dt}}}\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_{\mathrm{sc}}\end{array}\right]$$

(1)

$$J{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{dt}}}+D\hspace{0.17em}{\omega }_{\mathrm{m}}\hspace{0.33em}+\hspace{0.33em}{T}_{\mathrm{L}}=T_{\mathrm{e}}; T_{\mathrm{e}}={\displaystyle \frac{\partial E_{\mathrm{co}}(\theta \hspace{0.33em}\hspace{0.17em}{i}_1\hspace{0.33em}{i}_2\hspace{0.33em}{i}_3\hspace{0.33em}{i}_{sc})}{\partial \theta }}$$

(2)

where $T_{\mathrm{e}}=T_{\mathrm{e}}(\hspace{0.17em}\theta ,i_1,i_2,i_3)$ is electromagnetic torque. The following symbols are used in Equations (1) and (2): u_i—phase voltages; $\mathrm{i}\in (1,2,3)$, R_i—phase resistances, $\mathrm{i}\in (1,2,3)$; ${\omega }_{\mathrm{m}}$—The mechanical rotational speed; $d\theta /dt=p\hspace{0.17em}{\omega }_{\mathrm{m}}$—electrical angular speed; p—number of rotor pole pairs; J—total rotor moment of inertia; D—rotor damping of viscous friction coefficient; $T_{\mathrm{L}}$—load torque; $E_{\mathrm{co}}(\theta \hspace{0.33em}\hspace{0.17em}{i}_1\hspace{0.33em}{i}_2\hspace{0.33em}{i}_3\hspace{0.33em}{i}_{\mathrm{sc}})$—total magnetic field co-energy in the machine’s air gap. Taking into account the nonlinearity of the magnetic circuit and the magnetic couplings between particular phases, the fluxes of individual phases ${\psi }_{\mathrm{i}}={\psi }_{\mathrm{i}}(\theta \hspace{0.33em}\hspace{0.17em}{i}_1\hspace{0.33em}{i}_2\hspace{0.33em}{i}_3\hspace{0.33em}{i}_{\mathrm{sc}})$, ${\psi }_{\mathrm{sc}}={\psi }_{\mathrm{sc}}(\theta \hspace{0.33em}\hspace{0.17em}{i}_1\hspace{0.33em}{i}_2\hspace{0.33em}{i}_3\hspace{0.33em}{i}_{\mathrm{sc}})$, $\mathrm{i}\in (1,2,3)$ in Equation (1) can be expressed as a sum of self and mutual fluxes, each depending on only one phase current, which can be written in the form:

$$\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_{s\mathrm{c}}\end{array}\right]=\left[\begin{array}{c}{L}_{1\mathsf{\sigma }}{i}_{1\hspace{0.17em}}+{\psi }_{11}(\theta ,i_1)+{\psi }_{12}(\theta ,i_2)+{\psi }_{13}(\theta ,i_3)+{\psi }_{1\mathrm{sc}}(\theta ,i_{\mathrm{sc}})\\ L_{2\mathsf{\sigma }}{i}_{2\hspace{0.17em}}+{\psi }_{21}(\theta ,i_1)+{\psi }_{22}(\theta ,i_2)+{\psi }_{23}(\theta ,i_3)+{\psi }_{2\mathrm{sc}}(\theta ,i_{\mathrm{sc}})\\ L_{3\mathsf{\sigma }}{i}_{3\hspace{0.17em}}+{\psi }_{31}(\theta ,i_1)+{\psi }_{32}(\theta ,i_2)+{\psi }_{33}(\theta ,i_3)+{\psi }_{3\mathrm{sc}}(\theta ,i_{\mathrm{sc}})\\ L_{\mathrm{sc}\mathsf{\sigma }}{i}_{\mathrm{sc}}+{\psi }_{\mathrm{sc}1}(\theta ,i_1)+{\psi }_{\mathrm{sc}2}(\theta ,i_2)+{\psi }_{\mathrm{sc}3}(\theta ,i_3)+{\psi }_{\mathrm{scsc}}(\theta ,i_{\mathrm{sc}})\end{array}\right]$$

(3)

where $L_{1\sigma },L_{2\sigma },L_{3\sigma },L_{\mathrm{sc}\mathsf{\sigma }}$ coefficients of end-turn self-inductances. The expression for total electromagnetic torque in Equation (2) can be written as the sum of two components, where the first $T_{\mathrm{eself}}$ is self-torque and the second $T_{\mathrm{emutual}}$ is mutual torque. In the form:

$$T_{\mathrm{e}}=T_{\mathrm{e}}(\theta \hspace{0.33em}{i}_1\hspace{0.33em}{i}_2\hspace{0.33em}{i}_3\hspace{0.33em}{i}_{sc})=T_{\mathrm{eself}}+T_{\mathrm{emutual}}$$

(4)

$$T_{\mathrm{eself}}={\displaystyle \sum _{i=1}^3{\displaystyle \underset{0}{\overset{i_i}{\int }}\frac{\partial {\psi }_{ii}(\theta ,i_i)}{\partial \theta }\hspace{0.33em}d{i}_i}}+{\displaystyle \underset{0}{\overset{i_{\mathrm{sc}}}{\int }}\frac{\partial {\psi }_{\mathrm{scsc}}(\theta ,i_{\mathrm{sc}})}{\partial \theta }\hspace{0.33em}d{i}_{\mathrm{sc}}}$$

(5)

$$T_{\mathrm{emutual}}={\displaystyle \sum _{i=2}^3{\displaystyle \sum _{j=1}^{i-1}\frac{\partial {\psi }_{ij}(\theta ,i_j)}{\partial \theta }{i}_i}}+{\displaystyle \sum _{j=1}^3\frac{\partial {\psi }_{\mathrm{sc}j}(\theta ,i_j)}{\partial \theta }\hspace{0.33em}}{i}_{\mathrm{sc}}$$

(6)

In special cases of the simple models, mutual fluxes may be constant or for simplest negligibly small, i.e., equal zero. Then, the electromagnetic torque (4) will be simplified to the expression (5).

3.2. SRM Model for Configuration with Two Parallel Winding Paths in a Phase

With four stator poles per phase, two parallel winding paths for each given phase are obtained (a = 2, Figure 2b). The structure of the mathematical model of the three-phase SRM machines with two parallel winding paths per phase (A and B) can be written in the following form:

$$\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ 0\end{array}\right]=\left[\begin{array}{cccc}{R}_1& 0& 0& 0\\ 0& R_2& 0& 0\\ 0& 0& R_3& 0\\ 0& 0& 0& R_{\mathrm{sc}}\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\\ i_{\mathrm{sc}}\end{array}\right]+{\displaystyle \frac{\mathrm{d}}{\mathrm{dt}}}\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_{\mathrm{sc}}\end{array}\right]; \left[\begin{array}{c}{i}_1\\ i_2\\ i_3\end{array}\right]=\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 1\end{array}\right]\hspace{0.33em}\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\end{array}\right]$$

(7)

The individual vectors of the voltages $u_{\mathrm{i}}$, currents $i_{\mathrm{i}}$, fluxes ${\psi }_{\mathrm{i}}$ and resistance matrices $R_{\mathrm{i}}$ (i = 1,2,3) in (7) have the form:

$$u_1={\left[u_{1\mathrm{A}}\hspace{0.33em}{u}_{1\mathrm{B}}\right]}^{\mathrm{T}}; i_1={\left[i_{1\mathrm{A}}\hspace{0.33em}{i}_{1\mathrm{B}}\right]}^{\mathrm{T}}; {\psi }_1={\left[{\psi }_{1\mathrm{A}}\hspace{0.33em}{\psi }_{1\mathrm{B}}\right]}^{\mathrm{T}}; R_1=\mathrm{diag}({\mathrm{R}}_{1\mathrm{A}}\hspace{0.33em}{\mathrm{R}}_{1\mathrm{B}})$$

(8)

$$u_2={\left[u_{2\mathrm{A}}\hspace{0.33em}{u}_{2\mathrm{B}}\right]}^{\mathrm{T}}; i_2={\left[i_{2\mathrm{A}}\hspace{0.33em}{i}_{2\mathrm{B}}\right]}^{\mathrm{T}}; {\psi }_2={\left[{\psi }_{2\mathrm{A}}\hspace{0.33em}{\psi }_{2\mathrm{B}}\right]}^{\mathrm{T}}; R_2=\mathrm{diag}({\mathrm{R}}_{2\mathrm{A}}\hspace{0.33em}{\mathrm{R}}_{2\mathrm{B}})$$

(9)

$$u_3={\left[u_{3\mathrm{A}}\hspace{0.33em}{u}_{3\mathrm{B}}\right]}^{\mathrm{T}}; i_3={\left[i_{3\mathrm{A}}\hspace{0.33em}{i}_{3\mathrm{B}}\right]}^{\mathrm{T}}; {\psi }_3={\left[{\psi }_{3\mathrm{A}}\hspace{0.33em}{\psi }_{3\mathrm{B}}\right]}^{\mathrm{T}}; R_3=\mathrm{diag}({\mathrm{R}}_{3\mathrm{A}}\hspace{0.33em}{\mathrm{R}}_{3\mathrm{B}})$$

(10)

$${\psi }_1=\left[\begin{array}{c}{L}_{1\mathrm{A}\mathsf{\sigma }}{i}_{1\mathrm{A}}+{\displaystyle \sum _{j=1}^3{\psi }_{1\mathrm{A}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})+{\displaystyle \sum _{j=1}^3{\psi }_{1\mathrm{A}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})+{\psi }_{1\mathrm{Asc}}(\theta ,i_{\mathrm{sc}})}}\\ L_{1\mathrm{B}\mathsf{\sigma }}{i}_{1\mathrm{B}}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{j=1}^3{\psi }_{1\mathrm{B}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{j=1}^3{\psi }_{1\mathrm{B}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})+{\psi }_{1\mathrm{Bsc}}(\theta ,i_{\mathrm{sc}})}}\end{array}\right]$$

(11)

$${\psi }_2=\left[\begin{array}{c}{L}_{2\mathrm{A}\mathsf{\sigma }}{i}_{2\mathrm{A}}+{\displaystyle \sum _{j=1}^3{\psi }_{2\mathrm{A}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})+{\displaystyle \sum _{j=1}^3{\psi }_{2\mathrm{A}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})+{\psi }_{2\mathrm{Asc}}(\theta ,i_{\mathrm{sc}})}}\\ L_{2\mathrm{B}\mathsf{\sigma }}{i}_{2\mathrm{B}}+{\displaystyle \sum _{j=1}^3{\psi }_{2\mathrm{B}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{j=1}^3{\psi }_{2\mathrm{B}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})+{\psi }_{2\mathrm{Bsc}}(\theta ,i_{\mathrm{sc}})}}\end{array}\right]$$

(12)

$${\psi }_3=\left[\begin{array}{c}{L}_{3\mathrm{A}\mathsf{\sigma }}{i}_{3\mathrm{A}}+{\displaystyle \sum _{j=1}^3{\psi }_{3\mathrm{A}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})+{\displaystyle \sum _{j=1}^3{\psi }_{3\mathrm{A}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})+{\psi }_{3\mathrm{Asc}}(\theta ,i_{\mathrm{sc}})}}\\ L_{3\mathrm{B}\mathsf{\sigma }}{i}_{3\mathrm{B}}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{j=1}^3{\psi }_{3\mathrm{B}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})+{\displaystyle \sum _{j=1}^3{\psi }_{3\mathrm{B}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})+{\psi }_{3\mathrm{Bsc}}(\theta ,i_{\mathrm{sc}})}}\end{array}\right]$$

(13)

$${\psi }_{\mathrm{sc}}=L_{\mathrm{sc}\mathsf{\sigma }}{i}_{\mathrm{sc}}+{\displaystyle \sum _{j=1}^3{\psi }_{\mathrm{sc}j}(\theta ,i_{j\mathrm{A}})}+{\displaystyle \sum _{j=1}^3{\psi }_{\mathrm{sc}j}(\theta ,i_{j\mathrm{B}})}+{\psi }_{\mathrm{scsc}}(\theta ,i_{\mathrm{sc}})$$

(14)

The electromagnetic torque (2) is the sum of two components where the first $T_{\mathrm{eself}}$ is self-torque and the second $T_{\mathrm{emutual}}$ is mutual torque.

$$T_{\mathrm{e}}=T_{\mathrm{e}}(\theta \hspace{0.33em}\hspace{0.17em}{i}_1\hspace{0.33em}{i}_2\hspace{0.33em}{i}_3\hspace{0.33em}{i}_{sc})=T_{\mathrm{eself}}+T_{\mathrm{emutual}}$$

(15)

$$T_{\mathrm{eself}}={\displaystyle \sum _{i=1}^3{\displaystyle \underset{0}{\overset{i_{i\mathrm{A}}}{\int }}\frac{\partial {\psi }_{i\mathrm{A}i\mathrm{A}}(\theta ,i_{i\mathrm{A}})}{\partial \theta }\hspace{0.33em}d{i}_{i\mathrm{A}}}}+{\displaystyle \sum _{i=1}^3{\displaystyle \underset{0}{\overset{i_{i\mathrm{B}}}{\int }}\frac{\partial {\psi }_{i\mathrm{B}i\mathrm{B}}(\theta ,i_{i\mathrm{B}})}{\partial \theta }\hspace{0.33em}d{i}_{i\mathrm{B}}}}+{\displaystyle \underset{0}{\overset{i_{\mathrm{sc}}}{\int }}\frac{\partial {\psi }_{\mathrm{scsc}}(\theta ,i_{\mathrm{sc}})}{\partial \theta }\hspace{0.33em}d{i}_{\mathrm{sc}}}$$

(16)

$$\begin{array}{l}{T}_{\mathrm{emutual}}={\displaystyle \sum _{i=2}^3{\displaystyle \sum _{j=1}^{i-1}\frac{\partial {\psi }_{i\mathrm{A}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})}{\partial \theta }\hspace{0.33em}{i}_{i\mathrm{A}}}}+{\displaystyle \sum _{i=2}^3{\displaystyle \sum _{j=1}^{i-1}\frac{\partial {\psi }_{i\mathrm{B}j\mathrm{B}}(\theta ,i_{j\mathrm{B}})}{\partial \theta }\hspace{0.33em}{i}_{i\mathrm{B}}}}+\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^3\frac{\partial {\psi }_{i\mathrm{B}j\mathrm{A}}(\theta ,i_{j\mathrm{A}})}{\partial \theta }\hspace{0.33em}{i}_{i\mathrm{B}}}}+{\displaystyle \sum _{j=1}^3\frac{\partial {\psi }_{\mathrm{sc}j}(\theta ,i_{j\mathrm{A}})}{\partial \theta }\hspace{0.33em}}{i}_{\mathrm{sc}}+{\displaystyle \sum _{j=1}^3\frac{\partial {\psi }_{\mathrm{sc}j}(\theta ,i_{j\mathrm{B}})}{\partial \theta }\hspace{0.33em}}{i}_{\mathrm{sc}}\end{array}$$

(17)

Equations (7)–(17) generally constitute a mathematical model of the SRM of the short circuit stator fault for configuration with two parallel winding paths per phase (A and B). The structure of Equations (7)–(17) can be extended to the mathematical model of the SRM through a configuration of the four current branches in each phase (A, B, C, D, Figure 2c). The elimination of the equation for the shorted circuit of the given winding in (1)–(6) or (7)–(17) along with the adjustment of the value of the equation coefficients allows equations to be obtained for special cases, i.e., one without a partial short circuit in the first phase. In practice, instead of a complex analytical method, a model for SRM based on the Look Up Table (LUT) is often used. Three-dimensional or two-dimensional LUTs are built based on FEM analyses or measurements. The models presented can be simplified in special cases by assuming the omission of selected magnetic couplings or the linearity of the magnetisation characteristics.

4. Numerical Calculations

If the windings are reconfigured (a = var) while keeping the same maximum winding path current in each path and ignoring the path resistance, the SRM speed will depend on the following:

$$n={\displaystyle \frac{U_{\mathrm{DC}}}{a}}$$

(18)

For the same operating point, a change in the number of parallel winding paths results in a change in the supply voltage. The SRM was designed for a supply voltage of U_DC = 270 V. For the calculations, it was assumed that for a = 1 the supply voltage U_DC would be 270 V. The calculations were performed for one operating point of the SRM. The control parameters were assumed to be, respectively, θ_on = −3.25° (turn-on angle) and θ_off = 7° (turn-off angle) at the SRM speed of n = 1000 rpm.

4.1. Waveforms

For each of the cases analysed (Table 1), the waveforms of the electromagnetic torque, source current, phase/path currents and short-circuit current were determined. The waveforms for a = 1 are shown in Figure 4 and Figure 5.

When all phase poles are connected in series, the SC state has a relatively small effect. Due to the lack of current limitation, partial loss of the SRM winding is compensated for by increasing the phase current (Figure 5). Consequently, the generated electromagnetic torque is not reduced (Figure 4), but losses from the faulty winding increase. The SC analysed does not give rise to a significant i_SC value (Figure 5). In the short-circuit current, i_SC reaches an rms value of 0.08 A. The effect of SC on the other phases is practically negligible. The bigger issue is the OC state. For a = 1, this eliminates an entire phase. The specifics of the SRM allow it to continue operating, but at the cost of a large increase in the electromagnetic torque ripple. The issue is the same when starting from any rotor position. For a = 1, there will be rotor positions for which a start-up is not possible.

For a parallel configuration of a = 2, the number of poles connected in series on each winding path decreases (Figure 2b). This is one of the configurations used in practice. Figure 6 and Figure 7 show the waveforms of electromagnetic torque, source current, and phase currents. Figure 8 shows the winding path currents and the short circuit current for the SC state.

The use of a = 2 has little effect on the state of SC. The short-circuit current for a = 2 is virtually the same as for a = 1(I_SCrms = 0.08 A). This means that there is an identical level of tolerance to that for a = 1. Once an SC has occurred in phase 1 winding path A, it has virtually no effect on the winding path currents in the other phases (the effect is marginal). In the case of the OC state, the configuration a = 2 makes it possible to continue further SRM operation with a reduced load torque. It is possible to start the SRM from any rotor position (with a reduced starting torque).

The a = 4 configuration is no longer typical. In practice, it is used mostly in low-voltage systems. The resulting waveforms are shown in Figure 9, Figure 10 and Figure 11.

As with a = 1 or a = 2, the configuration analysed tolerates SCs very well (ignoring the vibration and acoustic aspects). A much higher current of i_1A will flow in the faulty winding path (with lower R and L). With single-pulse control, the SRM will produce a higher electromagnetic torque at the expense of a poorer efficiency. The short circuit current itself, as for a = 1 and a = 2, has a marginal value (I_SCrms = 0.08 A). The SC state has virtually no effect on the winding path currents for Ph2 and Ph3. The a = 4 configuration tolerates OC very well. Performance is not significantly affected.

4.2. FFT Analysis of DC Current

In the case of an SRM, a good diagnostic signal is the source current, i_DC (Figure 3). SRM operation monitoring and diagnostics can be performed based on frequency analysis of the source current. The first harmonic of the diagnostic signal is determined from this equation:

$$f_1={\displaystyle \frac{n\cdot p}{60}}$$

(19)

where p is the number of rotor poles.

For the SYM 24/16 operation, the source current should include the zero component, the third harmonic, and its multiples. Deviations from symmetrical operation will be reflected in the waveform of the source current, i_DC. Consequently, there will be a first harmonic, a second harmonic, etc., in the source current.

For the classic configuration (a = 1), Figure 12 shows the diagnostic signal under analysis. The content of the major harmonics for the source current a = 1 is shown in Figure 13.

For an ideal symmetrical operation state, the first and second harmonics are absent. In the SC state, the first and second harmonics are present. For a = 1, the SC state represents a short circuit of 12.5% of the total number of phase windings. Once an SC has occurred, the first harmonic makes up several percent of the DC component (13%). With the higher number of winding turns shorted for a = 1, the amplitude of the first and second harmonics (etc.) will increase. A bigger issue is the detection of single-turn short circuits. For a = 1, the OC state indicates the loss of one phase. This is clearly manifested by a significant increase in the first harmonic. It reaches a value higher than the DC component. This is characteristic of the complete loss of one of the motor phases.

The use of the a = 2 configuration somewhat increases the resistance of the SRM to the effects of certain fault conditions. The diagnostic signal is shown in Figure 14. The content of the major harmonics is shown in Figure 15.

In the SYM state, changing the configuration from a = 1 to a = 2 makes no difference. The same harmonics are present and their interrelationships are very similar. Diagnostic harmonics, e.g., f₁, appear in the SC state. An SC state for a = 2 means that 25% of the number of winding turns on the path in question are shorted. The first and second harmonics increase by 16% and 9% with respect to the DC component. However, this is an increase of a few percent. A significant difference can be seen for the OC state. For a = 2 and the OC state, the first harmonic is no longer predominant. However, the first harmonic still reaches a much higher value than in the SC state.

The a = 4 case tolerates fault states best. Figure 16 shows the diagnostic signals obtained. The content of the higher harmonics of the diagnostic signal is shown in Figure 17.

The emergence of SC and OC states has the least effect on the diagnostic signal. The SC state for a = 4 means that 50% of the winding turns of the path are already shorted. This is a relatively severe short circuit. The first and second harmonics of the diagnostic signal represent the highest percentages of the DC component (23% and 14%). However, the effect of a strong SC state is significantly limited by the configuration a = 4. The same is true for the OC state. The loss of one of the four phase paths is not very problematic. Good fault tolerance in this operating state has certain consequences. As can be seen in Figure 17, the main harmonics of the diagnostic signal (f₁, f₂) achieve similar amplitude values for both the SC and OC states. From this point of view, it is the most difficult configuration to diagnose. However, this does not affect the monitoring of the SRM operation. The appearance of the diagnostic signal harmonics f₁ and f₂ is an unambiguous symptom of an internal SRM fault.

5. Laboratory Test

The laboratory test stand is shown in Figure 18. The laboratory stand consists of the following: a DC power supply (TDK Lambda (Tokyo, Japan) 300 V, 17 A), a six-channel power analyser (Yokogawa (Tokyo, Japan) WT1600), torque transducer (Magtrol (Rossens, Switzerland) TS 107 10 N·m, 15,000 rpm), a programmable DC load (ITECH (New Taipei City, Taiwan) 500 V, 5.0 kW), an oscilloscope recorder (Yokogawa DL850) with current probes, a prototype BLPM machine acting as a load and a three-phase diode bridge. The tested SRM prototype is powered by a dedicated half-bridge power converter (noncommercial). The power converter was made in such a way that it is possible to measure the i_DC current. The test conditions were identical to the numerical calculations.

The power converter was used as shown in Figure 3. The control system used an encoder with a resolution of 0.25° mech. Depending on the configuration, one or two power analysers were used. They allowed 12 current (and voltage) signals to be recorded. For a = 4, 21 channels are needed to record all currents (for the SC state). Due to the technical limitations of the test stand, the number of recorded currents was reduced to six (in simultaneous recording).

The scope of the results presented was limited to phase currents, the source current, the short circuit current (for the SC state) and selected winding path currents (for a = 2 and a = 4).

For a = 1 (a series connection), the waveforms of the phase currents and the source current were determined. For the operating states analysed, these values are shown in Figure 19, Figure 20 and Figure 21. For i_DC, the DC component and the first three harmonics were determined. These are shown in Figure 22.

In general, the waveforms obtained from the phase currents are similar to those obtained in the numerical tests (Figure 5). The slight differences between the phase currents are reflected in the source current. It is not perfectly symmetrical. For the SYM state, small harmonic values for f₁ and f₂ appear due to the internal asymmetry of the SRM (Figure 22). The emergence of SC or OC fault states results in an increase in the harmonics f₁ and f₂. These are similar to the results obtained in the numerical test results (Figure 13). The SC state increases the harmonics f₁ and f₂. This is a significant increase, but less than for the OC state. For the a = 1 configuration, the OC state always generates high values of f₁ and f₂. The values of f₁ and f₂ depend on the number of shorted winding turns, but in general, for a = 1, they will always be lower than in the OC state. This is a characteristic of a = 1. The fault diagnosis for a = 1 is unambiguous.

For a = 2, the waveforms of the SYM, SC and OC state currents are shown in Figure 23, Figure 24 and Figure 25. The harmonics of the diagnostic signal from the iDC are shown in Figure 26.

For a = 1, the waveforms obtained from the currents under laboratory conditions are close to those derived from numerical calculations. The winding path currents did not reveal a significant effect of the change induced by a faulty operating state. In the case of the harmonic analysis of the diagnostic signal, the configuration a = 2 confirms the existence of the diagnostic harmonics f₁ and f₂. Their values are close to those obtained in the numerical calculations. The effect of the SC state is clearly seen. At the same time, for a = 2, the OC state generates high harmonic values for f₁ and f₂. Their amplitudes are no longer higher than the amplitudes of the DC component. For a = 2, the resulting fault type can be determined unambiguously from the amplitude values obtained.

The case a = 4 gives the highest resistance to the emergence of fault states during operation. The waveforms of the phase currents and the source current are shown in Figure 27, Figure 28 and Figure 29. The winding path currents for SC are shown in Figure 30, Figure 31 and Figure 32. The harmonic current distribution of the i_DC is shown in Figure 33.

The laboratory tests for the a = 4 case confirmed the results of the numerical calculations. This applies to both phase and winding path current waveforms. Under laboratory conditions, no effect of the fault state on the winding path currents was found. This applies to both the failed phase (Figure 30) and the adjacent phases (Figure 31 and Figure 32). This is a significant advantage of a = 4. This solution provides a very high fault tolerance. The laboratory tests also confirmed that monitoring alone, based on FFT analysis of the source current, unambiguously indicates the emergence of a fault (by increasing harmonics f₁ and f₂). The problem is an unambiguous diagnosis of the type of fault that has occurred. In the case of a = 4, both SC and OC do not have a significant negative impact on the performance of SRM.

6. Conclusions

The use of a larger number of parallel SRM winding paths increases the model’s resistance to fault states in operation. Partial short-circuiting (only one pole) is relatively well tolerated in all configurations analysed. The classic configuration (serial connection) for the SC state is the most favourable. This does not mean that the other configurations tolerate this fault state significantly worse. If the entire pole winding is short-circuited, the number of parallel winding paths has no effect on the short-circuit current level. For the OC state, increasing the number of parallel winding paths increases the tolerance of the fault state in operation. For an OC state with more than one parallel path, starting from any rotor position is possible (with limited starting torque). In the event of loss of one phase, starting from any position may not be possible.

The SC state analysed (only one pole) is a completely safe operating condition. If fewer windings are short-circuited in a pole, thermal damage (transition to the OC state) must be expected.

The SC state for more than one parallel path shows that for SRM 24/16, there is very little effect on the remaining winding path currents. For SRMs, this favours the use of solutions with as many parallel winding paths as possible. This also applies to the OC state.

Monitoring based on a single diagnostic signal being the source current is acceptable for SRMs. The SRM diagnostics for up to two parallel paths are not problematic. In the case of four parallel paths, the monitoring itself unambiguously detects the operating fault state. However, determining the cause of the fault (diagnosis) is very difficult. Both the SC and the OC states can, under certain conditions, give faults with very similar symptoms. For four parallel paths, the SC and OC states are not critical operating states. These are states that are well tolerated by this configuration.