**High Resistance Fault-Detection and Fault-Tolerance for Asymmetrical Six-Phase Surface-Mounted AC Permanent Magnet Synchronous Motor Drives**

**Claudio Rossi 1, Yasser Gritli 1,2,\*, Alessio Pilati 1, Gabriele Rizzoli 1, Angelo Tani <sup>1</sup> and Domenico Casadei <sup>1</sup>**


Received: 19 April 2020; Accepted: 12 June 2020; Published: 15 June 2020

**Abstract:** In the last decade, the interest for higher reliability in several industrial applications has boosted the research activities in multiphase permanent magnet synchronous motors realized by multiple three-phase winding sets. In this study, a mathematical model of an asymmetric surface-mounted six-phase permanent magnet synchronous motor under high resistance connections was developed. By exploiting the intrinsic properties of multiphase machines in terms of degrees of freedom, an improved field-oriented control scheme is presented that allows online fault detection and a quite undisturbed operating condition of the machine under high resistance connections. More specifically, the proposed strategies for online fault-detection and fault-tolerance are based on the use of multi-reference frame current regulators. The feasibility of the proposed approach was theoretically analyzed, then confirmed by numerical simulations. In order to validate experimentally the proposed strategies, the entire control system was implemented using TMS-320F2812 based platform.

**Keywords:** stator fault; high resistance connection; fault-detection; fault-tolerant control; six-phase permanent magnet synchronous machines; field-oriented control

#### **1. Introduction**

Multiphase permanent magnet synchronous machines (m-φ PMSMs) have gained significant attention, in variable-speed drives and generation systems, during the last decade. They have attracted much interest becoming a viable solution for a wide range of modern industry applications such as aerospace applications, naval propulsion, energy generation and transportation electrification [1–3].

The main reasons of this interest are justified by combining the well-known advantages of permanent magnet synchronous machines (PMSMs) in terms of high efficiency, high power density and high dynamic performances, with the strengths of multiphase machines, which provide lower torque ripple, lower current harmonics, fault tolerance capabilities and higher torque/power switch rating [1,4].

As mentioned in [5], it is known that stator windings faults account for 40% of the overall electric machine failures in different industrial applications. As with any rotating electrical machine in healthy conditions, PMSMs undergo mechanical and electrical stresses symmetrically distributed inside the machine. In particular, the stator windings are subjected to several stresses induced by a combination of several factors, including thermal effects, vibrations, voltage spikes caused by adjustable-speed drives and environmental conditions [5,6].

Effectively, under healthy operating conditions, the phase impedances of the stator windings are identical, leading to balanced phase currents. When a stator fault occurs, phase currents are no longer balanced determining too high peak values, which may affect the performance and reliability of the motor.

Stator winding faults can be roughly classified as open-circuit or short-circuit, both affecting the phases and/or the terminal connections. Some other anomalies are not destructive at incipient stage but can evolve and initiate serious damages to the motors. During the last decade, high resistance connection (HRC) has been clearly identified as the main initiator of the stator failures. In fact, HRC is a progressive failure mode that can affect any power connection and/or end winding and is mostly caused by a combination of excessive vibration levels, poor workmanship, metal fatigue, overheating and/or corrosion of the power contact surfaces. Comprehensive descriptions of HRC mechanism propagation, initiated by the above factors, are available in [6–8].

Although the advantageous performances of m-φ PMSMs over their classical 3-φ PMSMs counterparts, they are more subjected to stator faults owing to the higher number of stator windings. Thus, if such a fault is not properly cleared in a timely manner, it spreads and may conduct to rotor magnets demagnetization and eventual dramatic damages with serious unexpected outages [9,10].

Several techniques have been developed for the diagnosis of stator faults for three-phase machines. Classical off-line techniques such as measurement and comparison of winding resistances or related voltage drops, analysis of the temperature distribution by infrared thermography and partial discharge analysis, have been successfully applied [11–13]. Even if standard off-line techniques provide reliable results for stator asymmetry, they are limited by the necessity of full or partial motor disassembly, and/or dedicated equipment or setup.

Actually, the four main diagnostic strategies, adopted to cope with the above limitations and to provide useful fault indexes and fault-tolerant control strategies, are known as model-based fault diagnosis, knowledge-based fault diagnosis, signal-based fault diagnosis and hybrid fault diagnosis approaches [14–16]. Although the advantages of each approach, in general signal-based approach is the preferred strategy.

Stator fault diagnosis for 3-φ PMSMs has been extensively investigated in the literature [5]. The main focus was on inter-turn short-circuit faults [17,18] and open-phase faults [19], while investigations on HRC are relatively few, except recent studies presented in [8,20,21]. Based on high order sliding mode principle, an interesting current-control scheme designed to simultaneously detect and tolerate the existence of HRC, was investigated in [8]. The fault compensation is obtained by canceling extra current dynamics, which provides more effective *d*–*q* currents components tracking. In [20], a full online diagnosis of HRC is developed for delta-connected PMSM using zero-sequence current component. The proposed approach has shown interesting performances for detecting and quantifying the extend of the fault. Another relevant approach dedicated to detecting and estimate the HRC severity, for vector-controlled PMSM drive system, is investigated in [21]. The proposed technique is based on a signal injection in the reference signals applied to the controlled PMSM under its normal operation, leading to the appearance of DC components in the stator phase currents, used thereby for detecting and estimating the propagation degree of HRC.

With reference to m-φ PMSMs, much more efforts have been directed toward effective fault tolerant strategies than diagnosis approaches. Among several stator configurations of m-φ PMSMs, multiple three-phase winding sets are probably the most preferred for numerous industrial applications. The interest for these configurations of m-φ PMSMs is mainly justified by the fact that each stator winding set can be separately supplied by standard three-phase inverters, which allows crucial power flow modularity control, particularly useful under stator fault conditions.

Different control strategies for 6-φ PMSMs can be found in the literature, namely, constant V/f control, field-oriented control (FOC) and direct torque control (DTC). A recent comprehensive review, including theory, simulations, and experimental tests is presented in [22]. Under the context of stator fault risks, several fault tolerant control techniques have been developed for exploiting the inherent

active redundancy due to the associated precious degrees of freedom, for ensuring a continuity of operation even in case of more than one phase affected by a stator fault. An open-phase fault-tolerant control for 6-φ PMSM is developed in [23], where the torque capability is maximized considering the overcurrent protection limits. In [24], two optimal current control modes that tolerate open-phase fault, with minimum stator losses and maximum torque output, have been analyzed. A novel optimized open-phase fault tolerant control strategy is developed in [25], where a genetic algorithm is used to maximize the average torque and minimize the torque ripple for post-fault operating condition. In [26], an intelligent complementary sliding-mode control approach was developed for effective open-phase fault tolerance. To maintain the stability of the fault-tolerant control of the 6-φ PMSM drive system, a Takagi–Sugeno–Kang–type fuzzy neural network with asymmetric membership function was developed to estimate unknown lumped uncertainty including parameter variations, external disturbances and nonlinear friction force online.

Recently, effective diagnostic techniques dedicated to 6-φ PMSMs under stator fault conditions have been presented in [26–28], for open-phase faults or inverter related-faults and in [29] for short-circuits.

In [27], both open-phase fault and open-switch fault are tolerated using a voltage compensationbased fault tolerant control. The detection process is based on real time monitoring of the current amplitude in a specific subspace, considering a predetermined threshold for alert. Diagnosis and fault tolerant approaches have been successfully developed in [28] for open-phase faults, open-switch faults and short-switch faults in T-type three-level inverter fed dual-three phase PMSM drives. After fault detection process, which is based on the amplitude variations of a specific current space vector, an effective open-phase fault compensation was achieved without changing the machine model, nor the control framework. Open-switch faults and short-switch faults are tolerated by making full use of the remaining healthy-phases after faults. Although the verified good performances in terms of detection and fault-tolerance, the use of current amplitude space vector for the proposed diagnosis technique may show some limitations when changing the operating conditions of the machine. In [29], a new magnetic equivalent circuit model for dual-three phase PMSM under winding short-circuit is proposed for accurate prediction of the fault impact.

Based on the above observations, the present contribution is aimed to present a new strategy of fault-detection and fault-tolerant control for 6-φ PMSMs affected by HRC. The existing papers on this type of fault are dealing with three-phase machines, but to the best of the author's knowledge, no recent papers were published investigating HRC in 6-φ PMSMs. The presented strategy is based on the use of multiple space vector transformations for developing a new mathematical model able to deal with stator winding affected by HRC, and on the use of multi-reference frame current regulators for implementing an improved field-oriented control (IFOC) scheme.

The proposed strategy allows online fault-detection and fault-tolerance to be achieved without the need of additional hardware, in both stationary and dynamic operating conditions, as it is based on detecting the DC component of a new variable representing the Fault Index. In this way it is possible to avoid the critical problem of detecting certain current harmonic components having variable frequency depending on the operating speed.

This study is organized as follows. Modeling of the investigated 6-φ PMSM under HRC, in terms of multiple space vector, is presented in Section 2. The proposed fault-detection and fault-tolerant control strategies are detailed in Section 3. Numerical simulations and experimental tests are presented and commented in Sections 4 and 5, respectively. The recommended Fault Index for quantifying the degree of HRC, as well as the corresponding simulation and experimental evaluations are presented in Section 6.

#### **2. Motor Modeling under HRC**

In this Section, the concept of multiple space vector transformations is presented for a set of six variables. Then, the mathematical modeling of an asymmetrical six-phase surface mounted permanent magnet synchronous motor, affected by HRC, is presented.

#### *2.1. Multiple Space Vector Transformations for Six-phase Systems*

Multiple space vector transformation concept is an effective approach for multiphase electrical systems representation [30]. It is particularly useful for modeling, analysis and control design for multiphase machines and drives. For a given electrical system composed by six real variables *xa*1, *xa*2, *xb*1, *xb*2, *xc*1, *xc*2, a new set of three complex variables *y*1, *y*3, *y*5, can be obtained by means of the symmetrical linear direct and inverse transformations expressed by Equations (1) and (2), respectively.

$$\begin{cases} \overline{y}\_{S1} = \frac{1}{3} \left[ \mathbf{x}\_{a1} + \mathbf{x}\_{a2} \,\overline{\alpha} + \mathbf{x}\_{b1} \,\overline{\alpha}^4 + \mathbf{x}\_{b2} \,\overline{\alpha}^5 + \mathbf{x}\_{c1} \,\overline{\alpha}^8 + \mathbf{x}\_{c2} \,\overline{\alpha}^9 \right] \\\ \overline{y}\_{S3} = \frac{1}{3} \left[ \mathbf{x}\_{a1} + \mathbf{x}\_{a2} \,\overline{\alpha}^3 + \mathbf{x}\_{b1} + \mathbf{x}\_{b2} \,\overline{\alpha}^3 + \mathbf{x}\_{c1} + \mathbf{x}\_{c2} \,\overline{\alpha}^3 \right] \\\ \overline{y}\_{S5} = \frac{1}{3} \left[ \mathbf{x}\_{a1} + \mathbf{x}\_{a2} \,\overline{\alpha}^5 + \mathbf{x}\_{b1} \,\overline{\alpha}^8 + \mathbf{x}\_{b2} \,\overline{\alpha} + \mathbf{x}\_{c1} \,\overline{\alpha}^4 + \mathbf{x}\_{c2} \,\overline{\alpha}^9 \right] \end{cases} \tag{1}$$

$$\begin{cases} \begin{array}{c} \mathbf{x}\_{d1} = \mathfrak{R}\_{\varepsilon}[\overline{y}\_{S3}] + \overline{\mathfrak{y}}\_{S1} \cdot \mathbf{1} + \overline{\mathfrak{y}}\_{S5} \cdot \mathbf{1} \\\ x\_{b1} = \mathfrak{R}\_{\varepsilon} \left[ \overline{y}\_{S3} \right] + \overline{\mathfrak{y}}\_{S1} \cdot \overline{\mathfrak{a}}^{4} + \overline{\mathfrak{y}}\_{S5} \cdot \overline{\mathfrak{a}}^{4} \\\ x\_{c1} = \mathfrak{R}\_{\varepsilon} \left[ \overline{y}\_{S3} \right] + \overline{\mathfrak{y}}\_{S1} \cdot \overline{\mathfrak{a}}^{8} + \overline{\mathfrak{y}}\_{S5} \cdot \overline{\mathfrak{a}}^{8} \\\ x\_{a2} = \mathfrak{R}\_{\mathfrak{m}} \left[ \overline{y}\_{S3} \right] + \overline{y}\_{S1} \cdot \overline{\mathfrak{a}} + \overline{\mathfrak{y}}\_{S5}^{} \cdot \overline{\mathfrak{a}}^{7} \\\ x\_{b2} = \mathfrak{R}\_{\mathfrak{m}} \left[ \overline{y}\_{S3} \right] + \overline{y}\_{S1} \cdot \overline{\mathfrak{a}}^{5} + \overline{y}\_{S5} \cdot \overline{\mathfrak{a}}^{11} \\\ x\_{c2} = \mathfrak{R}\_{\mathfrak{m}} \left[ \overline{y}\_{S3} \right] + \overline{y}\_{S1} \cdot \overline{\mathfrak{a}}^{9} + \overline{y}\_{S5} \cdot \overline{\mathfrak{a}}^{11} \end{array} \tag{2}$$

where <sup>α</sup> = *<sup>e</sup><sup>j</sup>* <sup>π</sup>/6, the symbol "·" represents the scalar product and "\*" the complex conjugate.

It is worth noting that the obtained three space vectors, involved in the transformations (1) and (2), can arbitrarily move in the respective independent subspaces, namely α1–β1, α3β<sup>3</sup> and α5–β5.

#### *2.2. Model of the 6-*φ *PMSM under HRC*

The multiple space vector transformation principle allows the modeling of six-phase AC PMSMs by means of vectors expressed in three α*–*β subspaces. The considered machine is a 30 asymmetrical 6-φ PMSM, as illustrated by Figure 1.

**Figure 1.** Six-phase surface-mounted permanent magnet synchronous motor with two sets of three phase windings and separated neutral points.

The model is developed under the conventional assumptions usually adopted for the analysis of AC rotating electrical machines and considers up to the eleventh spatial harmonic of the magnetic field in the air gap. Based on the concept of multiple space vector representation, the electrical quantities are developed in a stationary reference frame.

Assuming a set of six different stator phase resistances *RSa*1, *RSb*1, *RSc*1, *RSa*2, *RSb*<sup>2</sup> and *RSc*2, under the effect of HRC, the two sets (*k* = 1, 2) of three voltage equations can be expressed as

$$\begin{cases} v\_{\rm Ssk} = R\_{\rm Ssk} \ i\_{\rm Ssk} + \frac{dq\_{\rm Ssk}}{dt}; \\ v\_{\rm Sls} = R\_{\rm Skl} \ i\_{\rm Ssk} + \frac{dq\_{\rm Sks}}{dt}; \\ v\_{\rm Sck} = R\_{\rm Sck} \ i\_{\rm Sck} + \frac{dq\_{\rm Sck}}{dt}; \end{cases} \quad k = 1, 2. \tag{3}$$

Considering the direct transformation (1), the six stator voltage expressions given by Equation (3) can be reformulated, leading to the following new set of three stator voltage space vectors

$$\overline{w}\_{\rm S1} = R\_{\rm S0}^{+} \overline{i}\_{\rm S1} + \overline{R}\_{\rm S10}^{\*} \overline{i}\_{\rm S1}{}^{\*} + \overline{R}\_{\rm S10} \overline{i}\_{\rm S3} + \overline{R}\_{\rm S4} \overline{i}\_{\rm S3}{}^{\*} + \overline{R}\_{\rm S4}^{\*} \overline{i}\_{\rm S5} + R\_{\rm S0}^{-} \overline{i}\_{\rm S5} + \frac{d\overline{\rho}\_{\rm S1}}{dt} \tag{4}$$

$$
\overline{w}\_{\rm S3} = \overline{R}\_{\rm S10}^{\ast} \overline{i}\_{\rm S1} + \overline{R}\_{\rm S4} \, \overline{i}\_{\rm S1}{}^{\ast} + R\_{\rm S0}^{+} \, \overline{i}\_{\rm S3} + R\_{\rm S0}^{-} \, \overline{i}\_{\rm S3}{}^{\ast} + \overline{R}\_{\rm S10} \, \overline{i}\_{\rm S5} + \overline{R}\_{\rm S4}^{\ast} \, \overline{i}\_{\rm S5} + \frac{d\overline{\rho}\_{\rm S3}}{dt} \tag{5}
$$

$$
\overline{w}\_{\rm S5} = \overline{R}\_{\rm S4} \ \overline{i}\_{\rm S1} + R\_{\rm S0}^{-} \ \overline{i}\_{\rm S1}^{\*} + \overline{R}\_{\rm S10} \ \overline{i}\_{\rm S3} + \overline{R}\_{\rm S4}^{\*} \ \overline{i}\_{\rm S3}^{\*} + R\_{\rm S0}^{+} \ \overline{i}\_{\rm S5} + \overline{R}\_{\rm S10} \ \overline{i}\_{\rm S5} + \frac{d\overline{\rho}\_{\rm S5}}{dt} \tag{6}
$$

where,

$$R\_{S0}^{+} = \frac{1}{6} \left[ R\_{\text{Sd1}} + R\_{\text{Sd2}} + R\_{\text{Sb1}} + R\_{\text{Sb2}} + R\_{\text{Sc1}} + R\_{\text{Sc2}} \right] \tag{7}$$

$$R\_{S0}^{-} = \frac{1}{6} \left[ R\_{\text{Sd1}} - R\_{\text{Sd2}} + R\_{\text{Sb1}} - R\_{\text{Sb2}} + R\_{\text{Sc1}} - R\_{\text{Sc2}} \right] \tag{8}$$

$$\overline{R}\_{\text{S4}} = \frac{1}{6} \left[ R\_{\text{S4}1} + R\_{\text{S42}} \overline{\alpha}^4 + R\_{\text{S61}} \ \overline{\alpha}^4 + R\_{\text{S62}} \ \overline{\alpha}^8 + R\_{\text{S41}} \ \overline{\alpha}^8 + R\_{\text{S62}} \right] \tag{9}$$

$$\overline{R}\_{\rm S10} = \frac{1}{6} \left[ R\_{\rm Sd1} + R\_{\rm Sd2} \overline{\alpha}^{10} + R\_{\rm Sb1} \ \overline{\alpha}^{4} + R\_{\rm Sb2} \ \overline{\alpha}^{2} + R\_{\rm Sc1} \ \overline{\alpha}^{8} + R\_{\rm Sc2} \ \overline{\alpha}^{6} \right] \tag{10}$$

From the previous equations it is evident that in case of balanced stator resistances the only resistance component different from zero is *R*<sup>+</sup> *S*0.

The stator flux space vectors can be expressed by

$$\overline{\varphi}\_{S1} = L\_{S1} \hat{i}\_{S1} + 2 \,\, q\mu\_{\text{M1}} \, \cos(\chi) \, e^{j \, \, \theta} + 2 \,\, q\mu\_{\text{M1}} \, \cos(11 \, \, \chi) \, e^{-j \, \, 11 \, \, \theta} \tag{11}$$

$$\overline{\varphi}\_{\rm S3} = L\_{\rm S3} \, \overline{i}\_{\rm S3} + 2 \, \overline{\varphi}\_{\rm M3} \, \cos(3 \, \text{y}) \, e^{j \, \text{3 } \, \theta} + 2 \, \overline{\varphi}\_{\rm M9} \, \cos(9 \, \text{y}) \, e^{-j \, \text{9 } \, \text{0}} \tag{12}$$

$$\overline{\boldsymbol{\varphi}}\_{\text{S\!S}} = \boldsymbol{L}\_{\text{S\!S}} \hat{\boldsymbol{i}}\_{\text{S\!S}} + 2 \,\,\boldsymbol{q}\_{\text{M\!S}} \cos(\text{5 } \boldsymbol{\gamma}) \,\, \boldsymbol{e}^{j\ 5 } \,\, ^0 + 2 \,\, \boldsymbol{q}\_{\text{M\!T}} \cos(\text{7 } \boldsymbol{\gamma}) \,\, \boldsymbol{e}^{-j\ 7 } \,\, \boldsymbol{0} \tag{13}$$

In Equations (11)–(13), γ = (π − β)/2 and the constant values ϕ*M*1, ϕ*M*3, ϕ*M*5, ϕ*M*7, ϕ*M*9, ϕ*M*11, are expressed as in Equations (14)–(19), respectively.

$$\varphi\_{M1} = \frac{2\,\mu\_0\,\,N\_S\,\,L\,\,\pi\,\,H\_{R,\max}\,\,K\_{BS1}\,\,K\_{RS1}}{\pi^2} \tag{14}$$

$$
\varphi\_{M3} = \frac{2\,\mu\_0\,\,N\_S\,\,L\,\,\pi\,\,H\_{R,\max}}{\pi^2} \frac{\mathcal{K}\_{BS3}\,\,\mathcal{K}\_{BS3}}{9} \tag{15}
$$

$$\varphi\_{\rm M5} = \frac{2 \,\mu\_0 \,\, N\_S \,\, L \,\, \pi \,\, H\_{\rm R,max}}{\pi^2} \frac{K\_{\rm BS5} \,\, K\_{\rm RSF}}{25} \tag{16}$$

$$
\rho\_{M\mathcal{T}} = \frac{2\,\mu\_0 \, N\_S \, L \,\text{\textpi } H\_{\text{R,max}}}{\pi^2} \frac{K\_{BS\mathcal{T}} \, K\_{RS\mathcal{T}}}{49} \tag{17}
$$

$$
\rho\_{M9} = \frac{2\,\mu\_0\,\text{N}\_S\,L\,\text{\textpi}\,H\_{R,\text{max}}\,K\_{RS9}\,\text{\textdegree}\_{RS9}}{\pi^2} \frac{\text{K}\_{RS9}\,\text{K}\_{RS9}}{81} \tag{18}
$$

$$\varphi\_{M11} = \frac{2\ \mu\_0 \ N\_S \ L \ \pi \ H\_{R,\max}}{\pi^2} \frac{K\_{BS11} \ K\_{RS11}}{121} \tag{19}$$

Based on the previous equations, the electromagnetic torque can be formulated as

$$\begin{array}{llll} T\_{cm} = & \left\{ \begin{array}{c} \boldsymbol{\delta} \text{ } \boldsymbol{p} \text{ } q\_{M1} \left[ \over \boldsymbol{i}\_{S1} \cdot \boldsymbol{j} \text{ } \cos(\boldsymbol{\gamma}) \text{ } \boldsymbol{e}^{\boldsymbol{j}} \text{ } \boldsymbol{\delta} \right] + 18 \text{ } \boldsymbol{p} \text{ } q\_{M3} \left[ \over \boldsymbol{i}\_{S3} \cdot \boldsymbol{j} \text{ } \cos(\boldsymbol{3} \text{ } \boldsymbol{\gamma}) \text{ } \boldsymbol{e}^{\boldsymbol{j}} \text{ } \boldsymbol{\delta} \right] + \\ + 30 \text{ } \boldsymbol{p} \text{ } q\_{M5} \left[ \over \boldsymbol{i}\_{S5} \cdot \boldsymbol{j} \text{ } \cos(\boldsymbol{5} \text{ } \boldsymbol{\gamma}) \text{ } \boldsymbol{e}^{\boldsymbol{j}} \text{ } \boldsymbol{\delta} \right] + 42 \text{ } \boldsymbol{p} \text{ } q\_{M7} \left[ \over \boldsymbol{i}\_{S5} \cdot \boldsymbol{j} \text{ } \cos(\boldsymbol{7} \text{ } \boldsymbol{\gamma}) \text{ } \boldsymbol{e}^{\boldsymbol{j}} \text{ } \boldsymbol{\delta} \right] + \\ + 54 \text{ } \boldsymbol{p} \text{ } q\_{M9} \left[ \over \boldsymbol{i}\_{S3} \cdot \boldsymbol{j} \text{ } \cos(\boldsymbol{9} \text{ } \boldsymbol{\gamma}) \text{ } \boldsymbol{e}^{\boldsymbol{j}} \text{ } \boldsymbol{\delta} \right] + 66 \text{ } \boldsymbol{p} \text{ } q\_{M11} \left[ \over \boldsymbol{i}\_{S1} \cdot \boldsymbol{j} \text{ } \cos(11 \text{ } \boldsymbol{\gamma}) \text{ } \boldsymbol{e}^{\boldsymbol{j}} \text{ } \boldsymbol{1$$

It can be noted that the torque, besides the fundamental component, contains several additional oscillating contributions that can be compensated by using suitable machine design and current control techniques.

#### **3. Proposed Fault-Detection and Fault-Tolerant Strategy**

In this section, the proposed strategy for an online fault-detection of HRC in 6-φ PMSM is presented. Then, an improved field-oriented control (IFOC) scheme based on appropriate stator currents control, which provides fault-tolerance against the investigated stator fault, is presented.

In order to better understand the principle of the proposed IFOC scheme and the associated online fault detection algorithm, it is useful to analyze the fault effects in the rotating reference frames, where the needed current regulators are conventionally implemented.

Taking into account a stator windings design with isolated neutral points as shown in Figure 1, the current space vector in the α3–β<sup>3</sup> subspace is equal to zero. Thus, the voltage equations expressed in the stator reference frame by Equations (4)–(6), can be limited to the 1st and 5th subspaces, corresponding to Equations (4) and (6), respectively. It is worth noting that for control purposes the voltage equations in α1–β<sup>1</sup> subspace will be expressed in a reference frame (*d*1–*q*1) rotating at an angular speed of ω, whereas the equations in α5–β<sup>5</sup> subspace will be expressed in a reference frame (*d*5–*q*5) rotating at an angular speed of 5 ω.

The transformation of the space vectors from stator reference frame to the rotating reference frame, regarding the 1st and 5th subspaces, can be carried out by using the following relationships

$$
\overline{\mathfrak{X}}\_1^r = \overline{\mathfrak{X}}\_1 e^{-j\
f} \tag{21}
$$

$$
\overline{\mathfrak{x}}^r{}\_{\mathfrak{F}} = \overline{\mathfrak{x}}\_{\mathfrak{F}} e^{-j\frac{\pi}{3}\theta} \tag{22}
$$

where *xr* <sup>1</sup> and *<sup>x</sup><sup>r</sup>* <sup>5</sup> are the vectors in the new rotating reference frames, whereas *x*<sup>1</sup> and *x*<sup>5</sup> are the vectors expressed in the stationary reference frames.

Assuming isolated neutral points (*iS*<sup>3</sup> = 0), substituting Equations (11) and (13) into Equations (4) and (6), respectively and taking into account Equations (21) and (22) leads to the following voltage equations written in rotating reference frames

$$\overline{u}\_{S1}^{\prime} = R\_{S0}^{+} \overline{l}\_{S1}^{\prime} + \overline{v}\_{S1,HR}^{\prime} + L\_{S1} \frac{d \overline{l}\_{S1}^{\prime}}{dt} \ + j\omega \ L\_{S1} \overline{l}\_{S1}^{\prime} + j\ 2 \ \omega \ \operatorname{\boldsymbol{\uprho}}\_{m1} \cos \left( \mathbf{y} \right) - j\ 2 \ \operatorname{\boldsymbol{\uprho}} \operatorname{\boldsymbol{\uprho}}\_{m11} \cos(11 \mathbf{y}) \ e^{-j120} \tag{23}$$

$$\nabla\_{\rm S5}^{\prime} = R\_{\rm SO}^{+} \vec{i}\_{\rm S5}^{\prime} + \vec{v}\_{\rm S5,HR}^{\prime} + L\_{\rm S5} \frac{d \vec{i}\_{\rm S5}^{\prime}}{dt} + j \, 5 \, \omega \, L\_{\rm S5} \vec{i}\_{\rm S5}^{\prime} + j \, 10 \, \omega \, \upmu\_{m5} \cos(5\gamma) - j \, 14 \, \omega \, \upmu\_{m7} \cos(7\gamma) \, e^{-j120} \tag{24}$$

where,

$$
\overline{w}'\_{S1,HR} = \overline{R}^\*\_{S10} \stackrel{\tau}{l\_{S1}}^\* e^{-j2.0} + \overline{R}^\*\_{S4} \stackrel{\tau}{l\_{S5}}^\* e^{j40} + R^-\_{S0} \stackrel{\tau}{l\_{S5}}^\* e^{-j60} \tag{25}
$$

$$
\overline{\sigma}\_{\rm S5,HR}^r = \overline{\mathcal{R}}\_{\rm S4} \overline{\ }\_{\rm S1}^r e^{-j4\ell} + \mathcal{R}\_{\rm S0}^- \overline{\ }\_{\rm S1}^{r\*} e^{-j6\ell} + \overline{\mathcal{R}}\_{\rm S10} \overline{\ }\_{\rm S5}^{r\*} e^{-j100} \tag{26}
$$

The space vectors *v<sup>r</sup> <sup>S</sup>*1,*HR* and *vr <sup>S</sup>*5,*HR* represent the voltage drops due to the stator HRC, in the *d*1*–q*<sup>1</sup> and *d*5*–q*<sup>5</sup> subspaces, respectively. In case of healthy conditions, *vr <sup>S</sup>*1,*HR* and *vr <sup>S</sup>*5,*HR* will be equal to zero as the only resistance component different from zero is *R*<sup>+</sup> *<sup>S</sup>*0, which is not present in Equations (25) and (26).

#### *3.1. Proposed HRC Detection Approach*

The effects of stator HRC can be identified by the presence of voltage space vectors *vr <sup>S</sup>*1,*HR* and *vr <sup>S</sup>*5,*HR*, expressed by Equations (25) and (26), respectively. The sensitivity of these voltages with respect to the HRC is established by their dependence on the new stator resistance components *R*− *<sup>S</sup>*0, *RS*<sup>4</sup> and *RS*10. Assuming that, for the proposed IFOC scheme, the current space vector*i r <sup>S</sup>*<sup>5</sup> is set to be equal to zero for reducing the torque ripple, the voltage space vectors, expressed by Equations (25) and (26), become

$$
\overline{v}\_{S1,HR}^{r} = \overline{R}\_{S10}^{\*} \, \overline{i}\_{S1}^{r} \, e^{-j2\vartheta} \tag{27}
$$

$$
\overline{v}\_{S5,HR}^{r} = \overline{R}\_{S4} \overline{i}\_{S1}^{r} e^{-j4\phi} + R\_{S0}^{-} \overline{i}\_{S1}^{r} e^{-j6\phi} \tag{28}
$$

As can be seen, under the presence of unbalanced stator winding resistances, Equation (27) reveals the presence of a single fault harmonic component in the voltage space vector *vr <sup>S</sup>*1,*HR*, which rotates at an angular speed of −2 ω with respect to *d*1–*q*<sup>1</sup> plane.

In addition, for the considered stator fault, Equation (28) shows the presence of two additional harmonic components in the voltage space vector *vr <sup>S</sup>*5,*HR*, which rotate at the two angular speeds of −4 ω and −6 ω with respect to *d*5–*q*<sup>5</sup> plane.

At this point, it is very useful to recall that under healthy conditions, i.e., the six stator phase resistances are equal, the new stator resistance components *R*− *<sup>S</sup>*0, *RS*<sup>4</sup> and *RS*<sup>10</sup> become equal to zero, as well as the voltage space vectors *vr <sup>S</sup>*1,*HR* and *vr <sup>S</sup>*5,*HR*, leading to a clear identification of the healthy case.

Therefore, an effective fault-detection strategy, for 6-φ PMSM under HRC, can be based on tracking the contribution of the fault component at <sup>−</sup><sup>2</sup> <sup>ω</sup> in the voltage space vector *vr <sup>S</sup>*1,*HR*, as well as the fault components <sup>−</sup><sup>4</sup> <sup>ω</sup> or <sup>−</sup><sup>6</sup> <sup>ω</sup> in the voltage space vector *vr <sup>S</sup>*5,*HR*.

Taking into account the above conclusions, an effective control strategy that intends to tolerate such a stator HRC for 6-φ PMSM, should be based on monitoring the above tracked fault components, which is the subject of the next Subsection.

#### *3.2. Proposed Fault Tolerant Control Strategy for HRC*

Field-oriented control (FOC) strategy is a widely adopted control scheme for implementing closed-loop speed control for m-φ PMSMs. An improved field-oriented control (IFOC) scheme, based on appropriate stator currents control, which provides fault-tolerance against the investigated stator HRC, is presented in this subsection. The block scheme of the proposed control strategy is illustrated by Figure 2.

**Figure 2.** Block scheme of the proposed control strategy, with the capability to detect and tolerate a high resistance connection (HRC) fault.

The current regulators PI1.1 and PI5.1 are implemented in the synchronous reference frames *d*1–*q*<sup>1</sup> and *d*5–*q*5, respectively, to ensure the tracking of currents references for torque control.

The reference currents *iS*<sup>5</sup>*d,ref* and *iS*<sup>5</sup>*q,ref* , in the *d*5–*q*<sup>5</sup> plane, are set to zero for compensating possible torque oscillations due to the 5th and 7th harmonic of the field distribution generated by permanent magnets in the air–gap.

In order to compensate the negative effects of the seventh and eleventh harmonic of the back-emf, two further current regulators PI1.2 and PI5.2 are necessary. They are employed in two different reference frames synchronous with the corresponding back-emf harmonics. In particular, the space vectors of eleventh and seventh back-emf harmonics are both rotating at −12 ω, as shown by Equations (23) and (24), respectively.

The four current regulators PI1.1, PI5.1, PI1.2 and PI5.2 are used in both FOC and IFOC schemes. As highlighted in the voltage Equations (23)–(26), the presence of HRC introduces disturbing voltage vector drops *v<sup>r</sup> <sup>S</sup>*1,*HR* and *vr <sup>S</sup>*5,*HR* rotating at different angular speeds.

More specifically, with the adopted mathematical model, the latter quantities are exactly equal to Equations (27) and (28). Due to the disturbances introduced by these quantities, the current regulators PI1.1 and PI5.1 cannot track properly the reference values *iS*<sup>1</sup>*d,ref* , *iS*<sup>1</sup>*q,ref* , *iS*<sup>5</sup>*d,ref* and *iS*<sup>5</sup>*q,ref* .

To cope with these undesired effects, in IFOC scheme supplementary current regulators PI1.3 and PI5.3–PI5.4, have been implemented in reference frames synchronized with the different angular speeds of the voltage vectors drops *vr <sup>S</sup>*1,*HR* and *<sup>v</sup><sup>r</sup> <sup>S</sup>*5,*HR*, respectively. Thus, a correct stator current reference tracking is ensured.

Finally, it is opportune to note that the generated output voltage of any supplementary regulator among PI1.3, PI5.3 or PI5.4, can be used for online detection of HRC affecting the stator phases of the 6-φ PMSM drive. In fact, in healthy conditions these voltages are practically zero, becoming different from zero only in case of a stator phase resistance asymmetry and showing an amplitude variation proportional to the severity of the fault.

#### **4. Simulation Results**

In order to verify the effectiveness of the proposed approach, the control scheme illustrated by Figure 2 was implemented in Matlab/Simulink™ for numerical simulations. The parameters of the machine are reported in Table 1, which corresponds to the real machine used for the experimental validation. By combining the above control scheme with the 6-φ PMSM model, the implemented system allows a very detailed analysis of the whole drive under stator HRC, which is emulated by an additional resistance in series to Phase a1. The numerical simulations were realized at constant speed of 150 rpm, under the rated torque of 20 Nm. The reference signals *is*1*d,ref* , *is*5*d,ref* and *is*5*q,ref* were set to zero. Three different operating conditions were analyzed.


**Table 1.** Parameters of the 6-φ PMSM.

Initially, the healthy 6-φ PMSM is controlled by a conventional FOC scheme [31], where the six phase resistances are equal. The corresponding simulation results, in terms of stator phase currents, are reported in Figure 3a. The subsequent current space vectors, evaluated in the α1–β<sup>1</sup> and α5–β<sup>5</sup> planes, are reported in Figure 3b,c, respectively. As can be seen in the zoomed area of Figure 3a, the six-phase currents are balanced, leading to a circular behavior of the corresponding locus in plane α1–β1. The current space vector—evaluated in the α5–β<sup>5</sup> plane—is equal to zero.

**Figure 3.** Simulation results: behavior of the drive using conventional field-oriented control (FOC), under healthy conditions: (**a**) stator currents waveforms, (**b**) current space vector in α1–β<sup>1</sup> plane and (**c**) current space vector in α5–β<sup>5</sup> plane.

Under healthy condition where the six-phase resistances are identical, the resistance components *R*− *<sup>S</sup>*0, *RS*<sup>4</sup> and *RS*10, expressed in Equations (8)–(10), respectively, are equal to zero. As a consequence, the voltage drops expressed by the space vectors *vr <sup>S</sup>*1,*HR* and *<sup>v</sup><sup>r</sup> <sup>S</sup>*5,*HR* are equal to zero, leading to a healthy 6-φ PMSM. Thus, the circular trajectory observed in Figure 3b is clearly justified by the dominance of the fundamental harmonic and the zero value in Figure 3c is the absence proof of any resistance unbalance in the machine. From this starting point of the investigations, these results are considered as a reference for the next simulations under faulty conditions.

The second operating condition was realized with the same conventional FOC scheme, but with an additional resistance Radd = 250 mΩ (0.7 pu of healthy stator phase resistance Rs) in series with Phase a1, during 1.0 s of steady-state faulty conditions. The simulation results, in terms of stator phase currents and the corresponding current space vectors in planes α1–β<sup>1</sup> and α5–β5, are reported in Figure 4a–c, respectively. Under the considered stator fault, the six stator phase resistances are no more equal, leading to the existence of voltage drops expressed by the space vectors *vr <sup>S</sup>*1,*HR* and *vr <sup>S</sup>*5,*HR*. Thus, the stator symmetry of the machine is lost, which justify the current unbalance evidenced in the zoomed area of Figure 4a, when compared to the healthy case (Figure 3a).

**Figure 4.** Simulation results: behavior of the drive using conventional FOC, under faulty phase-a1 with an additional resistance (Radd = 0.7 pu): (**a**) Stator currents waveforms; (**b**) current space vector in α1–β<sup>1</sup> plane; (**c**) current space vector in α5–β<sup>5</sup> plane.

It is worth noting that the behavior of the current space vector (Figure 4b), evaluated in the α1–β<sup>1</sup> plane, is mainly determined by the dominance of the fundamental current harmonic component with respect to the small contribution of an inverse component due to the unbalanced conditions. As a result, the trajectory is practically circular. The behavior variation in the current locus evaluated in plane <sup>α</sup>5–β<sup>5</sup> (Figure 4c) can be justified by the non-zero value of the space vector *vr <sup>S</sup>*5,*HR* under the actual stator asymmetry.

Finally, the proposed IFOC scheme illustrated by Figure 2 was implemented under a faulty Phase a1 with an additional resistance (Radd = 0.7 pu) during 1.0 s of steady-state faulty conditions as in the previous simulation. The corresponding simulation results are reported in Figure 5. As can be seen in the zoomed area of Figure 5a, although the presence of the stator fault, the six phase currents show a balanced behavior.

In fact, this can be explained by the compensating effect assured by the current controller PI1.3 against the disturbance introduced by the voltage space vector *vr <sup>S</sup>*1,*HR* in the synchronous reference frame *<sup>d</sup>*1–*q*1. The disturbances introduced by the voltage space vector *vr <sup>S</sup>*5,*HR* are cleared by the current controllers PI5.3 and PI5.4, in the synchronous reference frame *d*5–*q*5, leading to a zero value of the current space vector in the α5–β<sup>5</sup> plane as can be seen in Figure 5c.

At this level of investigation, the established theoretical analysis corroborates with numerical simulations results, confirming the effectiveness of the proposed fault-tolerant control.

**Figure 5.** Simulation results: behavior of the drive using the proposed improved field-oriented control (IFOC), under faulty phase-a1 with an additional resistance (Radd = 0.7 pu). (**a**) Stator current waveforms; (**b**) current space vector in α1–β<sup>1</sup> plane; (**c**) current space vector in α5–β<sup>5</sup> plane.

#### **5. Experimental Results**

In order to experimentally validate the previous simulation results, a complete drive system was mounted in laboratory (see Figure 6) and some experimental tests were carried out to verify the effectiveness of the proposed fault-detection and fault-tolerance strategies.

**Figure 6.** View of the experimental setup.

The experimental setup is composed of a six-phase MOSFET inverter and a six-phase surface mounted PMSM. The 6-φ PMSM parameters are reported in Table 1. The controlled 6-φ PMSM is coupled to a three-phase induction machine used in generator mode to set the mechanical speed of the drive. During the experimental tests, the fault condition was emulated by an external resistor inserted in series with the machine Phase a1. The proposed IFOC scheme, presented in Figure 2, was implemented on a TMS-320F2812 DSP based platform. The experimental tests were realized in steady-state condition at 150 rpm, and setpoint of the torque is 20 Nm, as in simulations of Section 4. An oscilloscope with six channels was used for acquiring four phase currents and two control signals from the control platform, with a sampling rate of 100 kHz.

The behavior of the multiphase drive was assessed initially with healthy stator windings, then under a resistance increase of Phase a1 by 70%, during 1.0 s for each test. The obtained current waveforms under healthy, and faulty conditions are reported in Figure 7a,b, respectively.

**Figure 7.** Experimental results: behavior of the drive using the proposed IFOC. Stator currents waveforms under (**a**) healthy and (**b**) HRC (Radd = 0.7 pu).

As can be seen, under healthy and faulty conditions, the six phase currents are balanced. Aside the small differences in amplitude that are probably due to measurement and acquisition chain accuracy, the six currents show practically the same waveforms.

The corresponding current space vectors in planes α1–β<sup>1</sup> and α5–β5, under healthy and HRC, are reported in Figure 8. The current space vector in plane α1–β1, under healthy and HRC, shows a circular behavior (Figure 8a,b), which confirms the theoretical analysis and the simulation results presented in Figure 3b. The corresponding current loci in planes α5–β5, are reported in Figure 8c,d, where a quasi-zero value can be observed (note the different axis scale). In fact, the latter behavior slightly different from zero is mainly caused by non-perfectly balanced conditions among the six stator phases due to manufacturing process, switching effects and measurement noise during data acquisition.

**Figure 8.** Experimental results: behavior of the drive using the proposed IFOC. Loci of the current space vector on plane α1–β1, under (**a**) healthy and (**b**) HRC (Radd = 0.7 pu). Loci of the current space vector on plane α5–β5, under (**c**) healthy and (**d**) HRC (Radd = 0.7 pu).

As can be seen, despite the presence of the fault, the proposed IFOC is able to ensure balanced stator phase currents. This can be clearly evidenced when comparing the obtained loci under healthy and HRC conditions (Figure 8), which are practically identical.

Finally, considering the good performances of the proposed IFOC to maintain the system of six phase current balanced even under the presence of stator asymmetry, it is obvious that the use of the classical Motor Current Signature Analysis (MCSA) cannot be adopted for detecting the presence of HRC. Thus, a more appropriate fault index is necessary for online fault detection and quantification. This problem introduces the subject of the following Section.

#### **6. Quantitative HRC Evaluation**

Using the proposed IFOC, the 6-φ PMSM drive can maintain good operating conditions, even under degraded mode of the stator windings affected by HRC. In this context, the presence of an online fault-detection process which ensures not only the monitoring of the stator windings state, but also quantifying the severity of the stator winding resistance unbalance, is advisable.

As already anticipated in the previous theoretical analysis, a suitable online fault-detection strategy, for 6-φ PMSM under HRC, can be based on tracking the contribution of the fault component at <sup>−</sup><sup>2</sup> <sup>ω</sup> in the voltage space vector *<sup>v</sup><sup>r</sup> <sup>S</sup>*1,*HR*, as well as the fault components at −4 ω or −6 ω in the voltage space vector *vr <sup>S</sup>*5,*HR*. Here, a new fault index "*Fi*" is proposed, that is based on the *d*–*q* outputs of the controller *PI*5.4 defined as

$$F\_i = \sqrt{v\_{PI \ 5.4,d}^2 + v\_{PI \ 5.4,q}^2} \tag{29}$$

In order to test the sensitivity of the proposed fault index, different additional resistances (Radd) were tested in simulations, namely 0.10 Ω (0.28 pu), 0.25 Ω (0.70 pu), 0.75 Ω (2.09 pu) and 1.00 Ω (2.78 pu).

The spectra of the fault index, obtained from simulation results under healthy and faulty conditions, are reported in Figure 9. Observing the obtained spectra, one can see the dominance of a DC component, which has the advantage to be easily detected even during speed variations.

**Figure 9.** Simulation results: spectra of the proposed fault index under healthy and different HRC conditions affecting Phase a1.

It can be noticed that the tracked DC component shows a relevant increase in amplitude from healthy (Radd = 0.00 Ohms) to the first faulty case (Radd = 0.10 Ohms). It is also clearly evident that higher the severity of the fault (from 0.10 Ohms to 1.00 Ohms), higher the value of the fault index.

The corresponding spectra obtained from experimental tests for Radd (0.25 Ω, 0.75 Ω and 1.00 Ω) are reported in Figure 10, respectively. As can be seen, the obtained experimental spectra corroborate with those obtained by simulations. It is worth noting that the experimental signals in Figure 10 show a certain noise compared to Figure 9 owing to an unavoidable small noise present in the measurement and acquisition chain.

**Figure 10.** Experimental results: spectra of the proposed fault index under healthy and different HRC conditions affecting Phase a1.

In case of faulty conditions, the maximum difference from simulation to experimental results is lower than 2 dB. In healthy conditions, the differences appearing when comparing Figures 9 and 10 are mainly due to measurement uncertainties, noting that the fault index is practically equal to zero in both simulation and experimental tests. More specifically, the fault index shows a relevant variation in amplitude (~15 dB) from healthy to the first faulty condition (Radd = 0.25 Ω). The proposed fault index has shown relevant increments in agreement with the fault severity also in experimental tests, leading to an effective fault index particularly adapted not only for fault-detection, but also for fault-quantification.

Finally, in order to show the performance of the proposed fault index against the operating speed, additional numerical simulations were carried out. The results achieved using a setpoint of the torque equal to 20 Nm as in previous tests and varying the speed from 50 rpm to 250 rpm, are shown in

Figure 11. As can be noted the fault index is not dependent on the speed, allowing a further advantage of the proposed fault index to be emphasized.

**Figure 11.** Behavior of the DC component of the fault index under a HRC of 0.25 Ω in Phase a1, at different speed operating conditions.

#### **7. Conclusions**

The main contribution of the study is the presentation of a new mathematical model of an asymmetric six-phase PMSM under HRC fault, which exploits the additional degrees of freedom given by the use of multi α–β planes adopted for describing the behavior of the machine. The presented mathematical model has a general validity, showing the possibility to investigate new diagnostic techniques based on monitoring the signals available in the multi α–β and *d*–*q* planes. In agreement with the developed mathematical model, an improved FOC scheme for asymmetric six-phase PMSMs based on the presence of several current regulators in the different α–β planes, was also presented. The proposed control scheme was adapted to provide not only an online HRC fault-detection and quantification, but also ensures a nearly undisturbed behavior of the drive.

The main features of the proposed HRC fault detection method are as follows:


The results obtained, the theoretical analysis and the numerical simulations were compared with experimental results achieving a good agreement.

The presented results emphasize that the fault index was dependent on the severity of the fault, showing a good sensitivity. Adding a resistance of 0.25 Ω to a stator phase winding, a variation of about 15 dB was observed for the fault index magnitude.

Considering the above cited features of the proposed fault detection technique, the definition of the threshold should be adapted only to the torque demand.

Based on the results obtained the authors are encouraged to continue the experimental tests for verifying the effectiveness of the proposed diagnostic technique in dynamic conditions and to further develop the mathematical model for also achieving a reliable fault localization.

**Author Contributions:** Conceptualization, C.R., Y.G. and D.C.; Funding acquisition, C.R.; Investigation, Y.G. and A.P.; Methodology, Y.G., G.R. and A.T.; Project administration, C.R. and D.C.; Software, A.P. and A.T.; Writing—original draft, Y.G., G.R. and D.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the Italian Ministry for Education, University and Research (MIUR) under the program "*Dipartimenti di Eccellenza (2018–2022)*".

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **Nomenclature**


#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Comparative Analysis of Fault-Tolerant Dual-Channel BLDC and SR Motors**

#### **M. Korkosz \*, P. Bogusz, J. Prokop, B. Pakla and G. Podskarbi**

The Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, Al. Postancow Warszawy 12, 35-959 Rzeszow, Poland

**\*** Correspondence: mkosz@prz.edu.pl; Tel.: +48-178-651-389

Received: 20 May 2019; Accepted: 25 June 2019; Published: 28 June 2019

**Abstract:** This article presents the results of a comparative analysis of two electronically commutated brushless direct current machines intended for fault-tolerant drives. Two machines designed by the authors were compared: a 12/14 dual-channel brushless direct current motor (DCBLDCM) with permanent magnets and a 12/8 dual-channel switched reluctance motor (DCSRM). Information is provided here on the winding configuration, the parameters, and the power converters of both machines. We developed mathematical models of the DCBLDCM and DCSRM which accounted for the nonlinearity of their magnetization characteristics in dual-channel operation (DCO) and single-channel operation (SCO) modes. The static torque characteristics and flux characteristics of both machines were compared for operation in DCO and SCO modes. The waveforms of the current and the electromagnetic torque are presented for DCO and SCO operating conditions. For DCO mode, an analysis of the behavior of both machines under fault conditions (i.e., asymmetrical control, shorted coil, and open phase) was performed. The two designs were compared, and their strengths and weaknesses were indicated.

**Keywords:** multiphase machines; fault-tolerance; dual-channel; brushless direct current motor with permanent magnet (BLDCM); switched reluctance motor (SRM)

#### **1. Introduction**

In fault-tolerant electrical drives, two solutions are used: multiphase motors (with more than three phases) as the standard or three-phase motors with dual windings. Multiphase motors, regardless of the type of machine, usually require a more extensive power supply system and more complex control algorithms [1–4]. Three-phase motors with independent, three-phase dual windings require two independent power supply circuits [5–21]. As a result, two channels are obtained which, depending on the type of machine and the configuration of its windings, may be magnetically independent or partly independent. Full or significant magnetic independence between the channels makes control of such a motor much easier. This is particularly important in the case of fault-tolerant operation.

Direct current electronically commutated motors include brushless direct current motors (BLDCMs) with permanent magnets and switched reluctance motors (SRMs). BLDCMs and SRMs can be designed as standard three-phase or multiphase machines. In the case of a BLDCM, the use of more than three phases is not a typical approach [1,22,23]. In the case of switched reluctance motors, a four-phase solution is a borderline case for potential commercial applications [24]. Consequently, in fault-tolerant drives, the authors suggest using dual-channel BLDCM (DCBLDCM) or SRM (DCSRM) (i.e., ones with dual three-phase windings). In such machines, in order to achieve the same operating point, depending on the control strategy adopted, two channels or only one channel is supplied in normal operating conditions. Selection of a DCBLDCM or a DCSRM for a given drive is not self-evident.

In this study, a comparative analysis was performed of two designs of dual-channel brushless direct current electronically commutated motors with or without permanent magnets. The machines were designed for use in critical drive systems. The first machine, with permanent magnets, was designated DCBLDCM and the other, without permanent magnets, was designated DCSRM. Both machines, at the stages of their design and construction, were adapted for independent dual-channel supply, that is, for dual-channel operation (DCO) and single-channel operation (SCO).

The aim of this study was to compare the features of the DCBLDCM and the DCSRM designs in DCO and SCO, especially in the context of guaranteeing the continuation of motor operation after the occurrence of a fault condition. The rated power, rated torque and base speed of both machines are different. Therefore, the comparison of performance of both machines is difficult. For this reason, the results are presented as a ratio of value to the base values (presented in Table 1). Nevertheless, the aim of the paper is to compare the possibilities of dual-channel operation rather than the performance.

In Section 2, the DCBLDC and DCSR machine designs studied are presented and the configurations of their power supply systems, the distributions of their stator windings, and their parameters and characteristics are shown. Section 3 contains the authors' nonlinear mathematical models of DCBLDC and DCSR machines for DCO and SCO. These mathematical models take into account the nonlinearity of the magnetic circuit and the magnetic coupling between particular phases of channels A and B. The models include novel electromagnetic torque formulas of the DCBLDCM and the DCSRM for DCO and SCO proposed by the authors. It was proven that the structure of the formulas of the mathematical model of SRM was the particular simplification of the formulas of the model of the BLDC machine assuming that PM magnetization equivalent current was zero (PM flux was neglected). Section 4 presents the static characteristics of the motors studied (the electromagnetic torque and the linkage fluxes) for DCO and SCO modes. The results of the analysis of current and electromagnetic torque waveforms presented in Section 5 are divided into two parts. The first part contains the results of an analysis of DCO and SCO, assuming the electric and magnetic symmetry of both channels. The second part describes the analysis that was performed of nonstandard operating conditions in DCO mode, such as:


The results of the comparative analysis of the two studied machines and the conclusions are presented in Section 6.

#### **2. Analysis of the DCBLDC and DCSR Motors**

An analysis was performed of two three-phase designs of brushless direct current machines. In the case of the DCBLDC motor, this was a solution with 12 stator poles and 14 rotor poles (magnets) with an external rotor (Figure 1a). The machine was designed for a small unmanned aerial vehicle (UAV). In the case of the DCSRM, a solution with 12 stator poles and 8 rotor poles was selected (Figure 1b). The DCSRM was designed for a fan drive. Both motors have the same number of stator poles. At the design stage of both motors, provisions were made for the possibility of a dual-channel power supply. There were six stator poles per channel. The use of two independent power supply channels made it possible to achieve a dual three-phase power supply in both cases. The dual-channel power supply diagram is shown in Figure 1c for the BLDCM and in Figure 1d for the SRM.

In the case of the configuration of the windings of the DCBLDC motor, the layout shown in Figure 1a (ABABAB) was used. As demonstrated in [6,24], this is the most advantageous configuration, and not only with regard to the distribution of the magnetic pull forces. In the case of the SRM, a similar principle was adopted with regard to the positions of the windings of the different channels. The configuration used (ABABABABABAB) is shown in Figure 1b. In the case of an SRM with four poles per phase, the short-flux path solution (NSNS) is usually selected [8,9]. This configuration of the poles of a single phase may also be used in the case of a dual-channel power supply [8,9,11,12]. It ensures the most advantageous motor parameters for a dual-channel power supply. However, in the

authors' opinion, in the case of a machine intended for a dual-channel power supply, the configuration with a long-flux path (NNSS) should be selected [11]. For a classic or dual-channel power supply, this configuration results in slightly poorer machine performance [11]. At the same time, it is characterized by greater magnetic independence of the different channels. The NNSS configuration of the poles of an SRM meets this condition. This is demonstrated later in this article.

Selected geometric parameters of both motors are shown in Table 1, which also contains selected electric parameters. The electric parameters were specified for the SCO power supply. The base torque obtained for the DCBLDCM was equal to *T*bDCBLDCM = 1.0 N·m, while that obtained for the DCSRM was equal to *T*bDCSRM = 0.5 N·m. The base torque in the DCBLDC was obtained when the line current was equal to *I*refDCBLDCM = 36.5 A. In the case of the DCSRM, this value was *I*refDCSRM = 8.5 A. The above values were adopted as a reference for presentation of the waveform of the line current (Section 5) and the electromagnetic torque in the relative values (Sections 4 and 5) as a function of time.

Figure 2c,d show idealized torque–speed characteristics of the DCBLDCM (Figure 2c) and the DCSRM (Figure 2d). The torque–speed characteristics of the two motors were significantly different. In the case of the DCBLDCM in DCO and SCO modes, there were small differences in the base speeds. For DCO mode, the base speed was slightly higher (by several percent). In the case of the DCSRM, the differences between DCO and SCO modes were much more significant. In DCO operation mode, the base speed was up to two times higher than in SCO mode. In the case of the DCSRM, there was a constant power range regardless of the operation mode (DCO or SCO). In the case of the DCBLDCM, there was practically no constant power region.

(**b**)

**Figure 1.** *Cont*.

**Figure 1.** The geometry and prototypes of dual-channel power supplied brushless electronically commutated motors. (**a**) The geometry and prototype of a three-phase DCBLDCM. (**b**) The geometry and prototype of a DCSRM. (**c**) A scheme of a DCBLDC motor supply system in the DCO mode. (**d**) A scheme of a DCSRM supply system in the DCO mode.

**Figure 2.** A scheme of winding distributions of channels A and B on a stator of the three-phase DCBLDCM and DCSRM and theoretical torque vs. speed characteristics: (**a**) DCBLDCM, (**b**) DCSRM, (**c**) torque vs. speed for DCBLDCM, and (**d**) torque vs. speed for DCSRM.


**Table 1.** Main parameters of the BLDC motor and the SRM for SCO modes.

#### **3. A Mathematical Model of the DCBLDC and DCSR Motors Analyzed**

#### *3.1. Main Assumptions and General Equation Structure*

The subjects of the mathematical modeling were dual-channel three-phase BLDC and SR machines, for which the authors' circuit-based models, known as flux models, were proposed. The presented mathematical models of the DCBLDC and DCSR machines took into account the nonlinearity of the magnetic circuit and the magnetic couplings between particular phases within a given channel (A or B), as well as between channels (A and B). The following simplifying assumptions were made in the proposed models:


The general structure of the formulas of the circuit-based mathematical models of the three-phase dual-channel machines can be written in the following form:

$$
\begin{bmatrix} \mathbf{u}^{\boldsymbol{\Lambda}} \\ \mathbf{u}^{\boldsymbol{\mathsf{B}}} \end{bmatrix} = \begin{bmatrix} \mathbf{R}^{\boldsymbol{\Lambda}} & \mathbf{0} \\ \mathbf{0} & \mathbf{R}^{\boldsymbol{\mathsf{B}}} \end{bmatrix} \begin{bmatrix} \mathbf{i}^{\boldsymbol{\Lambda}} \\ \mathbf{i}^{\boldsymbol{\mathsf{B}}} \end{bmatrix} + \frac{\mathbf{d}}{\mathbf{d}t} \begin{bmatrix} \boldsymbol{\Psi}^{\boldsymbol{\Lambda}} \\ \boldsymbol{\Psi}^{\boldsymbol{\mathsf{B}}} \end{bmatrix}, \tag{1}
$$

$$J\frac{d\omega}{dt} + D\omega + T\_{\text{L}} = T\_{\text{e}} \tag{2}$$

$$\frac{\mathbf{d}\theta}{\mathbf{d}t} = \omega \tag{3}$$

where for channels A and B (k <sup>∈</sup> A, B), the vectors representing voltages **<sup>u</sup>**k, currents **<sup>i</sup>** k, fluxes ψk, as well as matrixes of resistances **R**<sup>k</sup> are defined as follows:

$$\mathbf{u}^{\mathbf{k}} = \left[u\_1^{\mathbf{k}}, u\_2^{\mathbf{k}}, u\_3^{\mathbf{k}}\right]^\mathbf{T}, \ \mathbf{i}^{\mathbf{k}} = \left[\mathbf{i}\_1^{\mathbf{k}}, \mathbf{i}\_2^{\mathbf{k}}, \mathbf{i}\_3^{\mathbf{k}}\right]^\mathbf{T}, \mathbf{u}^{\mathbf{k}} = \left[\boldsymbol{\psi}\_1^{\mathbf{k}}, \boldsymbol{\psi}\_2^{\mathbf{k}}, \boldsymbol{\psi}\_3^{\mathbf{k}}\right]^\mathbf{T}, \ \mathbf{R}^\mathbf{k} = \text{diag}\left(\mathbf{R}\_1^\mathbf{k}, \mathbf{R}\_2^\mathbf{k}, \mathbf{R}\_3^\mathbf{k}\right)$$

The following symbols are used in Equations (1)–(3): θ—rotor position, ω—angular velocity, *J*—rotor moment of inertia, *D*—coefficient of viscous friction, *T*L—load torque, and *T*e—electromagnetic torque. Electromagnetic torque *T*<sup>e</sup> in Equation (2) can be calculated as a derivative of total magnetic field coenergy in the air gap with respect to rotor position.

#### *3.2. Mathematical Models of DCBLDC Motors*

#### 3.2.1. DCBLDCM—DCO Mode

In machines with permanent magnets, the fluxes in Equation (1) depend on the rotor position θ, the winding current, and the permanent magnets' magnetization equivalent current, designated as *i* PM. The voltage–current Equation (1) and the expression for electromagnetic torque in Equation (2) for DCO can be written in the following form:

$$
\begin{bmatrix} \mathbf{u}^{\rm A} \\ \mathbf{u}^{\rm B} \end{bmatrix} = \begin{bmatrix} \mathbf{R}^{\rm A} & \mathbf{0} \\ \mathbf{0} & \mathbf{R}^{\rm B} \end{bmatrix} \begin{bmatrix} \mathbf{i}^{\rm A} \\ \mathbf{i}^{\rm B} \end{bmatrix} + \frac{\mathbf{d}}{\mathbf{d}t} \begin{bmatrix} \boldsymbol{\Psi}^{\rm A}(\boldsymbol{\theta}, \mathbf{i}^{\rm A}, \mathbf{i}^{\rm B}, \mathbf{i}^{\rm PM}) \\ \boldsymbol{\Psi}^{\rm B}(\boldsymbol{\theta}, \mathbf{i}^{\rm A}, \mathbf{i}^{\rm B}, \mathbf{i}^{\rm PM}) \end{bmatrix} \tag{4}
$$

$$T\_{\text{e}} = T\_{\text{e}}(\,\,\theta, \mathbf{i}^{\text{A}}, \mathbf{i}^{\text{B}}, \mathbf{i}^{\text{PM}}) \tag{5}$$

where flux linkages caused by windings' currents and permanent magnets for both channels A and B are defined as follows:

$$\boldsymbol{\Psi}^{\rm k}(\boldsymbol{\theta},\dot{\mathbf{i}}^{\rm A},\dot{\mathbf{i}}^{\rm B},\dot{\mathbf{i}}^{\rm PM}) = \begin{bmatrix} \boldsymbol{\psi}\_{1}^{\rm kPM}(\boldsymbol{\theta},\dot{\mathbf{i}}^{\rm PM}) + L\_{1\sigma}^{\rm kk}\dot{\mathbf{i}}\_{1}^{\rm k} + \sum\_{j=1}^{3} \left( \sum\_{\mathbf{l}=\mathbf{A}}^{\rm B} \boldsymbol{\psi}\_{1\mathbf{j}}^{\rm kL}(\boldsymbol{\theta},\dot{\mathbf{i}}\_{j}^{\rm k},\dot{\mathbf{i}}^{\rm PM}) \right) \\\\ \boldsymbol{\psi}\_{2}^{\rm kPM}(\boldsymbol{\theta},\dot{\mathbf{i}}^{\rm PM}) + L\_{2\sigma}^{\rm kk}\dot{\mathbf{i}}\_{2}^{\rm k} + \sum\_{j=1}^{3} \left( \sum\_{\mathbf{l}=\mathbf{A}}^{\rm B} \boldsymbol{\psi}\_{2\mathbf{j}}^{\rm kL}(\boldsymbol{\theta},\dot{\mathbf{i}}\_{j}^{\rm k},\dot{\mathbf{i}}^{\rm PM}) \right) \\\\ \boldsymbol{\psi}\_{3}^{\rm kPM}(\boldsymbol{\theta},\dot{\mathbf{i}}^{\rm PM}) + L\_{3\sigma}^{\rm kk}\dot{\mathbf{i}}\_{3}^{\rm k} + \sum\_{j=1}^{3} \left( \sum\_{\mathbf{l}=\mathbf{A}}^{\rm B} \boldsymbol{\psi}\_{3\mathbf{j}}^{\rm kM}(\boldsymbol{\theta},\dot{\mathbf{i}}\_{j}^{\rm k},\dot{\mathbf{i}}^{\rm PM}) \right) \end{bmatrix} \tag{6}$$

In Expression (6), ψkPM <sup>i</sup> (θ, *i* PM) are the fluxes generated by permanent magnets, and *L*kk <sup>i</sup><sup>σ</sup> are the leakage fluxes (for k ∈ A, B and *i* = 1,2,3).

The expression for electromagnetic torque (5), with assumptions (6) taken into account, can be written in the following form:

*T*e( θ, **i** A, **i** B, *i* PM) = <sup>3</sup> i=1 *i* A i ∂ψAPM <sup>i</sup> (θ, *<sup>i</sup>* PM) ∂θ + *i* B i ∂ψBPM <sup>i</sup> (θ, *<sup>i</sup>* PM) ∂θ <sup>+</sup> <sup>3</sup> i=1 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ *i* A .i 0 ∂ψAA ii (θ,*<sup>i</sup>* A <sup>i</sup> , *i* PM) ∂θ *di* A <sup>i</sup> + *i* B .i 0 ∂ψBB ii (θ,*<sup>i</sup>* B <sup>i</sup> , *i* PM) ∂θ *di* B i ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ <sup>+</sup> <sup>3</sup> i=2 i−1 j=1 *i* A i ∂ψAA ij (θ,*<sup>i</sup>* A j , *i* PM) ∂θ + *i* B i ∂ψBB ij (θ,*<sup>i</sup>* B j , *i* PM) ∂θ <sup>+</sup> <sup>3</sup> i=1 3 j=1 *i* B i ∂ψBA ij (θ,*<sup>i</sup>* A j , *i* PM) ∂θ + *T*cog(θ, *i* PM) (7)

Electromagnetic torque is the sum of torques from the fluxes of magnets, windings' currents, and the cogging torque *T*cog. Equations (2) and (4) with Expressions (6) and (7) constitute the nonlinear mathematical model of the DCBLDC machine in DCO mode.

#### 3.2.2. DCBLDCM—SCO Mode

In this particular case, when the DCBLDC machine is operating in SCO mode (e.g., only channel A or B is supplied), Equations (4) and (6), as well as the expression for torque (7), can be simplified (for k ∈ A, B):

⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *u*k 1 *u*k 2 *u*k 3 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *R*k <sup>1</sup> 0 0 0 *R*<sup>k</sup> <sup>2</sup> 0 0 0 *R*<sup>k</sup> 3 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *i* k 1 *i* k 2 *i* k 3 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ + d d*t* ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ψkPM <sup>1</sup> (θ, *i* PM) + *L*kk 1σ*i* k <sup>1</sup> <sup>+</sup> <sup>3</sup> j=1 ψkk 1j (θ, *i* k j , *i* PM) ψkPM <sup>2</sup> (θ, *i* PM) + *L*kk 2σ*i* k <sup>2</sup> <sup>+</sup> <sup>3</sup> j=1 ψkk 2j (θ, *i* k j , *i* PM) ψkPM <sup>3</sup> (θ, *i* PM) + *L*kk 3σ*i* k <sup>3</sup> <sup>+</sup> <sup>3</sup> j=1 ψkk 3j (θ, *i* k j , *i* PM) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)

$$\begin{split} T\_{\mathbf{e}}(\boldsymbol{\theta},\boldsymbol{l}\_{1}^{\mathbf{k}},\boldsymbol{l}\_{2}^{\mathbf{k}},\boldsymbol{l}\_{3}^{\mathbf{k}},\boldsymbol{l}^{\mathbf{PM}}) &= \sum\_{\mathbf{i}=1}^{3} \Big( \mathbf{i}\_{\mathbf{i}}^{\mathbf{k}} \frac{\partial \psi\_{\mathbf{i}}^{\mathbf{k}\mathbf{PM}}(\boldsymbol{\theta},\boldsymbol{l}^{\mathbf{PM}})}{\partial \boldsymbol{\theta}} \Big) + \sum\_{\mathbf{i}=1}^{3} \Big( \int \frac{\partial \psi\_{\mathbf{i}}^{\mathbf{k}}(\boldsymbol{\theta}) \boldsymbol{J}\_{\mathbf{i}}^{\mathbf{k}} \boldsymbol{l}^{\mathbf{p}}}{\partial \boldsymbol{\theta}} d\mathbf{i}^{\mathbf{k}}\_{1} \Big) + \sum\_{\mathbf{i}=2}^{3} \sum\_{\mathbf{j}=1}^{i-1} \Big( \boldsymbol{l}\_{1}^{\mathbf{k}} \frac{\partial \psi\_{\mathbf{i}}^{\mathbf{k}\mathbf{k}}(\boldsymbol{\theta}) \boldsymbol{J}\_{\mathbf{i}}^{\mathbf{k}} \boldsymbol{l}^{\mathbf{p}}}{\partial \boldsymbol{\theta}} \Big) \\ &+ \, T\_{\mathbf{c}\mathbf{g}}(\boldsymbol{\theta},\boldsymbol{l}^{\mathbf{PM}}) \end{split} \tag{9}$$

Equations (8) and (2) with Expression (9) constitute the nonlinear mathematical model of the DCBLDC machine in SCO mode.

#### *3.3. Mathematical Models of DCSR Motors*

#### 3.3.1. DCSRM—DCO Mode

The equations of the mathematical model of DCSR machines can also be derived from the equations of DCBLDC machines by eliminating the relevant fluxes produced by permanent magnets. The voltage–current Equation (1) and the equation of motion of a circuit-based mathematical model of the DCSRM for DCO mode can be written in the following form:

$$
\begin{bmatrix} \mathbf{u}^{\rm A} \\ \mathbf{u}^{\rm B} \end{bmatrix} = \begin{bmatrix} \mathbf{R}^{\rm A} & \mathbf{0} \\ \mathbf{0} & \mathbf{R}^{\rm B} \end{bmatrix} \begin{bmatrix} \mathbf{i}^{\rm A} \\ \mathbf{i}^{\rm B} \end{bmatrix} + \frac{\mathbf{d}}{\mathbf{d}t} \begin{bmatrix} \boldsymbol{\Psi}^{\rm A}(\boldsymbol{\theta}, \mathbf{i}^{\rm A}, \mathbf{i}^{\rm B}) \\ \boldsymbol{\Psi}^{\rm B}(\boldsymbol{\theta}, \mathbf{i}^{\rm A}, \mathbf{i}^{\rm B}) \end{bmatrix} \tag{10}
$$

$$T\_{\mathfrak{e}} = T\_{\mathfrak{e}}(\theta, \mathbf{i}^{\mathsf{A}}, \mathbf{i}^{\mathsf{B}}) \tag{11}$$

where for channels <sup>k</sup> <sup>∈</sup> A, B, flux linkages caused only by winding currents <sup>ψ</sup>k(θ, **<sup>i</sup>** A, **i** <sup>B</sup>) are defined as follows:

$$\boldsymbol{\Psi}^{\rm k}(\boldsymbol{\theta},\mathbf{i}^{\rm A},\mathbf{i}^{\rm B}) = \begin{bmatrix} L\_{1\sigma}^{\rm kk}\boldsymbol{i}\_{1}^{\rm k} + \sum\_{j=1}^{3} \left( \sum\_{l=\mathbf{A}}^{\rm B} \boldsymbol{\psi}\_{1\mathbf{j}}^{\rm k}(\boldsymbol{\theta},\boldsymbol{i}\_{j}^{\rm l}) \right) \\\ L\_{2\sigma}^{\rm kk}\boldsymbol{i}\_{2}^{\rm k} + \sum\_{j=1}^{3} \left( \sum\_{l=\mathbf{A}}^{\rm B} \boldsymbol{\psi}\_{2\mathbf{j}}^{\rm k}(\boldsymbol{\theta},\boldsymbol{i}\_{j}^{\rm l}) \right) \\\ L\_{3\sigma}^{\rm kk}\boldsymbol{i}\_{3}^{\rm k} + \sum\_{j=1}^{3} \left( \sum\_{l=\mathbf{A}}^{\rm B} \boldsymbol{\psi}\_{3\mathbf{j}}^{\rm k}(\boldsymbol{\theta},\boldsymbol{i}\_{j}^{\rm l}) \right) \end{bmatrix} \tag{12}$$

For the DCO mode of the DCSRM, the expression for electromagnetic torque, with assumptions taken into account, can be written in the following form:

$$\begin{split} T\_{\mathbf{e}}(\boldsymbol{\theta},\mathbf{i}^{\mathbf{A}},\mathbf{i}^{\mathbf{B}}) &= \sum\_{\mathbf{i}=1}^{3} \left( \int\_{0}^{\mathbf{i}^{\mathbf{A}}\_{\mathbf{i}}} \frac{\partial \psi^{\mathbf{A}\mathbf{i}}\_{\mathbf{i}}(\boldsymbol{\theta})\tilde{\boldsymbol{\mu}}^{\mathbf{A}}\_{\mathbf{i}}}{\partial \boldsymbol{\theta}} d\mathbf{i}^{\mathbf{A}}\_{\mathbf{i}} + \int\_{0}^{\mathbf{i}^{\mathbf{B}}\_{\mathbf{i}\mathbf{i}}} \frac{\partial \psi^{\mathbf{B}\mathbf{i}}\_{\mathbf{i}\mathbf{i}}(\boldsymbol{\theta})\tilde{\boldsymbol{\mu}}^{\mathbf{B}}\_{\mathbf{i}}}{\partial \boldsymbol{\theta}} d\mathbf{i}^{\mathbf{B}}\_{\mathbf{i}} \right) \\ &+ \sum\_{\mathbf{i}=2}^{3} \sum\_{\mathbf{j}=1}^{i-1} \Big( \dot{\mathbf{i}}^{\mathbf{A}}\_{\mathbf{i}} \frac{\partial \psi^{\mathbf{A}\mathbf{i}}\_{\mathbf{j}}(\boldsymbol{\theta},\mathbf{i}^{\mathbf{A}}\_{\mathbf{j}})}{\partial \boldsymbol{\theta}} + \dot{\mathbf{i}}^{\mathbf{B}}\_{\mathbf{i}} \frac{\partial \psi^{\mathbf{B}\mathbf{i}}\_{\mathbf{i}}(\boldsymbol{\theta},\mathbf{i}^{\mathbf{B}}\_{\mathbf{j}})}{\partial \boldsymbol{\theta}} \Big) + \sum\_{\mathbf{i}=1}^{3} \sum\_{\mathbf{j}=1}^{3} \Big( \dot{\mathbf{i}}^{\mathbf{B}}\_{\mathbf{i}} \frac{\partial \psi^{\mathbf{B}\mathbf{i}}\_{\mathbf{j}}(\boldsymbol{\theta},\mathbf{i}^{\mathbf{A}}\_{\mathbf{j}})}{\partial \boldsymbol{\theta}} \Big) \end{split} \tag{13}$$

#### 3.3.2. DCSRM—SCO Mode

In the particular case where the DCSRM is operating in SCO mode (e.g., only channel A or B is supplied), Equation (10) with the expression for torque (13) can be simplified (k ∈ A or B):

$$
\begin{bmatrix} u\_1^{\mathbf{k}} \\ u\_2^{\mathbf{k}} \\ u\_3^{\mathbf{k}} \end{bmatrix} = \begin{bmatrix} R\_1^{\mathbf{k}} & 0 & 0 \\ 0 & R\_2^{\mathbf{k}} & 0 \\ 0 & 0 & R\_3^{\mathbf{k}} \end{bmatrix} \begin{bmatrix} r\_1^{\mathbf{k}} \\ r\_2^{\mathbf{k}} \\ r\_3^{\mathbf{k}} \end{bmatrix} + \frac{\mathbf{d}}{\mathbf{d}t} \begin{bmatrix} L\_{1o}^{\mathbf{k}} t\_1^{\mathbf{k}} + \sum\_{j=1}^3 \left( \psi\_{1j}^{\mathbf{k}\mathbf{k}} (\boldsymbol{\theta}, t\_j^{\mathbf{k}}) \right) \\ L\_{2o}^{\mathbf{k}} t\_2^{\mathbf{k}} + \sum\_{j=1}^3 \left( \psi\_{2j}^{\mathbf{k}\mathbf{k}} (\boldsymbol{\theta}, t\_j^{\mathbf{k}}) \right) \\ L\_{3o}^{\mathbf{k}} t\_3^{\mathbf{k}} + \sum\_{j=1}^3 \left( \psi\_{3j}^{\mathbf{k}\mathbf{k}} (\boldsymbol{\theta}, t\_j^{\mathbf{k}}) \right) \end{bmatrix} \tag{14}
$$

$$T\_{\mathbf{e}}(\boldsymbol{\theta}, \mathbf{i}\_1^{\mathbf{k}}, \mathbf{i}\_2^{\mathbf{k}}, \mathbf{i}\_3^{\mathbf{k}}) = \sum\_{\mathbf{i}=1}^3 \left( \int\_0^{\mathbf{i}\_\mathbf{i}^\mathbf{k}} \frac{\partial \psi\_{\mathbf{ii}}^{\mathbf{k}\mathbf{k}}(\boldsymbol{\theta}, \mathbf{i}\_\mathbf{i}^\mathbf{k})}{\partial \boldsymbol{\theta}} d\mathbf{i}\_\mathbf{i}^\mathbf{k} \right) + \sum\_{\mathbf{i}=2}^3 \sum\_{\mathbf{j}=1}^{i-1} \left( \mathbf{i}\_\mathbf{i}^\mathbf{k} \frac{\partial \psi\_{\mathbf{ij}}^{\mathbf{k}\mathbf{k}}(\boldsymbol{\theta}, \mathbf{i}\_\mathbf{j}^\mathbf{k})}{\partial \boldsymbol{\theta}} \right) \tag{15}$$

Equations (14) and (2) with Expression (15) constitute the nonlinear mathematical model of the DCSRM in SCO mode.

Based on the presented equations for the BLDCM in DCO and SCO modes, it is possible to obtain models for special simplifying assumptions, for example, by omitting couplings between particular phases or channels A and B.

#### *3.4. Flux Characteristics for Simulation Models*

The relationships of fluxes as a function of rotor position and phase current were determined by means of 2D field methods (finite element method) and then the obtained set of relationships was used in the circuit models. For example, Figure 3 shows 3D views of flux linkage ψ<sup>A</sup> <sup>1</sup> of the first phase of channel A as a function of rotor position θ and current *i* A <sup>1</sup> for a DCBLDCM (Figure 3a) and a DCSRM (Figure 3b).

**Figure 3.** *Cont*.

**Figure 3.** Flux linkage ψ<sup>A</sup> <sup>1</sup> vs. rotor position θ and current *i* A <sup>1</sup> : (**a**) DCBLDCM; (**b**) DCSRM.

#### **4. Static Analysis**

Static calculations were performed for the reference values of currents given in Table 1 that were required for SCO mode. For DCO mode, the reference values of currents were reduced in order to obtain a comparable shape of the static torque characteristic.

#### *4.1. Characteristics of the DCBLDCM*

Figure 4a shows the determined static characteristics of electromagnetic torque as a function of rotor position for DCO and SCO modes. In the case of DCO mode, the reference value of the current was reduced to 50% of the value specified in Table 1. The results of laboratory tests are shown in Figure 4b. Laboratory tests were performed for *I* = 0.5 *I*ref (i.e., *i* <sup>A</sup> = *i* <sup>B</sup> = 0.25 *I*ref for DCO mode and *i* <sup>A</sup> = 0.5 *I*ref, *i* <sup>B</sup> = 0 for SCO mode (channel A) and *i* <sup>A</sup> = 0 *I*ref, *i* <sup>B</sup> = 0.5 *I*ref for SCO mode (channel B)). The fluxes linked with the different windings for DCO mode are shown in Figure 4c. The linkage fluxes for SCO mode are shown in Figure 4d.

In dual-channel mode, the brushless motor with a permanent magnet required half of the value of the reference current of SCO mode in order to obtain the required value of the base torque. This was quite beneficial from the point of view of the control algorithm of the operation of a motor supplied from two channels. The configuration used was characterized by very high magnetic independence between the two channels. The magnetic couplings between the two channels in normal operating conditions were minimal. The linkage fluxes of the windings of the channel that was not being supplied (Figure 4d) came mainly from permanent magnets. The assumption made in the mathematical model that the two channels of the presented design are magnetically independent significantly simplified the equations of the mathematical model of the DCBLDCM.

**Figure 4.** *Cont*.

**Figure 4.** Static torque and flux linkage characteristics of the DCBLDCM. (**a**) Electromagnetic torque *T*e vs. rotor position θ in the DCO mode (*i* <sup>A</sup> = *i* <sup>B</sup> = 0.5 *I*ref) and SCO mode (*i* <sup>A</sup> = *I*ref, *i* <sup>B</sup> = 0)—simulation. (**b**) Electromagnetic torque *T*<sup>e</sup> vs. rotor position θ in DCO mode (*i* <sup>A</sup> = *i* <sup>B</sup> = 0.25 *I*ref), channel A SCO mode (*i* <sup>A</sup> = 0.5 *I*ref, *i* <sup>B</sup> = 0), and channel B SCO mode (*i* <sup>A</sup> = 0, *i* <sup>B</sup> = 0.5 *I*ref)—laboratory test. (**c**) Flux linkage vs. rotor position θ in DCO mode (*i* <sup>A</sup> = *i* <sup>B</sup> = 0.5 *I*ref)—simulation. (**d**) Flux linkage vs. rotor position θ in SCO mode (*i* <sup>A</sup> = *I*ref, *i* <sup>B</sup> = 0)—simulation.

#### *4.2. Characteristics of the DCSRM*

In the case of a switched reluctance motor, the relationship between DCO and SCO modes is more complex. This is largely due to this machine's principle of operation. Unfortunately, the principle that, in DCO mode, the reference value of the current should be equal to 50% of the value in SCO mode, which applies to the DCBLDCM, could not be adopted here. Due to the nonlinear relationship between the value of the generated electromagnetic torque and the current, achieving the same value of electromagnetic torque in DCO mode requires a current greater than 50% of the reference value of SCO mode. In the analyzed case, the required level was 70%. Figure 5a shows the relationship between the electromagnetic torque as a function of rotor position for DCO and SCO modes of the SRM. Examples of characteristics determined in laboratory conditions are shown in Figure 5b. Laboratory tests were performed for *I* = 0.5 *I*ref (i.e., *i* <sup>A</sup> = 0.5 *I*ref, *i* <sup>B</sup> = 0 for SCO mode (channel A); *i* <sup>A</sup> = 0, *i* <sup>B</sup> = 0.5 *I*ref for SCO mode (channel B); and *i* <sup>A</sup> = *i* <sup>B</sup> = 0.35 *I*ref for DCO mode). The characteristics of the linkage flux for DCO mode are shown in Figure 5c, and those for SCO mode in Figure 5d.

In the case of the SRM, the saturation of a magnetic circuit affected the relationship between DCO and SCO modes. This can be seen in the torque characteristics (Figure 5a) and the flux characteristics (Figure 5c,d). However, the impact of magnetic couplings between the two channels was small (Figure 5d). For this configuration, it can also be assumed that both channels were characterized by a very big magnetic separation. This is why this specific configuration of pole windings was selected for the dual-channel power supply (Figure 1b).

**Figure 5.** *Cont*.

**Figure 5.** Static torque and flux linkage characteristics of the DCSRM. (**a**) Electromagnetic torque *T*e vs. rotor position θ in DCO mode (*i* <sup>A</sup> = *i* <sup>B</sup> = 0.7 *I*ref) and SCO mode (*i* <sup>A</sup> = *I*ref, *i* <sup>B</sup> = 0)—simulation. (**b**) Electromagnetic torque *T*<sup>e</sup> vs. rotor position θ in DCO mode (*i* <sup>A</sup> = *i* <sup>B</sup> = 0.35 *I*ref), channel A SCO mode (*i* <sup>A</sup> = 0.5 *I*ref, *i* <sup>B</sup> = 0), and channel B SCO mode (*i* <sup>A</sup> = 0, *i* <sup>B</sup> = 0.5 *I*ref)—laboratory test. (**c**) Flux linkage vs. rotor position θ in DCO mode (*i* <sup>A</sup> = *i* <sup>B</sup> = 0.7 *I*ref)—simulation. (**d**) Flux linkage vs. rotor position θ in SCO mode (*i* <sup>A</sup> = *I*ref, *i* <sup>B</sup> = 0)—simulation.

#### **5. Transient Analysis**

#### *5.1. DCO and SCO*

An analysis of DCO and SCO of both motors was performed for two cases: operation at low rotational speed and operation at high rotational speed. In the first operating point, both motors worked with constant torque, which required use of a current controller. The second operating point was located on the natural characteristic. Like in the previous section, the obtained waveforms of the electromagnetic torque and the line currents were compared with the base values specified in Table 1. In SCO mode, it was assumed that only channel A of the machine would be supplied.

#### 5.1.1. Constant Torque Operation

For this operating point, the numerical calculations were performed for the speed of *n* = 2000 r/min. It was assumed that the electromagnetic torque *Te* was equal to the base value given in Table 1. The obtained results are shown in Figure 6.

The comparison of the waveforms of the electromagnetic torque and the currents led to the conclusion that, in this range of operation, there were no significant differences between DCO and SCO modes for the DCBLDCM (Figure 6a,b). In DCO mode, half of the value of the reference current for SCO mode was required. The torque ripple in the case of DCBLDCM was 32% in DCO mode and 31% in SCO mode. Slightly greater differences were observed for DCO and SCO modes of the DCSRM (Figure 6d) in the waveforms of both the electromagnetic torque and the currents. In the case of SCO mode, a greater electromagnetic torque ripple (34%) was observed in comparison with DCO mode (27%). The main cause of the increase in electromagnetic torque ripple was the changed shape of the static torque characteristic (Figure 5a). In order to obtain the same value of electromagnetic torque in DCO mode, the DCSRM required a reduction of the value of the reference current (to 70%). In the case of DCO mode, the impact of magnetic couplings between the channels was unnoticeable.

**Figure 6.** *Cont*.

**Figure 6.** Relative electromagnetic torque and line currents in DCO and SCO for the DCBLDCM and the DCSRM at low speed. (**a**) Electromagnetic torque *T*e for a DCBLDCM. (**b**) Line currents for the DCBLDCM. (**c**) Electromagnetic torque *T*e for the DCSRM. (**d**) Line currents for the DCSRM.

#### 5.1.2. Operation without Current Control

In the case of operation at the rated speed (*n* = 8000 r/min), there was no control of the line current. In the calculations, it was assumed that at this operating point, both motors would generate the rated torque equal to half of the value of the base torque. However, achieving the same value of electromagnetic torque in SCO mode is more complex. In the case of the DCSRM, a change of the turn-on and dwell angle provides great possibilities. In the case of the DCBLDCM, one can also change the turn-on angle in order to increase the value of the generated electromagnetic torque [25]. Another way to achieve the same value of electromagnetic torque in SCO and DCO modes in this part of the characteristic is to use PWM control. This applies to both motors. In the numerical calculations, the turn-on angle was changed in both motors. In the case of the DCBLDCM, in DCO mode, the turn-on angle was not changed. Figure 7 shows the results of the numerical calculations in the analyzed operating point.

In the case of operation at high rotational speed (without current control), the differences between DCO and SCO modes were more noticeable. This applied to both motors. In SCO mode, the electromagnetic torque ripple increased (42% (Figure 7a) and 65% (Figure 7c)). In DCO mode, the torque ripple was reduced (DCBLDCM—13% and DCSRM—39%). In the case of the DCSRM, torque ripple was always greater, regardless of the mode of operation. In the case of the DCSRM, in DCO mode, there was a noticeable small impact of magnetic couplings between the channels (Figure 7d).

Figure 8 shows examples of waveforms of line currents in DCO and SCO modes for both motors, measured in laboratory conditions. The laboratory system used in the tests is discussed in [25].

**Figure 7.** *Cont*.

**Figure 7.** Relative electromagnetic torque and line currents in DCO and SCO modes of the DCBLDCM and the DCSRM at high speed. (**a**) Electromagnetic torque *T*e for the DCBLDCM. (**b**) Line currents for the DCBLDCM. (**c**) Electromagnetic torque *T*e for the DCSRM. (**d**) Line currents for the DCSRM.

In the real system, the differences between DCO and SCO modes were greater. This was due to the differences in the electric and magnetic parameters of the different phases of both channels. However, the waveforms of the current confirmed the results of the numerical calculations shown in Figure 7b,d.

**Figure 8.** *Cont*.

**Figure 8.** Relative line currents in DCO and SCO modes for the DCBLDCM and the DCSRM in a high-speed laboratory test. (**a**) Line currents for the DCBLDCM in DCO mode. (**b**) Line currents for the DCBLDCM in SCO mode. (**c**) Line currents for the DCSRM in DCO mode. (**d**) Line currents for the DCSRM in SCO mode.

#### *5.2. Influence of Asymmetrical Control in DCO Operation*

In the analysis of DCO mode described in Section 5.1, both channels were controlled symmetrically. The test of magnetic independence for both machines was an analysis of a case where they are controlled asymmetrically. Use of two independent power supply channels (Figure 1c,d) allowed independent control of each channel. An analysis of the impact of asymmetric control was performed for a high rotational speed (*n* = 8000 r/min). In the calculations, it was assumed that the control parameters of channel A were the same as those specified in Section 5.1. In the case of channel B, the value of the turn-on angle was changed, and in the case of the DCSRM, the dwell angle was additionally changed. The results of the calculations are shown in Figure 9. Examples of results of laboratory tests are shown in Figure 10.

**Figure 9.** *Cont*.

**Figure 9.** Relative electromagnetic torque and line currents in DCO mode for the DCBLDCM and the DCSRM at asymmetrical control. (**a**) Electromagnetic torque *T*e for the DCBLDCM. (**b**) Line currents for the DCBLDCM. (**c**) Electromagnetic torque *T*e for the DCSRM. (**d**) Line currents for the DCSRM.

Application of the asymmetrical control for the DCBLDCM caused an increase of the average value of electromagnetic torque and the torque ripple (23%). For the DCSRM at asymmetrical control, the trend of the average value of electromagnetic torque depends on the control angles. At the same time, it is feasible to suppress the torque ripple. In the case presented in Figure 9c, the torque ripple was 29%.

The results of the laboratory test showed that both designs were characterized by very high magnetic independence between the channels. This was confirmed by the results of laboratory tests shown in Figure 10. In the case of the DCSRM, this justified the adoption of the "NNSS" configuration. Such high magnetic independence between the channels in both solutions provides great possibilities, also in the case of generator operation.

**Figure 10.** Relative line currents in DCO mode for the DCBLDCM and the DCSRM at asymmetrical control—a laboratory test. (**a**) Line currents for the DCBLDCM. (**b**) Line currents for the DCSRM.

#### *5.3. Influence of Shorted Coil in DCO Mode*

Defects inside the machine are one of the typical fault conditions. Usually, these are various short circuits inside the windings of the machine. This article includes an analysis of a short circuit in part of the winding of channel B of phase 1*<sup>B</sup>* <sup>2</sup> (the entire coil is shown in Figure 1c,d). The analysis was performed during operation at a high rotational speed (*n* = 8000 r/min). The shorted coil was simulated using an additional switch S2 (Figure 1c,d). At the selected time, a short circuit in the coil occurred. In the case of the DCBLDCM, the short circuit occurred at *t* = 0.66 s. In the case of the DCSRM, the short circuit occurred at *t* = 1.0 s. Figure 11 shows the results of the numerical calculations.

Occurrence of a fault operating condition in the form of a short circuit in a part of the winding has a negative impact on the service life of machines. This applies in particular to DCBLDCM machines. In the shorted part of the winding, a very high current is present (Figure 11b), which causes thermal damage to the winding. However, this is not the only negative impact of this defect. Disconnecting the defective channel has practically no influence on the value of the short-circuit current. Consequently, further operation of a DCBLDCM machine with this defect is practically impossible. Moreover, the fault operating condition causes substantial torque ripple (497% in the presented case). Under certain

conditions, it is possible to suppress it by application of appropriate drive system topology [21]. Nevertheless, it has no influence on the value of the short-circuit current.

**Figure 11.** *Cont*.

**Figure 11.** Relative electromagnetic torque and line currents in DCO mode for the DCBLDCM and the DCSRM at a shorted coil in channel B. (**a**) Electromagnetic torque *T*e for the DCBLDCM. (**b**) Line currents for the DCBLDCM. (**c**) Electromagnetic torque *T*e for the DCSRM. (**d**) Line currents for the DCSRM.

Unlike a DCBLDCM, in the case of a DCSRM, occurrence of a partial short circuit in the winding is not a critical condition. After an electromagnetic balance is reached, a small short-circuit current flows in the shorted part of the winding (Figure 11d). Much more problematic are the consequences of an unbalanced magnetic pull and the torque ripple (92%). However, the machine can continue to operate with a defective channel. Here, the experimental results of operation under the pole 1*<sup>B</sup>* <sup>2</sup> short-circuit condition are presented (Figure 12). The used motor controller could not operate in current-control mode. A small current flowed in the shorted part of the winding. In the rest of the winding, the current became two times greater. However, it had a noticeable influence on the other current waveforms. In comparison with the simulation results, this influence was more significant. The impact of the control parameters and the second channel on the short-circuit current was minor.

**Figure 12.** Line currents in DCO mode for the DCSRM at a shorted coil in channel B—laboratory test.

#### *5.4. Influence of an Open Phase in DCO Mode*

Another analyzed case of failure operation was an open winding in phase 1<sup>B</sup> of channel B. Like the short-circuit condition described in Section 5.2, the defect of phase 1<sup>B</sup> was simulated. At a certain point in time, the switch S1 was opened (Figure 1c,d). The defect of phase 1<sup>B</sup> in the DCBLDM occurred at *t* = 0.7 s, and in the DCSRM, at *t* = 1 s. The results of the numerical calculations are shown in Figure 13.

The failure condition caused by an open winding in one of the phases of channel B prevents further operation of machines in SCO mode. In DCO mode, after this defect occurs, a DCSR machine can continue to operate. In the case of a DCBLDC machine with delta-connected windings, this is also possible. Here, in both cases, the torque ripple increased (DCBLDCM—64% and DCSRM—165%). In the case of the DCSRM, the analyzed fault condition was identical to a defect of one or even both transistors in the power supply system. In laboratory conditions, in the case of the DCBLDCM, the detection of rotor position was based on a sensorless control algorithm. Failure to supply one of the phases of channel B resulted in errors in detection of rotor position in the defective channel (Figure 14a). Start-up of the DCBLDCM was possible with defective channel B (and with channel A not being supplied).

**Figure 13.** *Cont*.

**Figure 13.** Relative electromagnetic torque and line currents in DCO mode for the DCBLDCM and the DCSRM at open phase in channel B. (**a**) Electromagnetic torque *T*e for the DCBLDCM. (**b**) Line currents for the DCBLDCM. (**c**) Electromagnetic torque *T*e for the DCSRM. (**d**) Line currents for the DCSRM.

**Figure 14.** *Cont*.

**Figure 14.** Relative line currents in DCO mode at open phase in channel B—laboratory test: (**a**) DCBLDCM; (**b**) DCSRM.

#### **6. Conclusions**

This study compared two dual-channel electronically commutated motors: a DCBLDCM and a DCSRM. However, it was not a full comparison and it was focused on selected operation modes. The analysis excluding thermal, vibroacoustic and efficiency aspects was performed. We proposed a mathematical model for a three-phase DCBLDC machine that considered the nonlinearity of the magnetization characteristics and all couplings between the channels. We presented a model of a DCSRM machine and demonstrated that the structure of its equations can be derived from the DCBLDCM model by omitting all components related to fluxes generated by permanent magnets. In the case of both designs, it is possible to omit the couplings between the channels in symmetrical control conditions. As demonstrated, nontypical operating conditions result in a much greater impact of couplings between the channels in the analyzed DCBLDCM design. The DCBLDCM solution is better in symmetrical control conditions due to higher torque-to-mass ratio (Table 1). However, the DCSRM solution is characterized by much greater tolerance and reliability. In the case of a DCSRM, a short-circuit condition in a part of the winding of one of the phases does not prevent further operation of the drive system. In the case of a DCBLDCM, on the other hand, occurrence of such a failure condition is a critical failure that can bring the drive system to a stop. Switching a DCBLDCM into SCO mode does not reduce the value of the short-circuit current. This problem is not present in the case of a DCSRM. A DCSRM switches into SCO mode without any problems and the current in the shorted winding of one channel does not influence the operation of the remaining channels of the drive system.

**Author Contributions:** Conceptualization, M.K., Formal analysis, J.P., Investigation, M.K., J.P., B.P. and G.P., Methodology, M.K. and J.P., Project administration, M.K. and B.P., Software, M.K. and B.P., Supervision, J.P. and P.B., Visualisation, M.K. and G.P., Writing-original draft, M.K., Writing-review & editing, J.P., P.B., B.P. and G.P.

**Funding:** This work is financed in part by the statutory funds of the Department of Electrodynamics and Electrical Machine Systems, Rzeszow University of Technology and in the part by Polish Ministry of Science and Higher Education under the program "Regional Initiative of Excellence" in 2019 – 2022. Project number 027/RID/2018/19, amount granted 11 999 900 PLN.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Parsa, L.; Toliyat, H.A. Five-Phase Permanent-Magnet Motor Drives. *IEEE Trans. Ind. Appl.* **2005**, *41*, 30–37. [CrossRef]


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **The Fault-Tolerant Quad-Channel Brushless Direct Current Motor**

#### **Mariusz Korkosz \*, Piotr Bogusz and Jan Prokop**

The Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, Al. Powstancow Warszawy 12, 35-959 Rzeszow, Poland; pbogu@prz.edu.pl (P.B.); jprokop@prz.edu.pl (J.P.)

**\*** Correspondence: mkosz@prz.edu.pl; Tel.: +48-178-651-389

Received: 12 August 2019; Accepted: 20 September 2019; Published: 25 September 2019

**Abstract:** In this study, a permanent magnet brushless direct current machine with multi-phase windings is proposed for critical drive systems. We have named the solution, which has four-stator winding, a quad-channel permanent magnet brushless direct current (QCBLDC) motor. The stator windings are supplied by four independent power converters under quad-channel operation (QCO) mode. After a fault in either one, two, or three channels, further operation of the machine can be continued in triple-channel operation (TCO) mode, dual-channel operation (DCO) mode, or single-channel operation (SCO) mode. In this paper, a novel mathematical model is proposed for a QCBLDC machine. This model takes into account the nonlinearity of a magnetic circuit and all of the couplings between the phases within a given channel, as well as between channels. Based on numerical calculations, the static electromagnetic moment and the coupled fluxes were determined for the individual windings of the variants and work modes being analyzed. A normal work condition can be achieved in the QCO or DCO modes. For the DCO mode, an acceptable case uses a balanced magnetic pull (A and C channels supplied). The DCO A and B type work mode is comparable to the DCO A and C mode with regard to its efficiency in processing electrical energy. The vibroacoustic parameters of this mode, however, are much worse. In fault states, TCO, DCO, and SCO work modes are possible. As the number of active channels decreases, the efficiency of energy processing also decreases. In a critical situation, the motor works in overload mode (SCO mode). Laboratory tests conducted for one of the variants demonstrated that the TCO work mode is characterized by worse vibroacoustic parameters than the DCO A and C mode.

**Keywords:** multi-channel; quad-channel operation (QCO); triple-channel operation (TCO); dual-channel operation (DCO); single-channel operation (SCO); permanent magnet brushless direct current motor; BLDCM

#### **1. Introduction**

Critical drives are characterized by improved reliability and are usually intended for special applications, such as airplanes, submarines, and electric cars [1–3]. The main property of critical drives is their ability to continue operation after the occurrence of a fault state or emergency state. There are several ways to improve the reliability of these drives. The first is to use, for example, two independent drive systems [4]. In this case, full independence and separation of the drives is achieved. This is a safe solution because when one drive becomes defective, the other remains in operation. However, this solution requires more space and is heavier. Another group of solutions that has recently been developed consists of using only a one drive system using multi-channel [5–8] or multi-phase [9,10] machines. Such systems make it possible to reduce the mass of the motor and the space occupied by the drives while maintaining high reliability. Electric machines used in high-reliability critical drives include induction machines [11–13], switched reluctance machines (SRM) [8,14], permanent magnet synchronous machines (PMSM) [15–17], and brushless direct current

machines with a permanent magnet (BLDCM) [7,18]. Unlike multi-phase solutions, multi-channels have clearly separated channels that are usually supplied by separate power supply sources [7,8]. In the case of multi-channel three-phase machines, each channel consists of three windings that are electrically separated from the windings of the other channels [15]. In the case of multi-phase solutions, the occurrence of an emergency state caused by a break (e.g., in one of the phases) must lead to the occurrence of an asymmetric magnetic pull. In the case of a multi-channel solution, depending on the configuration adopted, operation of the drive can continue with a balanced magnetic pull.

This article analyzes the problem of operating a quad-channel brushless direct current machine with a permanent magnet (QCBLDC) in quad-channel operation (QCO) mode. The four channels of the machine can be supplied from one, two, or four supply sources. An example of supply from two sources (DC1 and DC2) in the QCO mode is shown in Figure 1.

**Figure 1.** A scheme of a quad-channel brushless permanent magnet direct current (QCBLDC) supply system under quad-channel operation (QCO) mode with two supply sources.

In the case of an emergency state in one of the channels, the system can switch to operating in the triple-channel operation (TCO) mode, the dual-channel operation (DCO) mode, or the single-channel operation (SCO) mode. This article assumes that a QCBLDC working in the QCO mode, after the occurrence of an emergency state in one of its channels, goes into the DCO mode. The third working channel in this case works as an emergency channel for the DCO mode. In the event of operation in the TCO mode, a balanced magnetic pull can be achieved only for the selected configurations of the channel location. Consequently, this operation status will not be analyzed at this stage of the research. The research problem formulated in this article is an answer to the question: which two channels, out of the three that are not defective, work in the DCO mode? So far, the literature on this topic has not presented an analysis of the properties of a four-channel permanent magnet brushless direct current (QCBLDC) machine, and there is no solution for the aforementioned research problem.

The objective of this article is to present the results of the research conducted by the authors on the properties of a QCBLDC motor working in two modes: QCO and DCO. Knowledge of the results of such an analysis is necessary to prepare a control algorithm that, in the event of the occurrence of an emergency state in one channel, would switch the motor from the QCO mode to the DCO mode. The analysis of the properties of a QCBLDC motor is conducted based on simulations and laboratory tests. Proprietary nonlinear circuit mathematical models for the QCBLDC motor are presented for the QCO and DCO modes. As part of the simulation tests, the results of a finite element method (FEM) two-dimensional analysis of the distribution of the magnetic fluxes and the distribution of stresses on the circumference of the stator are presented. The static characteristics, the current waveforms, and the electromagnetic torque are compared for the operating modes being analyzed. The results of the simulation tests were verified by conducting laboratory tests. The conclusions drawn from the results of the tests are presented in the summary.

#### **2. Model and Winding Configurations of a Quad-Channel BLDC Motor**

Figure 2a shows a proprietary prototype of the hybrid drive from a small unmanned aerial vehicle (UAV). The electric motor of this object is a three-phase brushless motor with permanent magnets (BLDCM). The stator of the motor with its windings is shown in Figure 2b. When designing the motor, the possibility to improve the reliability of the drive's operation is taken into account. The possibility of an independent multi-channel supply for the motor is provided. The tested motor has the possibility to use a quad-channel supply or, as shown in the literature [5], a dual-channel supply. In the case of a quad-channel supply, there are two possible configurations of the stator windings. These possibilities are shown in Figure 3a (variant I) and Figure 3b (variant II). What makes these configurations different is the location of the windings of the different channels on the circumference of the stator. In variant I, the phases of each channel are staggered by 120 mechanical degrees (Figure 3a). In variant II, the windings of each channel are staggered by 30 mechanical degrees (Figure 3b). The channels in both variants are distributed as follows:


Moreover, it was assumed that continuous operation was also possible using only two channels, i.e., in the DCO mode. In this case, the two remaining channels are redundant.

Regardless of the variant, there are six possible configurations of DCO: A and B, A and C, A and D, B and C, B and D, and C and D. In this paper, only two are analyzed; A and B and A and C. This analysis is shown in Table 1 and Figure 3c–f. Figure 3g,h show a phasor diagram of the induced voltages (BEMF) in the windings (1,2,3) of individual channels (A,B,C,D) for variants I and II. At the same time, under the conditions of a fully operational drive, it is possible to use all channels. This operating condition is identified as the QCO (quad-channel operation) mode. In this operating condition, each channel works with half of the required power. When analyzing the multi-channel supply, the impact of the location of the channels and the way they are supplied on the magnetic pull force was not considered. These aspects will be analyzed in subsequent stages of the research based on a coupled electromagnetic-mechanical analysis. Further, operations in odd channel number conditions (i.e., TCO and SCO) are not considered. The TCO mode may occur when one of the channels becomes

defective. SCO is a condition of the critical operation of the system. This means that the remaining three channels have already become defective.

**Figure 2.** The prototype of a three-phase quad-channel brushless machine with a permanent magnet (QCBLDC) machine; (**a**) the hybrid drive of a small unmanned aerial vehicle; (**b**) the stator with windings.


**Table 1.** The selected typical operation conductions of a QCBLDC motor.

Table legend: QCO — quad-channel operation, TCO — triple-channel operation, DCO — dual- channel operation, SCO — single-channel operation.

**Figure 3.** Scheme of the winding distribution of channels on the stator of a three-phase QCBLDC motor: variant I—quad-channel operation (QCO) (**a**), variant II—QCO (**b**), variant I—dual-channel operation (DCO) (channel A and C) (**c**), variant II—DCO (channel A and C) (**d**), variant I—DCO (channel A and B) (**e**), variant II—DCO (channel A and B) (**f**), variant I—phasor diagram (**g**), and variant II—phasor diagram (**h**).

#### **3. Mathematical Model of the QCBLDC Motors**

The subject of the mathematical modelling is the QCBLDC motor, for which the authors' circuit-based models, known as flux models, were proposed. The mathematical models are presented while taking into account the non-linearity of the magnetic circuit and the magnetic couplings between the particular phases within the given channel, as well as between the channels (A, B, C, D). Models

are included for a machine that works in two modes: the quad-channel operation (QCO) mode and the dual-channel operation (DCO) mode.

The following simplifying assumptions have been adopted in the proposed mathematical model of the QCBLDC machine:

1. Symmetry of the magnetic and electric circuit structure of both the stator and rotor;

2. Decomposition of the phase fluxes into a sum of fluxes induced by phase currents (leakage and main fluxes) and fluxes from permanent magnets;

3. Simplified leakage fluxes from currents in the end-turns of windings;

4. Omitting the influence of temperature on the fluxes generated by permanent magnets and stator resistance.

#### *3.1. Model for QCO Mode*

The general structure of the mathematical model of the three-phase QCBLDC motor in QCO mode can be written in the following form:

$$
\begin{bmatrix} \mathbf{u}^{\mathcal{A}} \\ \mathbf{u}^{\mathcal{B}} \\ \mathbf{u}^{\mathcal{C}} \end{bmatrix} = \begin{bmatrix} \mathbf{R}^{\mathcal{A}} & 0 & 0 & 0 \\ 0 & \mathbf{R}^{\mathcal{B}} & 0 & 0 \\ 0 & 0 & 0 & \mathbf{R}^{\mathcal{D}} \end{bmatrix} \begin{bmatrix} \mathbf{i}^{\mathcal{A}} \\ \mathbf{i}^{\mathcal{B}} \\ \mathbf{i}^{\mathcal{C}} \end{bmatrix} + \frac{\mathbf{q}}{\mathbf{d}t} \begin{bmatrix} \boldsymbol{\Psi}^{\mathcal{A}} \\ \boldsymbol{\Psi}^{\mathcal{B}} \\ \boldsymbol{\Psi}^{\mathcal{C}} \end{bmatrix} + \begin{bmatrix} \mathbf{e}^{\mathcal{A}} \\ \mathbf{e}^{\mathcal{B}} \\ \mathbf{e}^{\mathcal{C}} \\ \mathbf{e}^{\mathcal{D}} \end{bmatrix} \tag{1}
$$

$$J\frac{d\omega}{dt} + D\omega + T\_{\text{L}} = T\_{\text{e}} \tag{2}$$

$$\frac{\mathbf{d}\theta}{\mathbf{d}t} = \boldsymbol{\omega} \tag{3}$$

where for channels <sup>k</sup> <sup>∈</sup> (A, B, C, D), the vectors representing the phase voltages, **<sup>u</sup>**k, phase currents, **i** k, phase back-EMF voltages, **e**<sup>k</sup> = **e**k(θ, i PM), the flux linkages caused by the phase winding currents, ψ<sup>k</sup> = ψk(θ,**i** A,**i** B,**i** C,**i** D,**i** PM), as well as the matrixes of the stator resistances, **R**k, are defined as follows:

$$\begin{aligned} \mathbf{u}^{\mathbf{k}} &= \left[\mathbf{u}\_1^{\mathbf{k}}, \mathbf{u}\_2^{\mathbf{k}}, \mathbf{u}\_3^{\mathbf{k}}\right]^{\mathrm{T}}, \mathbf{i}^{\mathbf{k}} = \left[\mathbf{i}\_1^{\mathbf{k}}, \mathbf{i}\_2^{\mathbf{k}}, \mathbf{i}\_3^{\mathbf{k}}\right]^{\mathrm{T}}, \mathbf{e}^{\mathbf{k}} = \left[\mathbf{e}\_1^{\mathbf{k}}, \mathbf{e}\_2^{\mathbf{k}}, \mathbf{e}\_3^{\mathbf{k}}\right]^{\mathrm{T}}, \boldsymbol{\Psi}^{\mathbf{k}} = \left[\boldsymbol{\psi}\_1^{\mathbf{k}}, \boldsymbol{\psi}\_2^{\mathbf{k}}, \boldsymbol{\psi}\_3^{\mathbf{k}}\right]^{\mathrm{T}}, \\ \mathbf{R}^{\mathbf{k}} &= \mathrm{diag}(\mathbf{R}\_1^{\mathbf{k}}, \mathbf{R}\_2^{\mathbf{k}}, \mathbf{R}\_3^{\mathbf{k}}). \end{aligned}$$

The following symbols are used in Equations (1) to (3): θ—rotor angle position, ω—the rotor angular speed, i PM—the permanent magnet magnetization equivalent current, *J*—the rotor's (and load's) moments of inertia, *D*—the rotor damping of the viscous friction coefficient, *TL*—the load torque, *T*e—the total electromagnetic torque.

The phase back-EMF vectors in Equation (1) for channels k ∈ (A, B, C, D) are defined as follows:

$$\mathbf{e}^{\mathbf{k}} = \frac{\mathbf{d}}{\mathbf{d}t} \begin{bmatrix} \Psi\_1^{\text{kPM}}(\theta, \text{i}^{\text{PM}})\\ \Psi\_2^{\text{kPM}}(\theta, \text{i}^{\text{PM}})\\ \Psi\_3^{\text{kPM}}(\theta, \text{i}^{\text{PM}}) \end{bmatrix} = \omega \begin{bmatrix} \frac{\partial \Psi\_1^{\text{kPM}}(\theta, \text{i}^{\text{PM}})}{\partial \theta} \\\ \frac{\partial \Psi\_2^{\text{kPM}}(\theta, \text{i}^{\text{PM}})}{\partial \theta} \\\ \frac{\partial \Psi\_3^{\text{kPM}}(\theta, \text{i}^{\text{PM}})}{\partial \theta} \end{bmatrix} \tag{4}$$

where ψkPM <sup>i</sup> (θ, i PM) for <sup>i</sup> <sup>∈</sup> ( 1, 2, 3) are the permanent magnet fluxes linking the stator windings. The permanent magnet flux linking each stator winding of the QCBLDC motor follows the trapezoidal profile back-EMF. The real back-EMF is not a flat and ideal trapezoidal waveform. Other real back-EMF profiles can be defined in Equation (1). For example, the back-EMF waveform in the Fourier series for k ∈ (A, B, C, D) and i ∈ ( 1, 2, 3) is represented as

$$\mathbf{e}\_{\rm i}^{k} = \omega \left[ a\_{\rm i0}^{k} + \sum\_{\nu=1}^{\infty} \left( a\_{\rm i\nu}^{k} \sin(\nu \theta) + b\_{\rm i\nu}^{k} \cos(\nu \theta) \right) \right] \tag{5}$$

*Energies* **2019**, *12*, 3667

The flux linkages caused by the phase winding currents in Equation (1) for k ∈ (A, B, C, D) can be written in the following form:

$$\mathbf{\upmu}^{\mathbf{k}} = \begin{bmatrix} L\_{1\sigma}^{\mathbf{k}\mathbf{k}} \mathbf{i}\_1^{\mathbf{k}} + \sum\_{\mathbf{l}=\mathbf{A}}^{\mathbf{D}} \left( \sum\_{\mathbf{j}=1}^{3} \boldsymbol{\upupmu}\_{\mathbf{i}\mathbf{j}}^{\mathbf{k}l} (\boldsymbol{\uptheta}, \mathbf{i}\_{\mathbf{j}}^{\mathbf{l}}, \mathbf{i}^{\mathbf{P}\mathbf{M}}) \right) \\\\ L\_{2\sigma}^{\mathbf{k}\mathbf{k}} \mathbf{i}\_2^{\mathbf{k}} + \sum\_{\mathbf{l}=\mathbf{A}}^{\mathbf{D}} \left( \sum\_{\mathbf{j}=1}^{3} \boldsymbol{\upupmu}\_{\mathbf{j}\mathbf{j}}^{\mathbf{k}l} (\boldsymbol{\uptheta}, \mathbf{i}\_{\mathbf{j}}^{\mathbf{l}}, \mathbf{i}^{\mathbf{P}\mathbf{M}}) \right) \\\\ L\_{3\sigma}^{\mathbf{k}\mathbf{k}} \mathbf{i}\_3^{\mathbf{k}} + \sum\_{\mathbf{l}=\mathbf{A}}^{\mathbf{D}} \left( \sum\_{\mathbf{j}=1}^{3} \boldsymbol{\upupmu}\_{\mathbf{j}\mathbf{j}}^{\mathbf{k}l} (\boldsymbol{\uptheta}, \mathbf{i}\_{\mathbf{j}}^{\mathbf{l}}, \mathbf{i}^{\mathbf{P}\mathbf{M}}) \right) \end{bmatrix} \tag{6}$$

where k, l <sup>∈</sup> (A, B, C, D) is the stator channel index, i, j <sup>∈</sup> ( 1, 2, 3) is the stator phase number, and *<sup>L</sup>*kk 1σ are the coefficients of the end-turn self-inductances. The stator flux linking (the so-called self-flux) ψkk ii i-th i ∈ ( 1, 2, 3) phase for the k-th channel k ∈ (A, B, C, D) in Equation (6) is calculated based on the following dependencies:

$$\boldsymbol{\Psi}\_{\rm ii}^{\rm kk}(\boldsymbol{\theta}, \mathbf{i}\_{\rm i}^{\rm k}, \mathbf{i}^{\rm PM}) = \boldsymbol{\Psi}\_{\rm i}^{\rm k} - L\_{\rm i}^{\rm kk} \mathbf{i}\_{\rm i}^{\rm k} - \sum\_{\substack{\mathbf{l} = \mathbf{A} \\ \mathbf{l} \neq \mathbf{k} \\ \mathbf{l} \neq \mathbf{k}}}^{\rm D} \left( \sum\_{\substack{\mathbf{j} = \mathbf{l} \\ \mathbf{j} \neq \mathbf{i}}}^{\rm 3} \boldsymbol{\Psi}\_{\rm ij}^{\rm kl}(\boldsymbol{\theta}, \mathbf{i}\_{\rm j}^{\rm l}, \mathbf{i}^{\rm PM}) \right) \tag{7}$$

From non-linear dependence (Equation (7)), the phase current, *i* k <sup>i</sup> , is calculated:

$$\boldsymbol{\Psi}\_{\rm ii}^{\rm kk} = \boldsymbol{\Psi}\_{\rm ii}^{\rm kk}(\boldsymbol{\theta}, \mathbf{i}\_{\rm i}^{\rm k}, \mathbf{i}^{\rm PM}) \implies \mathbf{i}\_{\rm i}^{\rm k} = \mathbf{i}\_{\rm i}^{\rm k}(\boldsymbol{\theta}, \boldsymbol{\psi}\_{\rm ii}^{\rm kk}, \mathbf{i}^{\rm PM}) \tag{8}$$

The electromagnetic torque in Equation (2) can be calculated as a derivative of the total magnetic field co-energy in the air gap with respect to the rotor angle's position, θ. The expression *T*<sup>e</sup> = *T*e( θ,**i** A,**i** B,**i** C,**i** D, **i** PM) for electromagnetic torque for the QCO mode can be written in the following form:

$$\begin{split} T\_{\texttt{o}} &= \sum\_{\mathbf{k}=\mathbf{A}}^{\mathrm{D}} \sum\_{\mathbf{i}=1}^{3} \Biggl( \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\mathrm{kPM}}(\boldsymbol{0},\dot{\mathbf{i}}^{\mathrm{pM}})}{\partial \boldsymbol{\upphi}} \mathbf{i}^{\mathrm{k}}\_{\mathbf{i}} \Biggr) + \sum\_{\mathbf{k}=\mathbf{A}}^{\mathrm{D}} \sum\_{\mathbf{i}=1}^{3} \Biggl( \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}\bar{\mathbf{i}}}^{\mathrm{kk}}(\boldsymbol{0},\dot{\mathbf{i}}^{\mathrm{k}}\_{\mathbf{i}},\dot{\mathbf{i}}^{\mathrm{pM}})}{\partial \boldsymbol{\upphi}} \mathrm{d}\mathbf{i}^{\mathrm{k}}\_{\mathbf{i}} \Biggr) + \sum\_{\mathbf{k}=\mathbf{A}}^{\mathrm{D}} \sum\_{\mathbf{i}=\mathbf{A}}^{\mathrm{D}} \sum\_{\mathbf{i}=1}^{3} \Biggl( \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\mathrm{kk}}(\boldsymbol{0},\dot{\mathbf{i}}^{\mathrm{kM}}\_{\mathbf{i}},\dot{\mathbf{i}}^{\mathrm{pM}})}{\partial \boldsymbol{\upphi}} \mathbf{i}^{\mathrm{k}}\_{\mathbf{i}} \Biggr) \\ + \sum\_{\mathbf{k}=\mathbf{B}}^{\mathrm{D}} \sum\_{\mathbf{i}=1}^{3} \sum\_{\mathbf{j}=1}^{3} \Biggl( \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}\bar{\mathbf{i}}}^{\mathrm{kk}}(\boldsymbol{0},\dot{\mathbf{i}}^{\mathrm{kM}}\_{\mathbf{i}},\dot{\mathbf{i}}^{\mathrm{PM}})}{\partial \boldsymbol{\upphi}} \mathbf{i}^{\mathrm{k}}\_{\mathbf{i}} \Biggr) + \sum\_{\mathbf{k}=\mathbf{C}}^{\mathrm{D}} \$$

Electromagnetic torque (Equation (9)) is the sum of the so-called cogging torque, *T*cog(θ, i PM), torques from fluxes linking permanent magnets and windings currents. The cogging torque of the permanent magnet (PM) machines, produced by magnets, can be expanded into a Fourier series:

$$T\_{\rm cog} = T\_{\rm cog}(\theta, \mathbf{i}^{\rm PM}) = \sum\_{\nu=1}^{\infty} T\_{\nu}(\mathbf{i}^{\rm PM}) \sin(\nu q \theta + \theta\_0) \tag{10}$$

where *T*ν(i PM) is the amplitude of the ν-th harmonic, *q* is the number of slots, and θ<sup>0</sup> is the initial angle.

The component's electromagnetic torque produced by the permanent magnets and currents can be acquired in the form:

$$T\_{\rm c}^{\rm PM} = \sum\_{\mathbf{k}=\mathbf{A}}^{\rm D} \sum\_{\mathbf{i}=1}^{3} \left( \frac{\partial \psi\_{\mathbf{i}}^{\rm kPM}(\boldsymbol{\theta}, \mathbf{i}^{\rm PM})}{\partial \boldsymbol{\theta}} \mathbf{i}\_{\mathbf{i}}^{\rm k} \right) = \frac{1}{\omega} \sum\_{\mathbf{k}=\mathbf{A}}^{\rm D} \sum\_{\mathbf{i}=1}^{3} \left( \mathbf{e}\_{\mathbf{i}}^{\rm k} \mathbf{i}\_{\mathbf{i}}^{\rm k} \right) \tag{11}$$

Equations (1) and (2) with Equations (4), (6), and (9) constitute the nonlinear mathematical model of the QCBLDC motors in the QCO mode.

#### *3.2. Model for DCO Mode*

The voltage Equation (1) for the DCO mode, i.e., where only channels A and B are supplied, can be written in the following form:

$$
\begin{bmatrix} \mathbf{u}^{\Lambda} \\ \mathbf{u}^{\rm B} \end{bmatrix} = \begin{bmatrix} \mathbf{R}^{\Lambda} & 0 \\ 0 & \mathbf{R}^{\rm B} \end{bmatrix} \begin{bmatrix} \mathbf{i}^{\Lambda} \\ \mathbf{i}^{\rm B} \end{bmatrix} + \frac{\mathbf{d}}{\mathbf{d}t} \begin{bmatrix} \boldsymbol{\Psi}^{\Lambda} \\ \boldsymbol{\Psi}^{\rm B} \end{bmatrix} + \begin{bmatrix} \mathbf{e}^{\Lambda} \\ \mathbf{e}^{\rm B} \end{bmatrix} \tag{12}
$$

where, for channels <sup>k</sup> <sup>∈</sup> (A, B), the phase back-EMFs voltages, **<sup>e</sup>**<sup>k</sup> = **<sup>e</sup>**k(θ, <sup>i</sup> PM), and vectors representing the flux linkages caused by phase winding currents, ψ<sup>k</sup> = ψk(θ, **i** A, **i** B, **i** PM), are defined as follows:

**e**<sup>A</sup> = ω ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ∂ψAPM <sup>1</sup> (θ,iPM) ∂θ ∂ψAPM <sup>2</sup> (θ,iPM) ∂θ ∂ψAPM <sup>3</sup> (θ,iPM) ∂θ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , **e**<sup>B</sup> = ω ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ∂ψBPM <sup>1</sup> (θ,iPM) ∂θ ∂ψBPM <sup>2</sup> (θ,iPM) ∂θ ∂ψBPM <sup>3</sup> (θ,iPM) ∂θ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (13) ψ<sup>A</sup> = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *L*AA <sup>1</sup><sup>σ</sup> i A <sup>1</sup> <sup>+</sup> <sup>3</sup> j=1 ψAA 1j (θ, iA <sup>j</sup> , iPM) + <sup>ψ</sup>AB 1j (θ, iB <sup>j</sup> , iPM) *vspace*3*pt L*AA <sup>2</sup><sup>σ</sup> i A <sup>2</sup> <sup>+</sup> <sup>3</sup> j=1 ψAA 2j (θ, iA <sup>j</sup> , iPM) + <sup>ψ</sup>AB 2j (θ, iB <sup>j</sup> , iPM) *L*AA <sup>3</sup><sup>σ</sup> i A <sup>3</sup> <sup>+</sup> <sup>3</sup> j=1 ψAA 3j (θ, iA <sup>j</sup> , iPM) + <sup>ψ</sup>AB 3j (θ, iB <sup>j</sup> , iPM) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ψ<sup>B</sup> = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *L*BB 1σ i B <sup>1</sup> <sup>+</sup> <sup>3</sup> j=1 ψBA 1j (θ, iA <sup>j</sup> , iPM) + <sup>ψ</sup>BB 1j (θ, iB <sup>j</sup> , iPM) *L*BB 2σ i B <sup>2</sup> <sup>+</sup> <sup>3</sup> j=1 ψBA 2j (θ, iA <sup>j</sup> , iPM) + <sup>ψ</sup>BB 2j (θ, iB <sup>j</sup> , iPM) *L*BB 3σ i B <sup>3</sup> <sup>+</sup> <sup>3</sup> j=1 ψBA 3j (θ, iA <sup>j</sup> , iPM) + <sup>ψ</sup>BB 3j (θ, iB <sup>j</sup> , iPM) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

The expression for the electromagnetic torque, *T*<sup>e</sup> = *T*e( θ,**i** A,**i** B, **i** PM), with Equation (14) taken into account, can be written in the following form:

$$\begin{split} T\_{\texttt{e}} &= \sum\_{\mathbf{i}=1}^{3} \Big( \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm RM}}(\boldsymbol{\uptheta}\_{\mathbf{i}}^{\texttt{\rm PM}})}{\partial \boldsymbol{\uptheta}} \mathbf{i}\_{\mathbf{i}}^{\boldsymbol{\up}} + \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm PM}}(\boldsymbol{\uptheta}\_{\mathbf{i}}^{\texttt{\rm PM}})}{\partial \boldsymbol{\uptheta}} \mathbf{i}\_{\mathbf{i}}^{\mathbf{B}} \Big) + \sum\_{\mathbf{i}=1}^{3} \Big( \int\_{0}^{\frac{\lambda}{\theta}} \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm MA}}(\boldsymbol{\uptheta}\_{\mathbf{i}}^{\texttt{\rm PM}}, \mathbf{i}^{\texttt{PM}})}{\partial \boldsymbol{\uptheta}} d\mathbf{i}\_{\mathbf{i}}^{\boldsymbol{\upalpha}} + \int\_{0}^{\frac{\mathsf{B}}{\mathsf{B}}} \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm RM}}(\boldsymbol{\uptheta}\_{\mathbf{i}}^{\texttt{\rm PM}}, \mathbf{i}^{\texttt{PM}})}{\partial \boldsymbol{\uptheta}} d\mathbf{i}\_{\mathbf{i}}^{\mathbf{B}} \Big) \\ + \sum\_{\mathbf{i}=1}^{3} \sum\_{\mathbf{j}=1}^{3} \Big( \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm PM}}(\boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm PM}}, \mathbf{i}^{\texttt{PM}})}{\partial \boldsymbol{\upphi}} \mathbf{i}\_{\mathbf{i}}^{\mathbf{B}} + \frac{\partial \boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm PM}}(\boldsymbol{\upphi}\_{\mathbf{i}}^{\texttt{\rm PM}}, \mathbf{i}^{\texttt{PM}})}{\partial \boldsymbol$$

The components of the electromagnetic torque produced by permanent magnets and currents (the first component of the right side of Equation (15)) can be determined in the form:

$$T\_{\mathbf{e}}^{\rm PM} = \frac{1}{\alpha \prime} \sum\_{i=1}^{3} \left( \mathbf{e}\_{\mathbf{i}}^{\rm A} \, \mathbf{i}\_{\mathbf{i}}^{\rm A} + \mathbf{e}\_{\mathbf{i}}^{\rm B} \, \mathbf{i}\_{\mathbf{i}}^{\rm B} \right) \tag{16}$$

Equation (12) with Equations (13) and (14), and Equation (2) with Equation (15) constitute the mathematical model of a QCBLDC machine in the DCO mode.

#### **4. Static Characteristics**

The following assumptions were made in the FEM two-dimensional numerical calculations:


#### *4.1. Electromagnetic Torque*

The characteristics of static electromagnetic torque were determined for the QCO operation, and for both the analyzed variants (variant I and variant II) of the DCO operation, using FEM two dimensional commercial software [19].

The calculations were performed for one electrical period (36 mechanical degrees) at *I* = constant, supplying the phases, *Ph*1 and *Ph*2, and speed, *n* = 0.167 r/min. The current was changed in the range of 0 to 25 A for QCO and 0 to 50 A for DCO. The average value of the electromagnetic torque, *T*eav, as a function of the current, *I*, flowing in the channel is shown in Figure 4a. Examples of the relationship between the electromagnetic torque, *T*e, and the rotor position are shown in Figure 4b. The numerical calculations were verified under laboratory conditions. A laboratory stand used to determine static characteristics is shown in Figure 4c. Examples of laboratory static torque characteristics are shown in Figure 4d.

In QCO operation, the configuration type (variant I, variant II) is completely unimportant. In DCO operation, the electromagnetic torque decreases (as a result of saturation). In the operating range (to the value of the rated torque), this influence is practically negligible. In overload operation (or in emergency operation), the constant torque in the DCO mode decreases. The difference between variant I and variant II is insignificant. A slightly smaller value of torque was generated in variant II. In both variants, no difference between the A and B configuration and the A and C configuration was identified in the DCO operation. This means that the type of variant and the configuration does not influence the value of the electromagnetic torque produced. There are differences between the QCO mode and the DCO mode in the stress within the magnetic circuit of the stator. Figure 5 shows examples of the surface force density of the magnetic circuit of the stator (of magnetic origin) for the selected positions of the rotor and for variant I of the stator winding. The results obtained for variant II are similar.

In the DCO mode, there is a significant increase in the stress compared to the QCO mode. In the case of the DCO A and B supply, the distribution of the stresses is non-symmetric, which is conducive to the occurrence of vibrations in the structure. In this regard, this configuration is not recommended. However, if operation needs to continue, e.g., after the C and D channels have become defective, the motor can continue to operate with increased asymmetry of the magnetic pull.

**Figure 4.** The static characteristics of electromagnetic torque: The average value of electromagnetic torque vs. current (**a**); electromagnetic torque vs. rotor positions—simulation tests (**b**); the stand to determine static characteristics (**c**); electromagnetic torque vs. rotor positions (laboratory test—variant II) (**d**).

**Figure 5.** Surface force density: QCO (**a**); DCO A and C (**b**); DCO A and B (**c**)—variant I.

#### *4.2. Flux Characteristics-Variant I*

Due to the division of the windings into four channels, there are twelve flux linkages with individual phases. Figures 6–8 show the flux linkages as a function of the rotor position for variant I in all analyzed configurations.

**Figure 6.** Flux linkages vs. rotor positions for QCO—variant I.

**Figure 8.** Flux linkages vs. rotor positions for DCO A and B—variant I.

In the case of the dual-channel supply in variant I, the A and C supply configuration (Figure 7) is more beneficial because of its minimal multi-channel magnetic coupling. The A and B configuration has a slightly greater magnetic coupling between the channels. However, the difference is not significant.

#### *4.3. Flux Characteristics-Variant II*

Figures 9 and 10 show the flux linkages for the analyzed configurations for variant II of the dual-channel supply.

**Figure 9.** Flux linkages vs. rotor positions for DCO A and C—variant II.

**Figure 10.** Flux linkages vs. rotor positions for DCO A and B—variant II.

In the case of variant II, it is not possible to indicate a more advantageous configuration due to the impact of the linkages or the impact of saturation of the magnetic circuit.

All determined characteristics were implemented in the simulation model as per the two-dimensional lookup table in the Matlab SISOTOOL system (R2019a, MathWorks, Natick, MA, USA) [20]. This has been explained in a previous paper [21].

#### **5. Waveforms, Current, Voltage, and Electromagnetic Torque**

#### *5.1. Numerical Calculations*

For the purpose of transient analysis, numerical calculations were performed for a constant rotor speed of *n* = 1000 r/min. In the calculations, it was assumed that, in the QCO mode, the value of the reference current set on the current control devices in all channels was equal to 10 A. In the dual-channel operation mode, the reference current was equal to 20 A for each of the analyzed configurations of both variants. Figures 11 and 12 show the electromagnetic torque of the motor for variant I (Figure 11) and variant II (Figure 12). Figure 13 shows the relationship between the flux linkage of the phase, *Ph*1, and the current for all cases analyzed.

**Figure 11.** Waveforms of electromagnetic torque—variant I.

**Figure 12.** Waveforms of electromagnetic torque—variant II.

**Figure 13.** Flux linkages vs. rotor positions for all configurations.

Selected results of the examinations are presented in Table 2.


**Table 2.** The selected results of the calculations for the analyzed operating conditions.

The variant types and configurations of the channels have little impact on the average value of the electromagnetic moment, *T*eav. Switching to the DCO mode results in an increase in copper losses (*P*cu), with a small reduction of iron losses (*P*Fe). In the case of dual-channel operation, the electromagnetic torque's ripple increases slightly. The efficiency of energy processing in the DCO mode is significantly deteriorated due to increased winding losses. The highest efficiency in the DCO mode was achieved for variant I of A and C. However, in general, the differences in the energy processing efficiency for each of the two variants of the DCO mode are small.

#### *5.2. Laboratory Test*

In laboratory conditions, a quad-channel supply for a QCBLDC motor was developed. Figure 14 shows the test stand. Laboratory tests were performed only for variant II.

**Figure 14.** Stand for transient test of a QCBLDC motor.

In laboratory conditions, the current waveforms were recorded during quad-channel, triple-channel, dual-channel, and single-channel operations (Figure 16): A switch from the QCO mode was connected to the TCO mode (Figure 16a), the DCO A and B mode (Figure 16b), the DCO A and C mode (Figure 16c), and to the SCO mode (Figure 16d) of variant II (*U*dc = 24 V, *T*<sup>L</sup> = 1.2 N·m).

(**c**)

**Figure 15.** *Cont.*

(**e**)

(**a**)

**Figure 16.** *Cont.*

#### *Energies* **2019**, *12*, 3667


(**b**)

(**c**)

**Figure 16.** Waveforms of the currents, load torque, and speed for QCO in the TCO mode (**a**), the DCO A and B mode (**b**), the DCO A and C mode (**c**), and the SCO mode (**d**).

During the transition from the QCO mode to the TCO, DCO, and SCO modes, channel D (Figure 16a), C and D (Figure 16b), B and D (Figure 16c), and A and B and C (Figure 16d) were disconnected, and channel A and B and C (Figure 16a), A and B (Figure 16b), A and C (Figure 16c), and D (Figure 16d) started to operate with a shaft load. To achieve the same moment, the currents in the active channels must increase by 133% in the TCO mode, by 200% in the DCO mode, and by 400% in the SCO mode. This increase leads to a decrease in speed in an open-loop control system and results in an output power decrease of a few percent.

The mechanical characteristics and the general efficiency were determined for the selected work mode in a stable state (without current regulation). The load torque was changed to 4 N·m (or 2 N·m for the SCO mode). The rotational speed as a function of the load torque is shown in Figure 17a. The general efficiency as a function of the load torque is shown in Figure 17b. This was determined using the direct method (η = *P*out/*P*in). For a load torque of *T*<sup>L</sup> = 2 N·m, the acceleration of vibration and the noise level were recorded. The results are given in Table 3.

**Figure 17.** Speed vs. torque load (**a**), overall efficiency vs. torque load (**b**)—variant II.

**Table 3.** Selected results of the acceleration and the noise level.


The QCO, TCO, and DCO work modes enable continuous operation of the machine. In the case of the SCO mode, the motor is usually already working in the overload range. This is a critical work condition that should enable operation of the device for a specific period of time. The QCO mode ensures the highest efficiency. In the TCO mode, the efficiency of energy processing is slightly reduced. In the DCO mode, a slightly higher efficiency is achieved in the A and C mode. This efficiency is smaller than that in the TCO mode.

The vibration and noise results presented in Table 3 indicate that the most advantageous work mode is QCO. The DCO A and C work condition was only slightly worse than QCO. The TCO work mode (regardless of the variant) was noticeably noisier than the DCO A and C mode. The DCO A and B work mode was significantly noisier than The DCO A and C mode and TCO mode. The SCO mode was comparable to the DCO A and B mode.

It is possible to maintain a constant speed after turning off one, two, or three channels under operation in a constant torque region.

In a practical layout, some differences between the channels were visible (Figure 16a–c). These differences are due to the differences between the voltages induced in the windings of the different channels. The induced voltages (BEMF) of the shut-down channels in the DCO of one phase are shown in Figure 18. These differences do not affect the reliability of the machine but, unfortunately, result in uneven loads on different channels.

**Figure 18.** The induced voltages: DCO A and B mode (**a**), DCO A and C mode (**b**).

The problem of operating in TCO, DCO, and SCO modes after the occurrence of a fault state (e.g., short-circuit in the winding) was initially analyzed. Under certain conditions, further motor operation is possible. This was confirmed by preliminary laboratory tests. This will be discussed in future publications.

#### **6. Conclusions**

Reliable operation is vitally important for critical drives. This article proposes a quad-channel design for a BLDC machine (QCBLDC) that allows operation with an independent supply from four inverter systems. During fault-free operation, this machine can work in one of the following two operating modes: quad-channel operation (QCO) or dual-channel operation (DCO). In the DCO mode, there are two possible configurations, which are not significantly different from each other with regards to their electrical parameters. The two variants analyzed in this article have very similar parameters. For mechanical reasons, it is more beneficial to supply channel windings staggered by 120 mechanical degrees (variant I). For technological reasons, it is easier to manufacture variant II (shorter winding connections). Compared to the QCO mode, the DCO mode for the same duty point is characterized by a slightly lower efficiency (greater copper losses) and only a slightly higher but balanced magnetic pull (A and C mode). In the case of the DCO A and C mode, the type of variant is not important. In the case of the DCO A and B variant II, a significant increase in vibrations and noise was observed. In the case of variant II of the TCO mode, there was also an insignificant (but noticeable) increase in the vibration and noise level compared to DCO A and C. The results of both the simulation tests and the laboratory tests confirm that the suggested design for the QCBLDC motor effectively works in both the QCO and DCO modes. The problem of operating in TCO, DCO, and SCO modes after the occurrence of a fault state, and the preparation of control algorithms that will facilitate the continued operation of a QCBLDC machine in such a situation, will be discussed in future publications.

**Author Contributions:** Conceptualization, J.P. and M.K.; methodology, J.P. and M.K.; software, M.K.; formal analysis, J.P. and M.K.; investigation, J.P.; resources, M.K.; writing—original draft preparation, M.K.; writing—review and editing, J.P. and P.B.; visualization, M.K.; supervision, J.P.; project administration, M.K.; funding acquisition, M.K.

**Funding:** This work is financed in part by the statutory funds of the Department of Electrodynamics and Electrical Machine Systems, Rzeszow University of Technology and in part by the Polish Ministry of Science and Higher Education under the program "Regional Initiative of Excellence" in 2019–2022. Project number 027/RID/2018/19, amount granted 11,999,900 PLN.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
