*Article* **Design and Implementation of a Speed-Loop-Periodic-Controller-Based Fault-Tolerant SPMSM Drive System**

#### **Tian-Hua Liu 1,\*, Muhammad Syahril Mubarok 1, Muhammad Ridwan 1,2,3 and Suwarno <sup>2</sup>**


Received: 1 June 2019; Accepted: 20 June 2019; Published: 20 September 2019

**Abstract:** This paper proposes a speed-loop periodic controller design for fault-tolerant surface permanent magnet synchronous motor (SPMSM) drive systems. Faulty conditions, including an open insulated-gate bipolar transistor (IGBT), a short-circuited IGBT, or a Hall-effect current sensor fault are investigated. The fault-tolerant SPMSM drive system using a speed-loop periodic controller has better performance than when using a speed-loop PI controller under normal or faulty conditions. The superiority of the proposed speed-loop-periodic-controller-based SPMSM drive system includes faster transient responses and better load disturbance responses. A detailed design of the speed-loop periodic controller and its related fault-tolerant method, including fault detection, diagnosis, isolation, and control are included. In addition, a current estimator is also proposed to estimate the stator current. When the Hall-effect current sensor is faulty, the estimated current is used to replace the current of the faulty sensor. A 32-bit digital signal processor, type TMS-320F-2808, is used to execute the fault-tolerant method and speed-loop periodic control. Measured experimental results validate the theoretical analysis. The proposed implementation of a fault-tolerant SPMSM drive system and speed-loop periodic controller design can be easily applied in industry due to its simplicity.

**Keywords:** periodic controller; surface permanent magnet synchronous motor; fault-tolerant system

#### **1. Introduction**

Electrical motors, including DC motors, induction motors, and permanent magnet synchronous motors, were used to for decades, enabling modern life. Electric motors and their related inverters are used to transform electric power into mechanical power. Electric motors are used in pumps, cranes, conveyors, mills, elevators, and transportation. The surface permanent magnet synchronous motor (SPMSM) became more popular due to its excellent characteristics: high-power density, high efficiency, and a simple control method [1]. The SPMSM is widely used in traction applications, including land and marine vehicles because the SPMSM does not require any brushes, and there is no slip frequency between its stator and rotor [1]. In addition, increased awareness of global warming and motivation to decrease carbon emissions further increased the attraction of electric vehicles, most of which are driven by SPMSMs, which have the benefits of high-power density, good dynamic responses, and simple control methods [2].

Failure of an SPMSM drive system can put drivers, operators, passengers, and people in the vicinity at risk of injury or even death. Failures can be divided into two main categories: motor faults and inverter faults. An inverter is far more fragile and more likely to suffer a fault than a motor due to its high PWM switching frequency, vulnerable power devices, and complicated control algorithm. Development of advanced fault-tolerant control methods is important to reduce the potential for accidents and huge financial losses incurred by them [3]. Research on advanced fault-tolerant control technology was successfully applied in motor drives, power supplies, transportation, and other industrial applications [4,5]. For example, Naidu et al. proposed fault-tolerant SPMSM drive topologies for automotive vehicles, which used X-by-wire systems to improve their safety, reliability, and performance [6]. Kontarcek et al. investigated a low-cost fault-tolerant SPMSM drive system for an open-phase fault in an SPMSM drive system, which was based on field orientation control. In addition, a prediction stator current for the next sampling interval and a new post-fault operation method of the SPMSM was investigated [7]. Jung et al. proposed a model reference adaptive technique-based diagnosis of an open-circuit fault. An observer was implemented to determine the faulty condition. Two major post-fault actions were discussed as well [8]. Cai et al. proposed a Bayesian network-based data-driven fault diagnosis methodology for three-phase inverters. Two output line-to-line voltages were measured to detect and diagnose faults, which could be used for multilevel inverter SPMSM drive systems [9]. Meinguet et al. used multiple fault indices to retrieve the most likely state of the AC drive systems. Based on the unbalance of the three-phase currents and instantaneous frequency, a fault-tolerant topology was derived [10]. Wang et al. proposed a fault-tolerant control for dual three-phase SPMSM drive systems under open-phase faults. The object of the research included two parts. The first part was to maximize the torque capability while protection was considered, and the second part was to minimize copper loss [11]. Tseng et al. proposed a fault-tolerant control for a dual-SPMSM drive system. Two simple methods, including a short-circuit fault-tolerant method and an open-circuit fault-tolerant method, were investigated. Experimental results showed that this dual-SPMSM drive system could maintain speed although one power device was open-circuited or short-circuited [12]. Wang et al. proposed a fault-tolerant control system of a parallel-voltage inverter-fed SPMSM drive system. Three fault-tolerant control strategies were proposed and compared. The proposed method not only provided smooth torque but also had less copper loss under open-circuit faults [13].

Recently, Nasiri et al. proposed a full digital current control of an SPMSM for vehicular applications. The objective of the control is to achieve a deadbeat dynamic response for the speed control of an SPMSM. The proposed method discussed a robust sensorless method; as a result, an encoder fault was allowed [14]. Bennett et al. investigated a fault-tolerant electric drive for an aircraft nose wheel steering actuator. The wheel steering actuator included two independent controllers. Each controller operated one-half of a dual three-phase SPMSM drive system. As a result, the other controller could control the aircraft nose when one controller failed [15]. Jeong proposed a fault detection and fault-tolerant control of the IPMSM drive system for electric vehicles. Once the fault was detected, the control scheme automatically reconfigured to provide post-fault operational capability [16]. Wang et al. implemented a fault-tolerant control with an active fault diagnosis for four-wheel independently driven electric ground vehicles. An adaptive control-based passive fault-tolerant controller was designed to ensure that the vehicle system was stable and tracked a desired vehicle motion when the in-wheel motor drive system failed [17]. Zhang et al. proposed an active fault-tolerant control for electric vehicles with independently driven rear in-wheel motors against actuator faults. After the fault was detected, a proper reconfigurable controller was switched on to achieve optimal post-fault performance [18]. Bolognan proposed remedial strategies against failures occurring in an inverter power device for an SPMSM drive system. Minimal redundant hardware was implemented [19]. Bai proposed a fault-tolerant control for a dual-winding SPMSM drive system based on the space vector pulse width modulation technique. The distribution of the space vector voltages was analyzed, and the vector control strategies under healthy and one-phase open-circuit faulty conditions were investigated to maintain the magnetomotive force of the SPMSM as a constant [20]. The papers mentioned above, however, only focused on the fault detection, diagnosis, and isolation [6–20]. None or only a few researchers focused on the controller design of fault-tolerant drive systems. When the SPMSM drive

system is operated in normal conditions, the three-phase currents are balanced. Thus, the torque pulsations are small. However, when the SPMSM drive system is operated in faulty conditions, the three-phase currents are seriously imbalanced, causing obvious torque pulsations. As a result, the drive system in a faulty condition is very difficult to control. To solve this challenge, this paper proposes a speed-loop periodic controller to improve the dynamic responses of the drive system under an open-circuit fault or short-circuit fault. To the authors' best knowledge, the ideas of this paper are original. No previously published papers covered this issue.

This paper proposes a speed-loop periodic controller to improve the transient responses and load disturbance responses for SPMSMs under normal and faulty conditions. This paper is divided into the following sections: firstly, a fault-tolerant inverter is presented. Secondly, the fault detection, diagnosis, isolation, and control methods are discussed. The methods use a back-up leg to replace the faulty leg in the inverter. After that, a speed-loop periodic controller and a current-loop PI controller are designed to improve the dynamic responses of the SPMSM drive system, including fast transient responses and good load disturbance responses. Next, the implementation of the drive system is discussed. Finally, several experimental results and conclusions are included.

#### **2. Fault-Tolerant SPMSM Drive System**

Failure of the SPMSM drive system can be categorized into two major types: motor failures and circuit failures. Motor failures includes bearing damage, open winding, and partially short-circuited winding. Circuit failures include inverter failure, current sensor failure, and encoder failure. The inverter is the most likely location of a fault and not the motor because, compared to the SPMSM, the inverter is more fragile and more likely to be open- or short-circuited. In addition, the current sensor and its circuit malfunction easily due to the offset voltage and aging of the circuit. As a result, a fault-tolerant control method is proposed here to use the estimated current to replace the measured current. This paper only focuses on the fault-tolerant method of the inverter and sensor and not the SPMSM. In this section, fault detection, diagnosis, isolation, and control of a fault-tolerant inverter are discussed first, and then fault detection, isolation, estimation, and control of a Hall-effect sensor are investigated.

#### *2.1. Fault Detection and Diagnosis of a Fault-Tolerant Inverter*

This research covers the situation when one power switch of the inverter is open- or short-circuited. The fault-tolerant inverter drive system is shown in Figure 1, which contains six IGBTs, *Sa*, *S <sup>a</sup>*, *Sb*, *S b* , *Sc*, and *S <sup>c</sup>*, and two back-up IGBTs, *St* and *S t* . At the output of the inverter, six TRIACs, including *Tat*, *Tbt*, *Tct*, *Ta f* , *Tb f* , and *Tc f* are added. In addition, three high-speed fuses *Fa*, *Fb*, and *Fc* are inserted into the inverter and a back-up leg, including *St* and *S t* , is added as well. This paper discusses an open-circuit fault and a short-circuit fault of one leg in the inverter.

**Figure 1.** Fault-tolerant inverter.

A performance index is established to identify if the SPMSM drive system failed [7]. During normal operation, the square magnitude error is calculated as follows:

$$
\varepsilon\_H(k) \cong \left(\frac{v\_d(k)}{L\_\text{s}} \Delta t\right)^2 + \left(\frac{v\_q(k) - \varepsilon\_q(k)}{L\_\text{s}} \Delta t\right)^2,\tag{1}
$$

where ε*n*(*k*) is the performance index under normal conditions. *vd*(*k*) is the d-axis voltage, *Ls* is the self-inductance, *vq*(*k*) is the *q*-axis voltage, *eq*(*k*) is the back-EMF, and Δ*t* is the time of each time interval. To avoid false detection, according to the authors' experiences, 10 times the normal error vector is an adequate threshold to determine the faulty condition. The performance index in the faulty condition, ε(*k*) can be expressed as

$$
\varepsilon(k) \succ 10 \text{ } \varepsilon\_n(k). \tag{2}
$$

In Equation (2), the performance index of the SPMSM in a faulty condition, ε(*k*), can be defined as

$$
\varepsilon(k) = \Delta \dot{i}\_d^2(k) + \Delta \dot{i}\_q^2(k), \tag{3}
$$

where ε(*k*) is the performance index in a faulty condition. Δ*id*(*k*) and Δ*iq*(*k*) are the current deviations in the *d*-axis and *q*-axis in a faulty condition. The DSP diagnoses the faulty condition by measuring the deviations of the *d*-axis and *q*-axis currents and then identifying whether the faulty condition occurred in either the *a*-phase, *b*-phase, or *c*-phase. The DSP transforms the *a*, *b*, *c* axis currents in the α–β axis currents, and then computes the current angle δ [21,22]. Taking the *a*-phase fault as an example, Figure 2a shows the *b*-phase and *c*-phase currents when the *a*-phase winding is open-circuited. The current can flow in either direction as shown in Figure 2b. The current may flow from the *b*-phase to the *c*-phase, which results in the current vector having a 270◦ angle, or the current may flow from the *c*-phase to the *b*-phase, which results in a 90◦ angle. The summarized results of the current angle δ when one phase is faulty are shown in Table 1. By computing the current angle δ, one can easily diagnose which phase is open. After that, an isolation and control method is executed to isolate the faulty part, and uses the back-up leg to replace the faulty leg. A fault-tolerant SPMSM drive system, thus, can be achieved.

**Figure 2.** Output current vector of *a*-phase fault: (**a**) three-phase winding; (**b**) current vector.

**Table 1.** Current angle at different faulty phases.


#### *2.2. Fault Detection and Control of a Current Sensor*

This paper also investigates the detection and control of a fault in a one-phase current sensor. Previous research used a current estimator to evaluate the current sensor error, and an adaptive threshold was used to detect and diagnose the faulty condition [23,24]. In a faulty condition, the estimated current replaces the faulty current. The α–β axis voltages and currents were obtained using a coordinate transformation, and the estimated α–β currents in the discrete time domain can be expressed as

$$
\hat{\mathbf{i}}\_a(k) = \hat{\mathbf{i}}\_a(k-1) + \frac{T\_s}{L\_s} [\upsilon\_a(k) - r\_s \dot{\mathbf{i}}\_a(k) - \omega\_\varepsilon(k) \lambda\_m \sin \theta\_\varepsilon(k)], \tag{4}
$$

and

$$
\hat{i}\_{\beta}(k) = \hat{i}\_{\beta}(k-1) + \frac{T\_s}{L\_s} [v\_{\beta}(k) - r\_s i\_{\beta}(k) + \omega\_c(k) \lambda\_m \cos \theta\_c(k)].\tag{5}
$$

where <sup>ˆ</sup>*i*α(*k*) and <sup>ˆ</sup>*i*β(*k*) are the estimated current, *<sup>v</sup>*α(*k*) and *<sup>v</sup>*β(*k*) are the <sup>α</sup>–<sup>β</sup> axis voltages, and *<sup>i</sup>*α(*k*) and *i*β(*k*) are α–β axis currents. ω*e*(*k*) and θ*e*(*k*) are the electrical speed and angle. The current waveform factor *Fx*(*k*) and *F*ˆ*x*(*k*) can be calculated as

$$F\_x(k) = \frac{|i\_x|\_{\text{RMS}}(k)}{|i\_x|\_{\text{AVG}}(k) + e'} \tag{6a}$$

and

$$\mathcal{F}\_x(k) = \frac{|i\_x + \varepsilon\_x|\_{\text{RMS}}(k)}{|i\_x + \varepsilon\_x|\_{\text{AVG}}(k) + e'} \tag{6b}$$

where *<sup>e</sup>* is a constant to prevent the denominator in the Equations (6a) and (6b) from reaching zero. *ix RMS*(*k*) is the absolute value of the measured RMS current, *ix AVG*(*k*) is the absolute value of the measured average current, and ε*x*(*k*) is the estimated error. The residual function *Rx*(*k*) is obtained by computing the difference between the estimated waveform factor *F*ˆ*x*(*k*) and the measured waveform factor *Fx*(*k*). It can be expressed as

$$R\_x(k) = \mathcal{F}\_x(k) - F\_x(k). \tag{7}$$

By substituting Equations (6a) and (6b) into Equation (7), the residual function *Rx*(*k*) in the equation can be transformed into

$$R\_x(k) = \frac{|i\_x + \varepsilon\_x|\_{\rm RMS}(k)}{|i\_x + \varepsilon\_x|\_{\rm AVG}(k) + \varepsilon} - \frac{|i\_x|\_{\rm RMS}(k)}{|i\_x|\_{\rm AVG}(k) + \varepsilon} = \frac{|i\_x + \varepsilon\_x|\_{\rm RMS}(k)}{|i\_x + \varepsilon\_x|\_{\rm AVG}(k) + \varepsilon} - \frac{|i\_x|\_{\rm RMS}(k)}{|i\_x|\_{\rm AVG}(k) + \varepsilon} \tag{8}$$

where ε*x*(*k*) is the estimated current error of the *a*, *b*, *c* phases. The numerator of the estimated absolute value of the RMS current *ix* <sup>+</sup> <sup>ε</sup>*<sup>x</sup> RMS*(*k*) is always lower than or equal to the total of *ix RMS*(*k*)+ ε*x RMS*(*k*) due to the triangular inequality rule. By using this relationship, the residual function *Rx*(*k*) can be rewritten as

$$R\_{\mathbf{x}}(k) \le \frac{|i\_{\rm x}|\_{\rm RMS}(k)}{|i\_{\rm x} + \varepsilon\_{\rm x}|\_{\rm AVG}(k) + \varepsilon} + \frac{|\varepsilon\_{\rm x}|\_{\rm RMS}(k)}{|i\_{\rm x} + \varepsilon\_{\rm x}|\_{\rm AVG}(k) + \varepsilon} - \frac{|i\_{\rm x}|\_{\rm RMS}(k)}{|i\_{\rm x}|\_{\rm AVG}(k) + \varepsilon}.\tag{9}$$

The difference between the threshold value and the residual value is used to determine if a faulty condition occurred. When the system is in a steady-state condition and the current sensor is in a normal condition, the measured and estimated waveform factors are constants and the residual value is near zero. On the other hand, when the current sensor is in a faulty condition, the residual value abruptly increases due to its large error. Finally, the estimated current ˆ*ix*(*k*) replaces the measured faulty current *ix*(*k*). However, in this paper, the estimated currents are near the measured currents only in steady-state conditions. The transient responses of the estimated currents are ignored to simplify the current estimating method.

#### **3. Speed-Loop Periodic Controller**

The speed-loop periodic controller for a fault-tolerant SPMSM in this paper is an original idea. The internal model principle states that perfect asymptotic tracking of persistent inputs can be attained by replicating the signal generator in a stable feedback loop [25]. The internal model of the inputs is a signal generator. Figure 3a shows the basic continuous s-domain structure of a periodic signal generator, which includes a delay device *e*−*sTO* and a positive feedback. According to Figure 3a and assuming *Q*(*s*) = 1, the transfer function of the periodic signal generator can be expressed as

$$G\_{rs}(s) = \frac{u\_{rc}(s)}{e(s)} = \frac{e^{-sT\_O}}{1 - e^{-sT\_O}},\tag{10}$$

where *Grs*(*s*) is the transfer function of the periodic signal generator, and *e*−*sTO* is a time-delay unit. From Equation (10), the periodic signal generator *Grs*(*s*) can be expanded as follows [26]:

$$G\_{\rm tr}(s) = \frac{e^{-sT\_O}}{1 - e^{-sT\_O}} = -\frac{1}{2} + \frac{1}{sT\_O} + \frac{1}{T\_O} \sum\_{n=1}^{\infty} \frac{2s}{s^2 + \left(n\alpha\_o\right)^2}.\tag{11}$$

In Equation (11), the first item is a transfer function of an impulse, the second item is a transfer function of a step input, and the third item is the transfer function of the harmonics. In the real world, a low-pass filter *Q*(*s*) is required to compensate for the related harmonics, and a phase-lead compensator *Gf*(*s*) is used for the entire system delay compensation. To simplify the problem, assuming *Q*(*s*) is 1, the classic periodic controller makes *Grs*(*s*) approach ∞ at poles s = ±*jn*ω*o*. In this research, a DSP is used to execute the control algorithm; as a result, the s-domain periodic signal generator needs to convert into the z-domain periodic signal generator shown in Figure 3b. The z-domain periodic signal generator is expressed in a discrete form as follows:

$$\mathcal{G}\_{\rm rs}(z) = k\_{\rm rc} \frac{Q(z)z^{-N}}{1 - Q(z)z^{-N}} \mathcal{G}\_f(z),\tag{12}$$

where *krc* is a constant control gain, *Q*(*z*) is a low-pass filter (LPF), *Gf*(*z*) is a phase-lead compensator that compensates for the time delay, and *N* is the number of delay steps.

In the discrete time domain, the *z*−*<sup>N</sup>* is added as shown in Figure 3b. *N* can be expressed as

$$N = \frac{T\_0}{T\_s} \tag{13}$$

where *T*<sup>0</sup> is the fundamental period and *Ts* is the sampling interval of the speed-loop control system. The fundamental period *T*<sup>0</sup> determines the delay of the periodic controller in *N* steps. The delay steps determine the settling time of the SPMSM drive system. If the delay time is set too short, the output generates obvious overshoot but has quick responses; however, if the delay time is too long, the periodic controller has slow responses. The choice of the parameter *N* depends on the designer's experience. In addition, the periodic controller is added into the speed-loop PI controller in the forward loop [26], which is shown in Figure 3c. The speed-loop PI controller is used to improve the transient responses and load disturbance responses for the normal operation speed dynamics; however, the speed-loop periodical controller is used when the SPMSM drive system is faulty, which causes three-phase current imbalance. In Figure 3c, *Gp*(*z*) is the transfer function of the SPMSM drive system, *Gc*(*z*) is the speed-loop PI controller, and *Grs*(*z*) is the periodic signal generator, which is used to reduce the current harmonics. After that, the speed command ω*<sup>r</sup>* ∗ is input into the closed-loop system. In this closed-loop system, a periodic signal output *urs*(*z*) is added to the speed error Δω*r*(*z*) to generate the total input of the PI controller to control the system.

**Figure 3.** Periodic controller in speed-loop: (**a**) s-domain periodic controller *Grs*(*s*); (**b**) z-domain periodic controller *Grs*(*z*); (**c**) closed-loop control system.

Compared to the traditional speed-loop PI controller, the proposed method uses a periodic controller to cascade to the traditional speed-loop PI controller, which increases the gain at certain frequencies. As a result, the transient responses and load disturbance responses of the SPMSM can be effectively improved. The computation of the periodic controller is very simple, which only includes a delay operation, a low-pass filter, a positive feedback operation, and phase-lead compensation. As a result, it is easy to implement the proposed control method by using a DSP.

The parameters of the periodic controller, including a control gain *krc* and a phase-lead compensation *Gf*(*z*), are determined by using stability analysis in the closed-loop control system. The detailed analysis and the stable condition of a closed-loop control system were previously discussed and can be expressed as follows [13]:

$$k\_{rc} < \frac{2\cos(\theta\_H + p\omega)}{\left|H(e^{j\omega})\right|} \text{ and } k\_{rc} \ge 0,\tag{14a}$$

and

$$H(z) = \frac{G\_c(z)G\_p(z)}{1 + G\_c(z)G\_p(z)},\tag{14b}$$

where θ*<sup>H</sup>* is the phase angle of *H(z)*, ω is the frequency, *p* is the order of the phase-lead compensation, and *H*(*z*) is the transfer function of the closed-loop control system. The control gain *krc* and the order *p* of the phase-lead compensation are determined as shown below. In the z-domain analysis, the phase-lead compensation *Gf*(*z*) is commonly expressed as follows [13]:

$$G\_f(z) = z^p \tag{15}$$

The characteristics of the closed-loop speed-control SPMSM drive system are discussed here. Figure 4 shows the relationship between the boundary of the phase angle and operating frequency of the closed-loop drive system. The phase lead step *p* includes steps 0, 1, 2, and 3, which are shown as *p* = 0, *p* = 1, *p* = 2, and *p* = 3 in Figure 4, respectively. In the physical system, the available range of the compensated phase is between <sup>−</sup>90o and 90o, which is shown as the dashed line in Figure 4. From Figure 4, to operate in the phase boundary between <sup>−</sup>90<sup>o</sup> and 90o, the maximum

operating frequency is 0.4 kHz for zero-step phase-lead compensation, 5 kHz for one-step phase-lead compensation, 3.3 kHz for two-step phase-lead compensation, and 1.9 kHz for three-step phase-lead compensation. In order to obtain the widest operating frequency of the closed-loop SPMSM drive system, the one-step phase lead (*p* = 1) is selected in this research. After that, the gain *krc* is chosen according to the stability analysis. The stability condition is shown in Equation (14a), which shows that the gain *krc* needs to satisfy the inequality equation. Figure 5 shows the relationship between the maximum boundary 2 cos(θ*H*+*p*ω) <sup>|</sup>*H*(*ej*ω)<sup>|</sup> and the operation frequency. In order to both satisfy Equation (14a) and obtain the widest operating frequency range, the one-step phase-lead compensation that provides a very smooth curve was chosen for this paper. By using the one-step phase-lead compensation and satisfying Equation (14a), *krc* was selected as 1.5 because 2 cos(θ*H*+*p*ω) <sup>|</sup>*H*(*ej*ω)<sup>|</sup> was varied between 1.5 and 150 when the operating frequency varied from 0 kHz to 5 kHz.

**Figure 4.** Compensated phase responses using a periodic controller.

**Figure 5.** Boundary of control gain under different compensated phases.

The low-pass filter, LPF *Q*(*z*), was designed by using finite impulse response (FIR). FIR was chosen here because it is commonly used in digital filter applications. The transfer function of an FIR LPF *Q*(*z*) can be expressed as

$$Q(z) = \sum\_{i=0}^{m} a\_i z^{-i}.\tag{16}$$

The components of the periodic controller are shown in Figure 6. The speed error Δω*e*(*z*) is multiplied with a control gain *krc*, and then added to the *z*<sup>−</sup>*purs*(*z*) to generate *es*(*z*). A low-pass filter, *a*<sup>0</sup> + *a*1*z*−<sup>1</sup> + *a*2*z*<sup>−</sup>2, is used to reduce the high-frequency noise. After that, the output of the low-pass filter, which is *z*<sup>−</sup>*purs*(*z*), is added to the *krc*Δω*r*(*k*) to obtain *es*(*z*). Finally, *urs*(*z* − *p*) passes through the phase-lead compensator *zp* to obtain the *urs*(*z*). By using *k* as the interval step number, the output before delay is expressed as *urs*(*z*)*z*−*p*, and then the system error *es*(*z*) can be transformed into

$$
\omega\_s(k) = k\_{\rm r5} \Delta \omega\_{\rm r}(k) + u\_{\rm r5}(k - p). \tag{17}
$$

By using the LPF *Q*(*z*) with *ai* as the coefficient, the *urc*(*k*) can be expressed as

$$u\_{rs}(k) = \sum\_{i=0}^{m} a\_i e\_s(k - N - i + p), \ i = 0, 1, 2. \tag{18}$$

The total of *urs*(*k*) and Δω*r*(*z*) becomes the control input of the PI controller.

**Figure 6.** The proposed periodic controller.

#### **4. Current-Loop Controller**

In general, the current-loop PI controller, which is a minor-loop of the SPMSM drive system, is cascaded with the speed-loop controller. Figure 7 shows the detailed block diagram of the speed-loop controller and current-loop controller in an SPMSM drive system. First, the speed ω*<sup>r</sup>* is subtracted from the speed command ω∗ *<sup>r</sup>* to obtain the speed error Δω*m*. Then, the speed-loop controller is executed to generate the *q*-axis current command *i* ∗ *<sup>q</sup>*. The *d*-axis current command is set at zero in this research. Next, two PI controllers are designed to compute the *d*-axis voltage command *v*∗ *<sup>d</sup>* from the *d*-axis current error, and also the *q*-axis voltage command *v*∗ *<sup>q</sup>* from the *q*-axis current error. After that, the SVPWM inverter generates *a*-, *b*-, *c*-axis voltages *va*, *vb*, and *vc* from the information of the *v*<sup>∗</sup> *d* , *v*∗ *<sup>q</sup>*, and electrical rotor position θ*e*. The *a*, *b*, *c* voltages are injected into the SPMSM to generate the *a*, *b*, *c* currents *ia*, *ib*, and *ic*. Finally, the SPMSM rotates and reports its mechanical angle θ*<sup>m</sup>* to the DSP.

**Figure 7.** Detailed block diagram of speed-loop and current-loop PI controllers in an SPMSM drive system.

The SPMSM drive system returns the signals from the encoder and two Hall-effect current sensors to the DSP. The encoder detects the rotor angle θ*m*, and then computes the electrical angle θ*<sup>e</sup>* by multiplying pole pairs. The rotor speed ω*<sup>r</sup>* is obtained by taking the difference operation from the θ*r*. Two Hall-effect current sensors are used to measure the *a*-phase and *b*-phase currents *ia* and *ib*, and then

the *c*-phase current *ic* can be calculated because it is a three-phase balanced system. The relationship between the *a*, *b*, *c* currents and the *d*-, *q*-axis currents is shown below.

$$
\begin{bmatrix} i\_d \\ i\_q \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos(\theta\_\varepsilon) & \cos(\theta\_\varepsilon - \frac{2\pi}{3}) & \cos(\theta\_\varepsilon + \frac{2\pi}{3}) \\\sin(\theta\_\varepsilon) & \sin(\theta\_\varepsilon - \frac{2\pi}{3}) & \sin(\theta\_\varepsilon + \frac{2\pi}{3}) \end{bmatrix} \begin{bmatrix} i\_d \\ i\_b \\ i\_c \end{bmatrix} . \tag{19}
$$

In the *d*-, *q*-axis synchronous frame, the dynamic equation of currents for an SPMSM is expressed as

$$\frac{d\dot{u}\_d(t)}{dt} = \frac{1}{L\_\text{g}} (\upsilon\_d(t) - r\_s \dot{u}\_d(t) + \alpha\_d(t) L\_\text{s} \dot{u}\_q(t)),\tag{20}$$

$$\frac{d\dot{\imath}\_{\emptyset}(t)}{dt} = \frac{1}{L\_{\text{s}}} (v\_{\emptyset}(t) - r\_{\text{s}}\dot{\imath}\_{\emptyset}(t) - \omega\_{\text{c}}(t)L\_{\text{s}}\dot{\imath}\_{\text{d}}(t) - \omega\_{\text{c}}(t)\lambda\_{\text{m}}).\tag{21}$$

The dynamic equation of the speed is

$$\frac{d\omega\_c(t)}{dt} = \frac{1}{f\_m}(T\_c - B\_m\omega\_c(t) - T\_L),\tag{22}$$

and the electromagnetic torque is

$$T\_{\mathfrak{c}} = \frac{3}{2} \frac{P}{2} \Big(\lambda\_{\mathfrak{m}} i\_{\mathfrak{q}}(t)\Big),\tag{23}$$

where *<sup>d</sup> dt* is the differential operator, *Ls* is the stator inductance, *rs* is the stator resistance, λ*<sup>m</sup>* is the flux linkage, *Jm* is the inertia, *Bm* is the friction coefficient, and *TL* is the external load. Assuming the resistance voltage is neglected and the decoupling forward method is used, then the *d*-, *q*-axis voltage *v*∗ *<sup>d</sup>* and *v*<sup>∗</sup> *<sup>q</sup>* can be expressed as

$$\frac{1}{L\_{\rm s}} \upsilon\_{d}^{\*} = \frac{1}{L\_{\rm s}} (\upsilon\_{d} + \omega\_{\rm c} L\_{\rm s} i\_{q}) \, \tag{24}$$

and

$$\frac{1}{L\_s} \upsilon\_q^\* = \frac{1}{L\_s} (\upsilon\_q - \alpha \iota\_c L\_s i\_d - \alpha \iota\_c \lambda\_m). \tag{25}$$

Substituting Equations (24) and (25) into Equations (20) and (21), the dynamics of the SPMSM can be rewritten as

$$\frac{di\_d}{dt} = -\frac{r\_s}{L\_s}i\_d + \frac{1}{L\_s}v\_{d'}^\* \tag{26}$$

and

$$\frac{d\dot{i}\_q}{dt} = -\frac{r\_s}{L\_s}\dot{i}\_q + \frac{1}{L\_s}v\_q^\*.\tag{27}$$

After using the current-loop PI controllers, the *d*-, *q*-axis voltage commands, *v*∗ *<sup>d</sup>* and *v*<sup>∗</sup> *<sup>q</sup>*, are expressed as

$$w\_d^\*(t) = K\_\mathbb{P} \{ \mathbf{i}\_d^\*(t) - \mathbf{i}\_d(t) \} + K\_I \int\_0^t \{ \mathbf{i}\_d^\*(\tau) - \mathbf{i}\_d(\tau) \} d\tau,\tag{28}$$

and

$$w\_q^\*(t) = K\_P \left( i\_q^\*(t) - i\_q(t) \right) + K\_I \int\_0^t \left( i\_q^\*(\tau) - i\_q(\tau) \right) d\tau. \tag{29}$$

The *d*-axis voltage is obtained by substituting Equation (28) into Equation (24), and the *q*-axis voltage is obtained by substituting Equation (29) into Equation (25). Finally, the output voltages can be expressed as

$$w\_d(t) = K\_P \{ i\_d^\*(t) - i\_d(t) \} + K\_I \int\_0^t \left( i\_d^\*(\tau) - i\_d(\tau) \right) d\tau - \alpha\_\ell(t) L\_\circ i\_q(t),\tag{30}$$

and

$$w\_q(t) = Kp\left(i\_q^\*(t) - i\_\emptyset(t)\right) + K\_l \int\_0^t \left(i\_q^\*(\tau) - i\_\emptyset(\tau)\right)d\tau + \omega\_\varepsilon(t)L\_\delta i\_d + \omega\_\varepsilon(t)\lambda\_{\text{W.}} \tag{31}$$

After transferring the continuous time domain into discrete time domain, one can obtain the *d*-, *q*-axis voltage commands as

$$w\_d(k) = K\_P \left( i\_d^\*(k) - i\_d(k) \right) + K\_I T\_c \sum\_{n=1}^k \left( i\_d^\*(k) - i\_d(k) \right) - \omega\_\ell(k) L\_\varepsilon i\_\emptyset(k),\tag{32}$$

and

$$w\_q(k) = K\_P \left( i\_q^\*(k) - i\_q(k) \right) + K\_I T\_\varepsilon \sum\_{n=1}^k \left( i\_q^\*(k) - i\_q(k) \right) + \omega\_\varepsilon(k) L\_s i\_d(k) + \omega\_\varepsilon(k) \lambda\_{m\prime} \tag{33}$$

where *Tc* is the sampling interval of the current loop. From Equations (32) and (33), a block diagram of the PI current-loop controller can be constructed as shown in Figure 8. In this research, the parameters of the PI controller were obtained by using the pole assignment technique.

**Figure 8.** Current-loop PI controller.

#### **5. Implementation**

A block diagram of the implemented SPMSM drive system is shown in Figure 9a. A DSP type TMS320F2808 was used as the control center. The SPMSM drive system includes a fault-tolerant inverter, a DSP, gate drivers, current-sensing circuits, an encoder circuit, and an overcurrent protection circuit. The speed-loop PI controller includes *Kp* = 0.5 and *KI* = 0.2, which are obtained by pole assignment with two poles, *p*<sup>1</sup> = 0.79 and *p*<sup>2</sup> = 0.93. The speed-loop periodic controller includes *krc* = 1.5, *Q*(*z*) = 0.2 + 0.45*z*−<sup>1</sup> + 0.2*z*<sup>−</sup>2, *N* = 50, and *Gf*(*z*) = *z*. The sampling interval of the speed loop is 1 ms. The current-loop PI controllers include *Kp* = 12.17 and *KI* = 0.0006, which determine the inner-loop current dynamics. The sampling interval of the current loop is 100 μs.

The SPMSM has the following parameters: *rs* = 0.73 Ω, *Ls* = 1.37 mH, λ*<sup>m</sup>* = 0.167 Wb, *Bm* = 0.003 N·m·s/rad, and *KT* = 1.0 N·m/A. Figure 9b shows a photograph of the implemented drive system, which includes an SPMSM and a dynamometer, which provides the external load for the SPMSM drive system.

(b)

**Figure 9.** The implemented system (**a**) block diagram, and (**b**) photograph.

#### **6. Simulated and Experimental Results**

The simulated and experimental results were measured under the following five conditions: a normal condition, an open-circuit condition, a short-circuit condition, a faulty current sensor condition, and a multiple faulty condition. The details are given below.

#### *6.1. Normal Condition Experimental Results*

Figure 10a shows the measured speed responses at 100 r/min, 300 r/min, and 500 r/min. The periodic controller has quicker transient responses than the PI controller. Figure 10b shows the measured *q*-axis currents. The periodic controller provides greater input power when compared to the PI controller. Figure 11a shows the speed responses at 500 r/min when an external load of 3.5 N·m was added. The periodic controller provides a lower speed drop and quicker recovery time than the PI controller. Figure 11b shows the *q*-axis current responses in the same case. The periodic controller shows better performance than the PI controller, including a lower overshoot and quicker recovery time when an external load is added.

(**a**)

**Figure 10.** Measured speeds at 100 r/min, 300 r/min, and 500 r/min: (**a**) speed responses; (**b**) *q*-axis currents.

**Figure 11.** Measured results at 500 rpm and 3.5 N·m load: (**a**) speed responses; (**b**) *q*-axis currents.

#### *6.2. Inverter Open-Circuit Faulty Condition Experimental Results*

Figure 12a–c show the simulated results of the *a*-phase open circuit at 300 r/min without using the fault-tolerant method. The simulated results include the three-phase currents, q-axis current, and speed. Figure 13a shows the measured three-phase currents without using the proposed fault-tolerant method of the SPMSM drive system when its *a*-phase upper leg was open-circuited at 300 r/min. The faulty condition occurs at 0.15 s. A manual switch was in series with the power device. When the switch was opened, the power device was instantaneously opened. Thus, the PMSM drive system became a three-phase unbalanced drive system. Figure 13b shows the *q*-axis current response when the *a*-phase upper leg is open. The *q*-axis current oscillated due to the unbalanced three-phase currents. Figure 13c shows the measured speed response. As we can observe, in this figure, the speed varied between 485 r/min to 510 r/min when the *a*-phase upper leg was open. Figure 14a–c show the simulated results using the fault-tolerant control when the *a*-phase leg was open-circuited. The *d*-axis inductance remained the same as its nominal value, but the *q*-axis inductance was reduced to 50% of its nominal value due to the influence of saturation. The simulated results include the currents of the speed-loop PI controller, the currents of the speed-loop periodic controller, and the speed responses of speed-loop PI and periodic controllers. Figure 15a–c show the measured results of fault-tolerant control at 300 r/min when the *a*-phase was open-circuited. The periodic controller had better performance than

the PI controller, including lower peak current and smaller speed variations during faulty intervals. Figure 15a shows the three-phase currents using the fault-tolerant method. Figure 15b shows the measured currents of the speed-loop periodic controller. The measured speed responses of both the PI controller and periodic controller are shown in Figure 15c. Figure 16a–c show the measured results of the fault-tolerant control when one switch of the *a*-phase leg was open-circuit at 1500 r/min.

**Figure 12.** Simulated results of the *a*-phase open-circuited without using the fault-tolerant method: (**a**) three-phase currents; (**b**) *q*-axis current; (**c**) speed.

**Figure 13.** Measured results of the *a*-phase open-circuited without using the fault-tolerant method: (**a**) three-phase currents; (**b**) *q*-axis current; (**c**) speed.

**Figure 14.** Simulated results of the fault-tolerant control when the *a*-phase was open-circuited in *d*–*q* inductance asymmetry conditions: (**a**) currents of the speed-loop PI controller; (**b**) currents of the speed-loop periodic controller; (**c**) speed responses.

**Figure 15.** Measured results of the fault-tolerant control when the *a*-phase was open-circuited: (**a**) currents of the speed-loop PI controller; (**b**) currents of the speed-loop periodic controller; (**c**) speed responses.

**Figure 16.** Measured results at 1500 r/min of the fault-tolerant control when the *a*-phase was open-circuited: (**a**) currents of the PI controller; (**b**) currents of the periodic controller; (**c**) speed-responses.

#### *6.3. Inverter Short-Circuit Faulty Condition Experimental Results*

Figure 17a–c show the simulated results of the *a*-phase short-circuited at 300 r/min without using the fault-tolerant method, including the three-phase currents, *q*-axis current, and speed response. Figure 18a–c show the measured results of the same case. Figure 18a shows the measured three-phase currents without using the fault-tolerant method. Figure 18b shows the measured *q*-axis current. Figure 18c shows the measured speed response that dropped quickly due to the trip of the inverter. Figure 19a–c show the simulated results of the *a*-phase short-circuited at 300 r/min using the fault-tolerant method. The simulated results include the current responses, speed responses, and performance index. Figure 20a–c show the measured results of the fault-tolerant control when the *a*-phase inverter was short-circuited at 300 r/min. The two power devices in the upper leg and lower leg were both turned on to have this leg short-circuited. Figure 20a shows the measured current responses when using a PI controller. Figure 20b shows that the measured speed variation was 80 r/min when using a speed PI controller, but it was 50 r/min when using the speed-loop periodic controller. These results show that the periodic controller has better transient response than the PI controller. Figure 20c shows the performance index before and after the fault. Yan et al. proposed a PWM voltage source inverter diagnosis method for a PMSM drive system based on a fuzzy logic approach [27]. By using the fuzzy logic diagnosis method, the DSP could identify the faulty condition in 0.09 s after the fault occurrence. Compared to Yan's method, in this paper, from Figure 20b, the short-circuit fault-tolerant control was finished in 0.01 s. As a result, this work reduced the time by approximately 89% when compared to Yan's method. The reason is that fuzzy logic is more complicated than the method proposed in this paper. Hang et al. proposed the detection and discrimination of an open-phase fault in an SPMSM drive system based on the zero-sequence voltage components [28]. For one switch open, Hang's proposed detection and discrimination method required 0.04 s. Compared to Hang's method, in this paper, from Figure 15c, the open-circuit fault-tolerant control was finished in 0.006 s. As a result, this work reduced the time by approximately 85% when compared to Hang's method. However, this paper may cause more conduction loss because six TRIACs were used to change the structure of the inverter. Figure 21a–c show the measured results of the short-circuit fault-tolerant control at one switch of the *a*-phase leg at 1500 r/min.

**Figure 17.** *Cont.*

**Figure 17.** Simulated results when the *a*-phase was short-circuited without using the fault-tolerant method: (**a**) three-phase currents; (**b**) *q*-axis current; (**c**) speed response.

**Figure 18.** *Cont.*

**Figure 18.** Measured results when the *a*-phase was short-circuited without using the fault-tolerant method: (**a**) three-phase currents; (**b**) *q*-axis current; (**c**) speed response.

(a)

**Figure 19.** *Cont.*

**Figure 19.** Simulated results of the fault-tolerant control when the *a*-phase was short-circuited: (**a**) currents of PI controller; (**b**) speed responses; (**c**) performance index.

**Figure 20.** *Cont.*

**Figure 20.** Measured results of the fault-tolerant control when the *a*-phase was short-circuited: (**a**) currents of the PI controller; (**b**) speed responses; (**c**) performance indexes.

**Figure 21.** *Cont.*

**Figure 21.** Measured results at 1500 r/min of the fault-tolerant control when the *a*-phase was short-circuited: (**a**) currents of the PI controller; (**b**) currents of the periodic controller; (**c**) speed responses.

#### *6.4. Current Sensor Faulty Condition Experimental Results*

In addition, when the *a*-phase Hall-effect current sensor is open, the *a*-phase measured current suddenly becomes zero. Then, the estimated current is used to replace the measured current. In the experiment, a manual switch was connected with the current-sensing circuit. When the switch was opened, the phase current became zero, resulting in a one-phase current fault. Figure 22a shows the measured *a*-phase current and its estimated current in normal operating conditions. As we can see in this figure, they were very close. Figure 22b shows the residual and adaptive threshold. The residual was always below its adaptive threshold because the system was operated in normal conditions. Figure 23a,b show the *b*-phase measured current and its estimated current when the *b*-phase current sensor was faulty at 0.15 s. The estimated *b*-phase current replaced the measured *b*-phase current at 0.154 s. Figure 24a–c show the measured three-phase currents when the *a*-phase current sensor was faulty. Figure 24a shows the measured three-phase currents using the PI controller. Figure 24b shows the measured three-phase current using the speed-loop periodic controller. Again, the periodic controller performed better than the PI controller. Figure 24c shows the measured speed responses using the speed-loop periodic controller and the speed-loop PI controller. The speed-loop periodic controller once again performed better than the speed-loop PI controller.

**Figure 22.** Measured results of the *a*-phase in normal operating conditions: (**a**) measured and estimated currents; (**b**) residual and adaptive threshold.

**Figure 23.** *Cont.*

**Figure 23.** Measured results of the *b*-phase when the *b*-phase current sensor was faulty: (**a**) measured and estimated current; (**b**) residual and adaptive threshold.

**Figure 24.** *Cont.*

**Figure 24.** Measured three-phase currents of the fault-tolerant control when the *a*-phase current sensor was faulty: (**a**) current using PI controller; (**b**) current using periodic controller; (**c**) speed.

#### *6.5. Multiple Faulty Conditions Experimental Results*

Figure 25a,b show the simulated multiple faults when the *a*-phase leg was open-circuited and the *a*-phase current sensor was faulty using the periodic speed-loop controller. Figure 25a shows the simulated currents, and Figure 25b shows the simulated speed.

**Figure 25.** *Cont.*

**Figure 25.** Simulated multiple faults when the *a*-phase leg was open-circuited and the *a*-phase current sensor was faulty using the periodic speed-loop controller: (**a**) current; (**b**) speed.

(b)

The proposed method required more computation time for a DSP. In addition, the proposed method also added two IGBTs for the back-up leg, and six TRIACs for changing the structure of the inverter. As a result, the proposed drive system required a higher cost, and generated more conduction losses. In addition, the proposed method required more CPU computation time. These were considered the overheads of the process. According to the experimental results, there were no faulty cases that the proposed design failed to detect. All faulty cases were successfully detected and controlled.

#### **7. Conclusions**

In this paper, the design of a speed-loop periodic controller for a fault-tolerant SPMSM drive system was investigated and discussed. A 32-bit DSP, TMS-320F-2808, was used to execute the speed-loop periodic controller and fault-tolerant algorithm. The detailed design procedures of the speed-loop periodic controller design were presented. The experimental results showed that the proposed periodic speed-loop controller provided better performance, including faster transient responses and better load disturbance responses, than the speed-loop PI controller under normal operating conditions and faulty conditions. The experimental results validated the theoretical analysis. The proposed method can be applied in industry due to its simplicity. This paper only focused on the faulty conditions that were clearly open- or short-circuited. Unclear faulty conditions, including resistance changing, noise interruption, overheating, and current or voltage derating of the IGBT, will be discussed in future research.

**Author Contributions:** Conceptualization, T.-H.L.; methodology, T.-H.L., M.S.M. and M.R.; software, M.S.M. and M.R.; hardware, M.R.; data curation, M.S.M. and M.R.; writing—review and editing, T.-H.L., M.S.M.; funding acquisition, T.-H.L.; supervision, T.-H.L. and S.

**Funding:** This research was funded by the Ministry of Science and Technology, Taiwan, under Grant MOST-105-2221-E-011-095-MY2.

**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/).

## *Article* **Performance Analysis of Synchronous Reluctance Motor with Limited Amount of Permanent Magnet**

**Duc-Kien Ngo <sup>1</sup> and Min-Fu Hsieh 2,\***


**\*** Correspondence: mfhsieh@mail.ncku.edu.tw; Tel.: +866-06-275-7575 (ext. 62366)

Received: 11 August 2019; Accepted: 9 September 2019; Published: 11 September 2019

**Abstract:** This paper analyzes the performance of a synchronous reluctance motor (SynRM) equipped with a limited amount of a permanent magnet (PM). This is conventionally implemented by inserting PMs in rotor flux barriers, and this is often called the PM-assisted SynRM (PMa-SynRM). However, common PMa-SynRMs could be vulnerable to irreversible demagnetization. Therefore, motor performance and PM demagnetization should be simultaneously considered, and this would require the PM to be properly arranged. In this paper, various rotor configurations are carefully studied and compared in order to maximize the motor performance, avoid irreversible demagnetization and achieve higher PM utilization. Moreover, the field weakening capability is investigated and improved by regulating armature excitation. A particular rotor type with flux intensification was found to possess higher PM utilization, lower demagnetization possibility with fairly high performance. Thus, suitable rotor configurations are recommended for certain applications.

**Keywords:** SynRM; irreversible demagnetization; PMa-SynRM; flux intensifying

#### **1. Introduction**

The synchronous reluctance motor (SynRM), with its robustness, high overload capability and low cost, has become a popular research target for many years [1–4]. However, the relatively lower torque/power density and power factor are the inherent disadvantages of SynRMs [5–7] compared to a permanent magnet synchronous machine (PMSM). To overcome such weaknesses, a permanent magnet (PM) can be inserted into the rotor of the SynRMs with a modest volume, which leads to the birth of a type of motor called the permanent magnet assisted synchronous reluctance motor (PMa-SynRM) [8–11]. With the increasing number of related research works, the PMa-SynRM has become a popular choice in some applications and can be an alternative to a SynRM or PMSM [12–15].

Generally, the PMs inserted inside the rotor flux barriers produce a negative flux linkage along the *q*-axis. The *q*-axis inductance *Lq* is usually low due to the multiple flux barriers. The permanent magnet (PM) flux linkage (the flux linkage due to PM solely) promotes the rotation of the flux linkage vector, and therefore the voltage vector goes close to the current vector to increase the power factor [9]. The PM flux linkage also contributes to torque production so that the total torque increases. However, the volume/size of the added PM needs to be limited to avoid the motor becoming an interior permanent magnet synchronous motor (IPMSM) [16], which could also increase the cost. Nevertheless, the volume/size of the PM should not be too small to achieve the desired torque and power density, or to be vulnerable to irreversible demagnetization [10].

For SynRMs or PMa-SynRMs, various design possibilities can be considered, e.g., the number of flux barriers, with or without PMs, rare earth or other types of PM materials or the amount of PM employed. In an effort to standardize the design process of SynRMs and PMa-SynRMs, Bianchi et al. [11] proposed a series of steps that are synthesized from some example studies [11,17–20], where the inward PMs (near the rotor shaft) are larger than the outward ones to improve flux flows and avoid demagnetization. This PM arrangement is considered as a common trend for the PMa-SynRM rotor design.

Some motor designs presented in previous research [10,11,21–23] using either rare-earth or ferrite PMs are summarized in Table 1, including their ratio of PM-to-motor volume and torque density. It can be observed that the variety of rotor designs and PM arrangements is rich in these motors. However, the first four motors listed in Table 1 [10,11,21,22] employ a relatively large PM volume compared to the motor studied in Reference [23]. Furthermore, the PM size in the motor in Reference [23] is purposefully made identical for all the PM layers to reduce the manufacturing cost, which is different from common designs. The multiple flux barrier design allows the torque density of this motor to reach 28.1 Nm/L with PM taking only 0.95% of the motor volume by assuming sufficient cooling is applied, as shown in Table 1. However, as mentioned in Reference [16], the low PM volume and high excitation current could lead to its negligible contribution in torque production due to the low PM-torque-to-total-torque ratio and high probability of irreversible demagnetization with field-weakening applied. In addition, since the armature current *Is* is far from the characteristic current *Ich* on *d*-axis [24], the constant power speed range (CPSR) could become relatively low for PMa-SynRM with a little amount of PM. From the above discussions, it is necessary to propose a solution to improve the performance of this type of motor in terms of PM utilization, demagnetization resistivity and field weakening capability.


**Table 1.** The reference motor parameters and torque production.

Therefore, in this paper, several motor models based on a prototyped PMa-SynRM [23] with various PM arrangements using limited amount of PM are analyzed in detail. The analysis concentrates on the effect of the PM position on the magnetic distribution, inductances, torque production, torque/power-speed curves and magnetization characteristics. The armature current is also adjusted for observation on the correlation between the electrical and magnetic parameters affecting the motor performance. From the above analysis, this paper aims to achieve a high PM utilization rate to produce torque though a limited amount of PMs in a more efficient way. Demagnetization can also be avoided under high performance operations. The analysis was conducted using finite element analysis (FEA), which has been partially validated using previous experimental studies [23]. Note that differing from Reference [23], where the evaluation was only conducted for a fixed rotor structure, this paper makes a complete analysis with a sufficient number of models in order to make proper suggestions for the improvement of SynRM performance. In Reference [25], the PM volume was optimized for predetermined field-intensified PM machines. Here, in the present study, the models investigated cover not only conventional PMa-SynRMs, but also the novel flux-intensifying PMa-SynRMs [26]. In what follows, the terms flux-intensification, flux-intensifying and flied-intensified are all abbreviated as FI. In addition, by investigating over an existing prototype, the analysis can be better convincing. The comparisons can also be made to highlight the novelty of the current analysis.

This paper is organized as follows. The mathematical model and the configuration of the motor models are presented first in Section 2. Then, the investigation for the influence of PM positions on motor characteristics is carried out in Section 3, followed by the comparison of some motor models in Section 4. Section 5 presents the discussions over these investigated models. Finally, the paper is concluded by making suggestions for the design of such motors in Section 6.

#### **2. Mathematical Model and Configuration of Investigated Motors**

#### *2.1. Mathematical Modeling of Investigated Motors*

A conventional SynRM with a limited amount of PM embedded along the flux barriers, i.e., facing the physical *q*-axis, is called the first PM arrangement (hereafter denoted Type 1), as shown in Figure 1a. In contrast, when PM is added crossing the flux barriers, i.e., facing the *d*-axis, it is called the second PM arrangement (hereafter denoted Type 2), as the example illustrated in Figure 1b. For the Type 1 rotor, the flux linkage produced by the PM is arranged against the *q*-axis armature flux linkage, while for the Type 2 rotor, the PM flux linkage complements the *d*-axis armature flux linkage. The Type 2 motor can thus be called the flux-intensifying PMa-SynRM (FI-PMa-SynRM) [26].

**Figure 1.** Rotor configurations: (**a**) The first permanent magnet (PM) arrangement (Type 1); (**b**) The second PM arrangement (Type 2).

Assuming that the iron saturation and the cross-coupling effect are neglected, the stator dynamic voltage equations for synchronous machines in the *d-q* frame [27,28] can be expressed as:

$$
\sigma\_d = R\_s i\_d + \frac{d\lambda\_d}{dt} - a\lambda\_q \tag{1}
$$

$$
\sigma\_q = R\_s i\_q + \frac{d\lambda\_q}{dt} + a\lambda\_d \tag{2}
$$

where the subscripts *d* and *q* represent the *d*- and *q*-axis, respectively, *id* and *iq* are the currents, λ*<sup>d</sup>* and λ*<sup>d</sup>* are the flux linkages, *Rs* is the phase resistance, and ω is the electrical angular speed.

Equations (1) and (2) are general voltage equations for synchronous machines. To be applied to the two types of motors mentioned above, the flux linkages λ*<sup>d</sup>* and λ*<sup>q</sup>* in (1) and (2) need to be further discussed since the PMs are arranged differently in these two types of motors. For the Type 1 motors, as previously mentioned, the PMs are arranged in *q*-axis against the stator flux due to the *q*-axis current (*id*), and therefore the flux linkages in the *d*-*q* frame can be expressed as:

$$
\lambda\_d = L\_d i\_d, \lambda\_q = L\_q i\_q - \lambda\_m \tag{3}
$$

where *Ld* and *Lq* are the stator inductances in the *d*-*q* frame, and λ*<sup>m</sup>* is the PM flux linkage. Note that *Ld* and *Lq* do not take into account the PM flux linkage but only the flux linkage produced by *id* and *iq*. For the Type 2 motors, the PMs are placed in the *d*-axis to complement the stator flux and thus the flux linkages are given as:

$$
\lambda\_d = L\_d i\_d + \lambda\_{m\prime} \lambda\_q = L\_q i\_q \tag{4}
$$

From (3) and (4), the flux-weakening nature for the Type 1 motors and the flux-intensifying characteristics for the Type 2 can be clearly observed.

Figure 2 presents the equivalent circuits for the Type 1 [27] and Type 2 motors. As can be seen, the two types of motors have a difference in PM flux linkages. The phasor diagrams for Type 1 [9] and Type 2 are illustrated in Figure 3a,b, respectively, with the winding resistance being neglected [9,29]. Therefore, the voltage equations can be further expressed as:

**Figure 2.** Equivalent circuits: (**a**) Type 1; (**b**) Type 2.

**Figure 3.** Phasor diagrams: (**a**) Type 1; (**b**) Type 2.

$$V = \omega \sqrt{\left(L\_d I\_d\right)^2 + \left(L\_q I\_q - \lambda\_m\right)^2} \tag{5}$$

for Type 1 [30], and:

$$V = \omega \sqrt{\left(L\_d I\_d + \lambda\_m\right)^2 + \left(L\_q I\_q\right)^2} \tag{6}$$

for Type 2.

Note that the "-" sign in front of λ*<sup>m</sup>* in (3) and (5) indicates that the direction of this quantity is opposing *LqIq*, differing from the definition in [30].

The torque equations for Type 1 and Type 2 are respectively expressed as:

$$T = \frac{3N\_m}{4} \left[ \lambda\_m I\_d + (L\_d - L\_q) I\_d I\_q \right] \tag{7}$$

for Type 1 [29], and

$$T = \frac{3N\_{\rm tr}}{4} \left[ \lambda\_m I\_q + (L\_d - L\_q) I\_d I\_q \right] \tag{8}$$

for Type 2 [26].

Figure 4 illustrates the circle diagrams of these two types of motors. For the Type 1 motor with the first PM arrangement shown in Figures 1a, 3a and 4a indicate that the flux linkage generated by the *q*-axis current could cause the PM to be irreversibly demagnetized, especially for the thin PM. In contrast, for the Type 2 motor with the second PM arrangement shown in Figure 1b, the demagnetization on the PM can be avoided during maximum torque per ampere (MTPA) operation but may be possibly locally demagnetized during field weakening operation (this can be avoided by careful design), as shown in Figures 3b and 4b. This configuration may be subject to a lower power factor at low speed, but for medium and high-speed operations, the current phase advance would improve the power factor. On the other hand, inserting PM in the *d*-axis (flux paths) can decrease *Ld* and then decrease the reluctance torque so that the Type 2 motors would become closer to surface PM synchronous motors (SPMSMs). However, this requires a further investigation [31] and is not discussed here.

**Figure 4.** Circle diagrams: (**a**) Type 1; (**b**) Type 2.

The field weakening theory has been discussed in References [32,33] where ideally, the infinite speed could be achieved when the value of the armature current *Is* is equal to the characteristic current *Ich*. Practically, to increase the CPSR, *Is* should be selected as close to *Ich* as possible [32,34]. The motors in this paper have *Is* greater than *Ich* due to low the PM flux linkage by the limited PM quantity, as shown in Figure 4, where *Ich* = λ*<sup>m</sup>* / *Lq* for Type 1 and *Ich* = λ*<sup>m</sup>* / *Ld* for Type 2. There are two potential methods for improving the CPSR. The first method is to increase *Ich* by using stronger or more

magnet (higher λ*m*) or changing the rotor configuration to reduce the *q*-axis or *d*-axis inductance for Type 1 or Type 2, respectively. However, this would lead to a redesign of the motors [35]. The second method is to reduce *Is* (active reduction), as indicated in Figure 4, which however, should face a direct reduction of motor torque and power [24]. On the other hand, these motors can be operated in the maximum torque per voltage (MTPV) mode in which the current is reduced (passive reduction) when the speed increases. This would result in a partial overlap in the power-speed curves between the active and passive reductions of the excitation current in the field weakening region at high speed. This is explained in Section 4.3.

#### *2.2. Configuration of Investigated Motors*

As previously illustrated in Figure 1, for the rotor configuration of Type 1, the PMs are embedded along the flux barriers and located in the central part of each rotor pole. As indicated in Figure 1, *Ppm* is the magnet position from 1 to 4, *Wpm* is the magnet width and *Tpm* is the magnet thickness. For the Type 2 configuration, the PMs are arranged along the *d*-axis. Both types have the same PM positions viewed from the motor shaft (e.g., the PMs at Position 1 of Types 1 and 2 keep the same distance to the shaft). These arrangements of PM positions help to evaluate the effect of the PM directions (i.e., facing *d*- or *q*-axis). Note that, the magnetization of the PMs is all in the parallel pattern. The motor specifications and parameters are listed in Table 2, where the analysis at the peak current condition is for the purpose of exploring the capacity of the motors.

**Table 2.** Main specifications/parameters of models.


The performance of the prototype motor with Type 1 arrangement has been investigated with both the experiments and FEA simulations in Reference [23], where the results show that although the torque is high, this motor could not maintain the power and presented a low CPSR. This reduces its practical usefulness.

According to the previous discussion and the mathematical models described in the first part of this section, it is worthwhile to study the Type 2 motor as a potential alternative. Note that Section 2 mainly provides the mathematical background and briefly introduces the basic topologies of the two types of motor.

#### **3. Comparative Analysis of Influence of PM Position**

In this Section, the analyses on the effect of individual PM position on Type 1 and Type 2 motors using finite element analysis (FEA) were conducted. In these analyses, the no-load analysis aims to investigate the contribution of PM at different positions in the rotor to motor flux, and the on-load analysis is used to study the correlation between the PM position and armature excitation. No complete motor models are involved in this Section.

#### *3.1. No-Load Operation Comparison*

Base on the prototype motor in Reference [23], the PM pieces with a cross-sectional dimension of 1.5 × 8 mm were chosen and inserted into the PM positions from 1 to 4 (from inmost to outmost) in this study. Note that, all the PMs were fixed inside the flux barriers embracing them. In the beginning, to investigate the influence of the PM position, each time only one PM piece was placed at one of the above positions and the armature current was removed (no-load operation). The PM flux linkage and air-gap flux density *Bg* for each PM position of both types of motors are illustrated in Figure 5. As can be seen in Figure 5a,c, although the waveforms of the PM flux linkage for Type 1 seem to be different at different PM positions and the trend is unclear, the amplitudes are similar. On the other hand, the waveforms of *Bg* tend to spread out and the peak value deceases as the PM moves towards the inmost position. As shown in Figure 5b,d, all the PM flux linkage waveforms for Type 2 are basically trapezoidal and the amplitudes increase when the PM moves towards the outmost position. In addition, the waveforms of *Bg* are the same but the peak *Bg* increases when the PM moves towards the outmost position. The differences between the two types are significant. The PM position can be used on Type 1 to adjust the waveform of the PM flux linkage and both the waveform and amplitude of the air-gap flux density. For Type 2, this can adjust the amplitude of both the PM flux linkage and air-gap flux density.

**Figure 5.** The PM flux linkage and air-gap flux density at no-load: (**a**) The PM flux linkage for Type 1; (**b**) the PM flux linkage for Type 2; (**c**) the air-gap flux density for Type 1; (**d**) the air-gap flux density for Type 2.

**Table 3.** Air-gap flux density for each PM position.


The no-load peak flux density in the air gap for each PM position of both types is summarized in Table 3. For Type 1, the flux density at position 1 is the lowest (0.056 T) and that at position 4, is the highest (0.075 T). Similarly, for Type 2, the flux density at position 1 is the lowest (0.025 T) and that at position 4, is the highest (0.083 T). However, the lowest flux density for Type 2 is lower than Type 1, while the highest flux density for Type 2 is higher than Type 1. Thus, the no-load air-gap flux density for the Type 2 rotor seems to be more sensitive to positions. This is possibly because for Type 2, the PM flux is not blocked by the outer flux barriers while the blockage appears for Type 1. The revious analyses imply that for Type 1, the room for the PM flux can be more at the inward positions, i.e., larger or stronger PM [23], while for Type 2, more PM for the outward position could be used. In addition, the improvement of the PM flux linkage and air-gap flux density can be anticipated when the number of PM layers or the PM width increases, which is analyzed later.

#### *3.2. On-Load Operation Comparison and Flux Balance Index*

To fully investigate the influence of PM positions, the on-load operation is considered. Figure 6 shows the flux density distribution in the rotor with an 80 A peak current and maximum torque for each PM position (single PM each case). For Type 1, the most unbalanced flux density distribution occurs at the flux segments near the PMs (red circle) since the PM flux is obstructed by the surrounding flux barriers. The heavily unbalanced flux distribution may cause the PMs not to be utilized efficiently and lead to problems, such as local saturation, torque ripple or risk of demagnetization in motors [36]. For Type 2, the unbalance also occurs but appears to be lighter (dark blue circle) and the condition is almost the same for every PM position. This paper develops an index called the flux balance index to rate the degree of balance of flux distribution, which is given as:

$$K\_{\rm u} = \frac{B\_{\rm u}}{B\_{\rm rotor}} \cdot 100\% \tag{9}$$

**Figure 6.** Comparison of the flux density distribution in rotors: (**a**) Type 1; (**b**) Type 2.

where *Bu* is the lowest flux density at the central point (CP) of the main unbalanced magnetic distribution zones as highlighted in Figure 6, and *Brotor* is the average flux density in the rotor core obtained from a number of selected points (40 points in this paper) evenly spread out on the rotor core, as illustrated in Figure 7. The selection of these points only aims to represent the average flux density in the rotor without a particular criterion. Note that the determination of *Bu* and *Brotor* does not take into account the singularities of the magnetic field, which do not represent the general magnetic distribution although these points are significant for saliency and the torque of motors [18,37]. The higher the flux balance index is, the better and more balanced magnetic distribution is in the rotor. The flux balance indices for various the PM positions of both types of motors are shown in Table 4. As can be seen, for Type 1, the flux balance index increases in the order of PM positions from 1 to 4 [Cases 1.1 to 1.4 in Figure 6a], indicating that the magnetic distribution would be better when the PM moves outward. In contrast, for Type 2, the unbalanced zones and the unbalanced condition do not seem to change as the PM position changes [Case 2.1 to 2.4 in Figure 6b]. This may be due to the fact that for such a configuration, the PM flux is not obstructed by the flux barriers. Furthermore, this unbalance is only caused by the armature reaction at load condition. As a result, Type 2 has a greater flux balance index than Type 1 does, meaning that the arrangement of PMs in Type 2 can be a decent choice to help improve the balance of the magnetic field.

**Figure 7.** The selected points for the rotor average flux density calculation: (**a**) Type 1; (**b**) Type 2, where the PM at position 4 is used as representative.



A brief comparison of the torque and torque ripple between the various PM positions of both types is shown in Figure 8. It can be seen that the torque ripple for Type 1 significantly changes with the PM positions, but this does not happen in Type 2. As previously mentioned, for Type 1, both the flux balance index and the locations of unbalance zones vary with the PM position, and

the most unbalanced area occurs on the segment near the PM wherever it is placed (Figure 6a and Table 4). For Type 2, in contrast, both the flux balance index and the unbalanced zones seem to be independent of PM position (Figure 6b and Table 4). This indicates that the magnetic distribution in a Type 2 rotor is insensitive to the PM position, and this accounts for the invariant torque ripple with PM positions. In other words, for Type 1, the selection of PM position in a rotor should consider its effect on the magnetic distribution and thus torque ripple. For Type 2 motors, the torque ripple should not be a major concern for the placement of the PM. Note that, the torque output level is not affected significantly by the PM positions and motor types.

**Figure 8.** Torque production for each PM position.

**Figure 9.** Torque production with Id for each PM position. (**a**) Type 1, (**b**) Type 2.

To fully compare the contribution of each PM position on torque production, the current is regulated in terms of the magnitude and phase advances with respect to the *q*-axis. The current *Iq* is kept at 70A and *Id* is changed from 10 to 40 A, which considers the cross influence between the *d*- and *q*-axes and limits the armature within the peak value, i.e., 80 A. As the result shown in Figure 9, for Type 1, the torque decreases when the PM advances to position 4. In Type 2, although the highest air-gap flux density at position 4 is much higher than position 1, the torque is not much different at any positions. Besides, the highest torque can be generated by the PM at position 1 where the PM is separated from the flux barriers (no intersections), as shown in Figure 1b. However, the width of the PM is limited in this case. In Type 1, the most effective PM position in the torque production is the inward one [23] but it is limited by the possible room given in the rotor, and therefore the configuration with multiple PM layers is suitable for Type 1 to enhance the torque output. In Type 2, with the torque production at each PM position being similar and with the highest air-gap flux density occurring at position 4, this implies that the effective PM configuration is outwards, and the multiple PM layers construction may not be necessary.

In this section, small PM pieces have been used in both Type 1 and 2 motors for the comparison of the influence of PM position. Generally, the placement of these small PM pieces does not significantly change the rotor structure, especially for Type 1. However, the larger PM dimension (e.g., wider) or more pieces (e.g., multiple layers) would be more practical, and thus more variation in rotor configuration may require further assessment.

#### **4. Comparative Analysis of Motor Characteristics**

From the previous discussions, the comparison for various numbers of the PM layers as well as various the PM positions is presented. In this section, to fully investigate the performance and characteristics, six models broken down into the two types of motors discussed in Section 2 are created and shown in Figure 10, where Models 1, 2, and 3 are categorized in Type 1 and Models 4, 5, and 6 belong to Type 2. In these models, Models 1 and 4 have the PMs installed at position 1, Models 2 and 5 at position 4, and Models 3 and 6 at all the positions. As the previous analysis, the PMs are all magnetized in the parallel pattern. Note that these models can be treated as proper motors and the effect of various PM layouts (inmost, outmost and multiple layers) on motor inductances and torque production can be investigated through these rotor arrangements. All the models have the same main specifications and the PM thickness/volume. Their PM positions and dimensions are given in Table 5. Note that, the PM thickness is 1.5 mm, which is considered to be manufacturable [23,26].

**Figure 10.** Configurations of motor models: (**a**) Model 1; (**b**) Model 2; (**c**) Model 3; (**d**) Model 4; (**e**) Model 5; (**f**) Model 6.

**Table 5.** Position and dimension of PMs.


#### *4.1. Motor Inductances*

The inductance variation with the phase angle and magnitude of the current are shown in Figure 11, where the current angle was set to be zero for Figure 11c,d. Note that, these are the motor *d*- and *q*-axis inductances (*Ldm* and *Lqm*) with the presence of PMs rather than the ones without considering the PM flux, i.e., *Ld* and *Lq* (stator inductances) in Equations (3)–(8).

First, as shown in Figure 11a,c, for the Type 1 models (Models 1, 2 and 3), the inductance *Ldm* is the highest for Model 3 but not much higher than the others. The *q*-axis inductance *Lqm* is similar for Models 1, 2 and 3, which indicates that PM position does not significantly influence the electromagnetic characteristics of the motors. Conversely, as shown in Figure 11b,d, for the Type 2 models (Models 4, 5 and 6), the PM position has a great impact on *Ldm*, with the lowest for Model 5 because of the stronger flux created by the PM that limits the armature flux linkage. The inductance *Lqm* is similar for Models 4, 5 and 6. The above analysis shows that the PM position would be the most important design key for Type 2 motors.

**Figure 11.** Inductance comparison: (**a**) Inductance versus current angle for Type 1; (**b**) Inductance versus current angle for Type 2; (**c**) Inductance versus current magnitude for Type 1; (**d**) Inductance versus current magnitude for Type 2.

Second, as the current magnitude increases, the inductance *Ldm* for the Type 1 models decreases rapidly while *Lqm* decreases gently, as shown in Figure 11c. However, as shown in Figure 11d, both *Ldm* and *Lqm* for Type 2 declines slightly with the current magnitude. This results in a small inductance difference (*Ldm* – *Lqm*) variation, especially for Model 5 whose *Ldm* and *Lqm* almost do not change. This may be attributed to the flux barriers that slightly brings down *Lqm*. Meanwhile, *Ldm* decreases mildly because of the alleviated *q*-axis magnetic saturation effects [38]. The insignificant inductance variation that may benefit sensorless control shows that fewer PM layers and placing the PM outwards would be more beneficial to the Type 2 motor design.

#### *4.2. Torque Production*

The torque production and torque components (i.e., PM/reluctance torques) are illustrated in Figure 12 and Table 6. To obtain the curves of torque versus the current angle using FEA, the initial rotor position can first be set up and based on that which the current is applied. Then, the current angle is regulated and swept through the prescribed range so that the output torque can be calculated for each current angle. As shown in Figure 12a, the torque production and torque components for all the models of Type 1 are almost similar. Of these cases, the multiple PM layers one, i.e., Model 3 is the best choice for the sake of its highest achieved total torque and PM torque ratio, as presented in Table 6. However, Model 1 should also be further considered since its torque is only slightly lower, taking the advantage of inmost PM arrangement, as discussed in the previous analysis. As shown in Figure 12b, the torque production and components are diverse for the models of Type 2, which agrees with the previous discussion. The case with the most outward PM and fewer PM layers, i.e. Model 5 would the best choice for its highest achieved total torque by making the most utilization of the PM compared to Models 4 and 6, as presented in Table 6. Furthermore, as presented in Table 6, the torque production of all the models of Type 1 is higher than every model of Type 2. This seems to indicate that the models of Type 1 have better torque production than Type 2. However, the high PM torque and its easy regulation by applying the various PM configurations (i.e., large PM torque difference between these investigated models) is a significant advantage of Type 2. On the other hand, for the Type 1 models, the torque is brought down to zero when the current angle is zero (i.e., *Id* is zero) while for Type 2, the torque can only reduce to zero at the high current angle since the PM torque and the reluctance torque offset each other. Therefore, for the Type 2 motors, the armature excitation, i.e., the stator current *Is* can be easily used to regulate the torque characteristics.

**Figure 12.** Torque production and components: (**a**) Type 1; (**b**) Type 2.



#### *4.3. Torque and Power-Speed Curves*

As previously mentioned, the armature current *Is* can be reduced to be close to *Ich* to improve the speed range. The relationship between *Is* and *Ich* as *Is* of Models 1, 3 and 5 varies and is presented in Table 7, where as expected, the *Is*/*Ich* ratio generally decreases as *Is* decreases. Furthermore, Figures 13 and 14 show the comparison of torque-speed and power-speed curves between Models 1, 3 and 5, where a 220 V DC voltage is applied. In these curves, the rhombuses and circles denote the start- and end-points of the constant power region. The pink ones are for Model 3 and the blue ones are for Models 1 or 5. As can be seen in Figures 13 and 14, for the three models, the power-speed curves tend to converge at high speed and are partially overlapping. When *Is* reduces to be close to *Ich*, the CPSR of the motors improves but the torque and the power suffer some reduction. In Figure 13, the torque and power-speed curves of Model 1 are constantly slightly lower than Model 3 for the same *Is* although the highest torque production is obtained when the PM is at the inmost position in the rotor. In contrast, although at a low speed operation, the torque production of Model 5 is lower than Model 3, but the difference decreases as *Is* decreases. This results in an enhancement of the power-speed curves and CPSR for Model 5, which then surpasses Model 3, as seen in Figure 14. However, if *Is* continues to reduce, e.g., *Is* = 20 A, the CPSR of Model 5 cannot maintain the superiority to that of Model 3. This can be observed via the power-speed curves where there is an intersection point between these curves. The power corresponding to this point is denoted the intersection power, *Pi*. For such an intersection, if the required output power is greater than *Pi*, the CPSR of Model 5 is better than Model 3 and vice versa. Note that, as can be seen in Figure 14, this intersection point locates at the overlap of the MTPV control [33] regions of the target models so that this point no longer depends on the required current, but on the electromagnetic properties or the motor construction, i.e., PM flux linkage, inductance and *Ich*.

**Figure 13.** FieldweakeningcomparisonbetweenModels 1 and 3: (**a**)Torque-speedcurves; (**b**)Power-speedcurves.

**Figure 14.** FieldweakeningcomparisonbetweenModels 3 and 5: (**a**)Torque-speedcurves; (**b**)Power-speedcurves.

From the above discussions, with limited PM volume (less than common IPM motors), the selection of torque and power is closely related to desired motor speed range and needs to be chosen carefully by simultaneously considering the mechanical and electromagnetic characteristics of the motors and their applications. Although these motors are ideally infinite speed [16], the actual operating speed is limited. In addition, the selection of the PM positions and directions, e.g., moving PM from *q*-axis to *d*-axis in this paper, is the key to improve the performance of SynRMs with limited PM used.


**Table 7.** The constant power speed range (CPSR) analysis.

#### *4.4. Demagnetization Analysis*

The no-load PM flux linkage and air-gap flux density of Models 1, 3 and 5 are shown in Figure 15. It can be seen that when the PM width increases, the PM flux linkage and air-gap flux density of Type 2 (Model 5) increases significantly. This demonstrates the intensification of the magnetic field that is enabled by placing the PM along the *d*-axis instead of the *q*-axis. This can also explain the high PM torque ratio of Model 5 as presented in Table 6.

**Figure 15.** No-load magnetic characteristic. (**a**) PM flux linkage, (**b**) Air-gap flux density.

The distribution of flux density in the PMs of Models 1, 3 and 5 under the peak current at 120 ◦C are shown in Figure 16. The average flux density in the PMs at various temperatures are presented in Table 8. Note that, the demagnetization curves of N35H (PM material) are illustrated in Figure 17 [16,39]. It can be seen that the flux density in the PMs of Model 1 is very low, meaning that the PMs can be easily demagnetized. Moreover, the flux density in the PM of Model 1 is lower than Model 3 where the PMs are inserted at all positions. This implies that if only the inward PM position is used, an appropriate thickness of the PM should be carefully chosen to avoid irreversible demagnetization. The PMs in Models 3 and 5 have better demagnetization resistance compared to Model 1 since they possess better operating points, as the flux density presented in Table 8. Particularly, Model 5 possesses an advantage with an average PM flux density over 0.8 T. In contrast, the PMs in Model 3 can be locally demagnetized since their flux density are low, as shown in Figure 16. The key is the direction of the flux. In Type 1, the *q*-axis armature flux is opposing the PM flux and then reducing the flux at the *q*-axis. Instead, in Type 2, the *d*-axis armature flux and PM flux are in the same direction

and then the *d*-axis flux is intensified. The motors that employ flux intensifying such as the Type 2 ones have been called the flux intensifying IPMSM [38,40,41]. However, in this paper, less PM is used for the rotor with a dominant reluctance torque, and therefore they should still be considered as a kind of SynRMs. They are named the FI-PMa-SynRM in this paper [26].

Overall, as investigated in this section, the layout with the PMs crossing the flux barriers have a decreased possibility to be demagnetized.

**Figure 16.** Flux density in PM at the 120 ◦C of temperature: (**a**) Model 1; (**b**) Model 3; (**c**) Model 5.


**Table 8.** Average flux density in PM.

**Figure 17.** Demagnetization curves of N35H.

#### **5. Discussion**

Based on the previous analyses, some brief summaries are listed as follows.


The experimental studies were reported in [23], where the simulations were conducted using the same software package (JMAG). Since the focus of this paper is on the analysis and comparison of several types of SynRM rotors, therefore the experiments are not provided here.

#### **6. Conclusions**

In this paper, the analyses for six models of SynRMs with two different categories of PM layouts have been conducted, and their performance and electromagnetic characteristics have been comprehensively compared. From these analyses, it can be observed that the layout with PMs being arranged along the *q*-axis or embedded into the flux barriers has better torque production capability. For the other layout with the PM facing the *d*-axis or across the flux barriers, the advantages of SynRMs using a limited PM amount can be maintained and the inherent drawbacks, such as irreversible

demagnetization, can be overcome. These indicate that, for SynRMs with a limited amount of PM added and placed along the *d*-axis would better make use of the PM. In addition, the PM arrangement at the inward position is a decent choice for SynRM with the *q*-axis PM, but the PM dimension should be calculated carefully to avoid irreversible demagnetization.

**Author Contributions:** D.-K.N. conceived and conducted the research and wrote the paper. M.-F.H. suggested the research topic, guided D.-K.N. to complete the research and helped edit and finalize the paper. M.-F.H. also provided the laboratory space and facilities for this study.

**Funding:** This work is supported by the Ministry of Science and Technology, Taiwan under project contracts MOST 108-2622-8-006-014 and 107-2622-E-006-005-CC2.

**Acknowledgments:** The authors would like to thank Thanh Anh Huynh for his help in this work. JSOL is also acknowledged for providing JMAG software.

**Conflicts of Interest:** The authors declare no conflicts 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/).

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