Pulse Width Modulation Methods for Minimizing Commutation Torque Ripples in Low Inductance Brushless DC Motor Drives

This article proposes pulse width modulation (PWM) methods for low inductance brushless dc (BLDC) motor drives that minimize the commutation torque ripples. An uneven current of noncommutation phase generates the commutation torque ripples. In the low inductance BLDC motor drives, two types of time delays in PWM also induce critical torque ripples. In this article, the analyses of the time delays in PWM and their effects are introduced. Then, two PWM methods are proposed to eliminate the time delays in the commutation region and the conduction region, respectively. The commutation period is controlled to be synchronized to the switching period. In addition, the proposed methods synchronize the switching period to the commutation interrupt. As a result, the proposed methods effectively maintain the minimum commutation period as well as minimize the commutation torque ripples in the low inductance BLDC motor drives. The validity of the proposed methods is demonstrated by the simulation and experimental results in various driving conditions. The results show that the commutation period is remarkably shortened and the commutation torque ripples are reduced by 27.6% in comparison to the conventional method.


I. INTRODUCTION
B RUSHLESS dc (BLDC) motor has been widely used in electric vehicles, robots, and industrial equipment as it has high efficiency, high torque density, and a simple control method [1]. Especially, the attention toward the slotless BLDC motor has continuously been increasing as it has no iron loss, magnetic saturation, or cogging torque [2], [3]. From these advantageous properties, the total output torque is only determined by the magnetic torque, which is a product of the phase current and back electromotive force (EMF). In addition, the slotless BLDC motor is suitable for weight reduction, miniaturization, and mass production at a low cost [4]. Accordingly, it is favorably adopted in the small-sized motor drive systems, such as partial prosthetic hand systems, as shown in Fig. 1. Meanwhile, since the value of stator inductance of the slotless BLDC motor is relatively low, a small control error produces large current ripples, which lead to the torque ripples [2]. As a result, a precise control of the phase current that interacts with the back EMF becomes necessary.
In the BLDC motor drive system, a two-phase control method is commonly used, as the conduction loss is reduced and the control method is simple. The commutation torque ripples are the main issue in the two-phase control method, since it becomes up to 50% of the average torque [5]. Until recently, various pulse width modulation (PWM) methods have been introduced to reduce the commutation torque ripples by keeping constant current of the noncommutation (NC) phase [6]- [14]. The vector selection method considering nonideal back EMF was proposed in [6]. Modified PWM methods using coordinate transformation [7] and PWM-OFF-PWM method [8] were introduced. Also, the methods of changing the current controller using current hysteresis control [9] and integral variable structure control [10] were proposed. However, the previous research neglected the effects of the time delays in PWM, which also generate critical torque ripples in the low inductance BLDC motor drives. In [11], an overlapping control for the slotless BLDC motor was proposed, but the time delay problems still remain. Numerous compensation circuits, such as double-layer C-dump converter [12], diode-assisted buck-boost converter [13], and capacitor-switching-based boost converter [14], were proposed to minimize torque ripples by modulating the dc-link voltage. However, these circuits cannot be applied to a small-sized BLDC motor drive system, since an additional space for power semiconductors, capacitors, and inductors is required.
Meanwhile, the commutation delay is another main factor of the torque ripples. It is generated due to the time delay in PWM and causes various losses in the motor drive system [15], [16]. In the case of BLDC motor drives, the commutating instant does not match with the end of the PWM switching due to the commutation delay [17]. Thus, several methods compensating for the commutation delay were proposed in [18]- [24]. The methods of modifying the carrier function by initializing [18] and changing the frequency [19] at the commutating instant were introduced. However, the PWM duty ratio became an inaccurate value and the current harmonic distortion was generated due to the carrier function, which are critical in the low inductance motor drives. Mixing the ratio of the commutation and conduction period was proposed in [20], but the modified PWM duty ratio was still not precise. To reduce the commutation delay, pulse-amplitude modulation methods using the buck converter [21] and phase-controlled rectifier [22] were proposed. In [23], the PWM generation method using hall signals with the logical circuit was proposed. However, the abovementioned methods required additional circuits that take up considerable space. The commutation error compensation method, which considers both phase delay and position error, was proposed in [24]. Although this method was effective for compensating the position error, there was still a limit to eliminating the commutation delay.
This article proposes new PWM methods to minimize the commutation torque ripples in the low inductance BLDC motor drives. At first, considering the effect of PWM, the time delays called commutation end delay (CED) and commutation start delay (CSD) are introduced and analyzed. The analyses represent that the torque ripples due to the time delays are critical to the low inductance BLDC motor drives. Then, to eliminate each delay, two types of PWM methods called the N-switching period commutation method and the variable switching period (VSP) conduction method are proposed. The former modifies the PWM duty ratio to synchronize the commutation period to the switching period, and the latter adjusts the switching period to synchronize it to the commutating instant. As a result, the proposed methods effectively remove the time delays and maintain the NC phase current constant. Accordingly, the commutation torque ripples are minimized, and the commutation period is kept to a minimum in the low inductance BLDC motor drives. Finally, the proposed methods are verified by the simulation software, PLECS of Plexim Inc., and experimental results from a partial prosthetic hand system.

A. Modeling of BLDC Motor
where the switching state S is 1 when the upper switch conducts and −1 when the lower switch conducts. In one switching period t sw , S becomes 1 during the PWM duty ratio Dt sw and −1 in the rest of the time. Using the state-space averaging, the switching state is expressed as S t sw = 2D − 1. The two-phase control method of the BLDC motor is shown in Fig. 3. The commutation interrupt, which is triggered by the hall sensor or the zero-crossing point of the back EMF, starts at every 60 • of the electrical angle. The commutation region occurs when the commutation interrupt is triggered and the conduction region occurs in the rest of the time. In the commutation region, the outgoing (OG) phase current falls to zero and the incoming (IC) phase current rises to the value of the current reference where the NC phase current is determined by the current of the other two phases. Using the Kirchhoff's current law at the neutral point, the total output torque is expressed as where E is the peak value of trapezoidal back EMF. From (2), the commutation torque ripples are generated when the NC phase current does not remain constant. Three typical cases of phase current are shown in Fig. 3. In case 1, the rising time of the IC phase current t rise and the falling time of the OG phase current t fall are same, where the NC phase current and the output torque are maintained constant. However, in case 2 or 3, t rise is longer and shorter than t fall , respectively, causing the ripple of NC phase current and output torque. Depending on the motor parameters, the phase current within the commutation region is determined to be one of the abovementioned three cases. Accordingly, the compensation methods are required to achieve the commutation region as case 1.

B. Commutation End Delay
The previous compensation methods only tried to keep t rise and t fall identical. To satisfy the condition, the PWM duty ratio of each phase was calculated. However, in the PWM VSI system, the PWM sequence is updated at every peak of the carrier function, as shown in Fig. 4(a). Therefore, there is a time delay t CED between the end of the commutation region and the next PWM sequence, which is called "CED". The CED has a value between zero and t sw . As the PWM duty ratio stays constant until the next peak of the carrier function, the PWM sequence remains unchanged even if the commutation region ends. As a result, the torque ripples occur during CED even though the compensation method is applied. Assuming constant back EMF during the commutation region, the CED and the torque ripples are expressed as Therefore, the torque ripples due to commutation delay become severe when the CED is long or the stator inductance is low. Assuming v L is identical as well as the phase voltage, the commutation torque ripples due to CED are much more critical in the low inductance BLDC motor system.

C. Commutation Start Delay
The commutation interrupt indicates the start of the commutation region and is related to the rotor position. As the rotor position and the PWM sequence are irrelevant, the commutation interrupt could be triggered in between the PWM sequence, as shown in Fig. 4(b). Therefore, a time delay t CSD occurs between the commutation interrupt and the next PWM sequence, which is called "CSD". The CSD has a value between zero and t sw . The lagged commutation due to the time delay induces the torque ripples that are proportional to CSD. Moreover, it generates circulating current and reduces efficiency and various losses of the motor drives [24], [25]. These negative effects are even more severe in the low inductance motor drives and high-speed drives.
The effects of CED and CSD are shown in Fig. 5. Due to the time delays, the PWM sequence in the commutation region is distorted, which induces the commutation torque ripples. Meanwhile, it is necessary to minimize the duration of the commutation region to reduce the torque ripples and enhance system response [7], [12]. As the back EMF of OG phase starts decreasing at the commutating instant, the commutation period should be as short as possible to assure constant output torque in the commutation region. In the following section, the novel PWM methods are proposed that minimize the duration of the commutation region as well as eliminate both CED and CSD.

III. PROPOSED PWM METHODS
In order to eliminate the CED and CSD with the minimum commutation period, this article proposes two PWM methods: 1) the N-switching period (NSP) commutation method; and 2) the VSP conduction method.

A. NSP Commutation Method
According to (3), the CED is produced by the time delay between the commutation period and the switching period. The period of commutation region varies in the conventional PWM methods; it depends on the parameters, such as phase current, resistance, and inductance. Therefore, the commutation period and duration of CED become nonperiodical. The prepresented research [26] synchronized the commutation period to a digital sampling period to eliminate CED, but the commutation period was not achieved to be a minimum, since the minimum value becomes shorter than a sampling period when the motor is driven with a higher switching frequency. In the proposed NSP commutation method, the commutation period is synchronized to the multiple of the switching period and N cm , which is called "commutation constant". Moreover, the commutation period and the commutation constant are controlled to be a minimum.
To control the commutation period, both the rising time of IC phase current and the falling time of OG phase current should be equalized to the multiples of the switching period as follows: As t rise and t fall are determined by the phase voltage and motor parameters, the PWM duty ratio of each phase is modified to satisfy (4). The proposed method is divided into the following two cases according to the relation between the commutation period and the motor electric constant, i.e., τ elec = L s /R s .
Short Commutation Period ( t cm < 2τ elec ): In the case of the short commutation period, the IC phase current rises rapidly to reach the reference value within a short commutation period. Therefore, the upper switch of the IC phase stays ON and the OG and NC phase switches are controlled, as shown in Fig. 6(a). Thus, the phase voltages and neutral voltage are expressed as As the IC phase current rises from zero to a reference value and the OG phase current falls from a reference value to zero, each phase current is calculated by solving the differential equations. Using the Taylor's expansion, the phase currents are derived as To satisfy (4), each phase current should meet Using (6)-(8), the rising time of the IC phase current and falling time of the OG phase current are calculated as And, the PWM duty ratios of OG and NC phases are derived as As the duty ratio exists between 0 and 1, the conditions of the commutation period are expressed as follows: In addition, the period of the commutation region is synchronized to the switching period in the NSP commutation method. Therefore, the minimum values of the commutation constant and the commutation period are expressed as Using the proposed method, the CED is eliminated, while the commutation period is kept minimized. However, if the switching period is so long that it is not able to satisfy (11) even if N cm = 1, another PWM method can be applied.
Long Commutation Period ( t cm > 2τ elec ): In the case of the long commutation period, the IC phase current rises slowly to reach the reference value within a long commutation period. Therefore, the upper switch of the OG phase stays ON and the IC and NC phase switches are controlled, as shown in Fig. 6(b). Thus, the phase voltages are expressed as The neutral voltage, phase current, rising time, and falling time are derived in the same way as before. As a result, the PWM duty ratio of IC and NC phases are expressed as And, the condition of the commutation period is derived as In the case of a long commutation period, N cm is always 1, so the commutation period is identical to the switching period.
From (12) and (16), the commutation period is synchronized to the multiples of the switching period in any case of the driving system using the proposed methods. Fig. 7 shows the PWM duty ratio according to the commutation period, where the duty ratio is always between 0 and 1.
In the conventional PWM methods, the commutation period keeps changing while driving. On the other hand, from (10) and (15), the parameters, such as current reference, back EMF, stator resistance, and inductance, are all reflected to calculate the PWM duty ratio. Therefore, the commutation period of the proposed methods is kept constant regardless of the parameters. In addition, the proposed method is implemented using a simple three-phase PWM VSI without any additional circuit. Fig. 8 shows the carrier function, commutation interrupt, and PWM sequence in the cases of ideal commutation and actual one using the NSP commutation method, respectively. As the timing of the PWM sequence and the commutation interrupt is different, a nonperiodical CSD occurs. To eliminate it, the VSP conduction method is proposed, which is based on the methods usually applied in the six-step control [27]- [29]. The VSP method was used in the case of low-switching frequency modulation, but has not been addressed in the two-phase control of the BLDC motor. In BLDC motor drives, the commutation interrupt occurs at every 60 • of the electrical angle. Therefore, the interval of the commutation interrupts t ci is expressed as

B. VSP Conduction Method
whereω r is an average speed between two interrupts, which is calculated using linear extrapolation. To eliminate CSD, the conduction region has to end exactly at the commutating instant. Therefore, the period of conduction region t cd is controlled to be the multiples of the switching period as where N cd is the conduction constant. In the conventional methods, the commutation period t cm in (18) varies and is complicated to calculate while driving. However, using the proposed NSP commutation method, t cm is fixed and predetermined by the user. Therefore, the conduction period t cd could be controlled by modifying the conduction constant and the switching period.
Considering the maximum value of the switching frequency determined by the motor drive system, the conduction constant and the VSP are expressed as Fig. 9 shows N cd and VSP according to the rotor electric speed. As the speed increase, the conduction constant N cd decreases, and the VSP is calculated using (19). Fig. 10 shows the switching sequence using both the NSP commutation method and the VSP conduction method. Compared to Fig. 8, both CED and CSD are eliminated with minimized commutation torque ripples.
The VSP conduction method is implemented only using a software algorithm without any additional circuit. In addition,   the PWM duty ratio is not distorted as the conduction region ends exactly at the commutating instant. As a result, both the time delay and the torque ripples are minimized at the transition from the conduction region to the commutation region.

IV. SIMULATION RESULTS
To verify the proposed PWM methods, the simulation is performed using PLECS. The parameters in Table I are used for the BLDC motor model. Also, the digital signal processor (DSP), the carrier function generator, and the three-phase PWM VSI are implemented.
The three-phase current waveforms using the conventional PWM method, the previous method in [11], and the proposed NSP commutation method are shown in Fig. 11(a). The switching period is modified to ignore the effect of CSD. The torque errors in the figure are expressed as Fig. 11. Simulation results of three-phase current using conventional, previous [11], and proposed NSP and VSP methods at the following. In the conventional method, the falling time is much shorter than the rising time, where the fluctuation of the NC phase current is generated. As addressed in (2), the ripple of the NC phase current induces the commutation torque ripples, which are more than half of the torque reference. In the previous method, in [11], the rising and falling times are equalized as shown in the second waveform of Fig. 11(a). However, there is still CED between the end of the commutation region and the next PWM sequence. Therefore, the commutation torque ripples around half of the torque reference remains even though the compensation method is applied. On the other hand, the torque ripples are minimized in the proposed NSP commutation method. In the last waveform of Fig. 11(a), both the rising and falling times are synchronized to the switching period. Using (13), the commutation constant N cm is derived as 1 and the commutation period t cm is synchronized to a switching period. Since 2 τ elec = 64.55 μH/Ω from Table I, the criterion of (11) is satisfied showing the validity of the analysis and the proposed methods. As a result, the current of the NC phase is maintained constant and the commutation torque ripples are reduced to 24.5% of the torque reference.
The phase current waveforms using the conventional PWM method, the NSP commutation method only, and the NSP commutation method and VSP conduction method simultaneously are shown in Fig. 11(b). In the conventional method, the CSD and CED exist and the rising and falling times are not equalized, resulting in large torque ripples. Using the NSP commutation method only, the CED is eliminated, while the CSD still remains, as shown in Fig. 11(b). Therefore, the torque ripples still exist, which are proportional to the CSD. Using the NSP commutation method and the VSP conduction method simultaneously, both CSD and CED are eliminated. In the last waveform of Fig. 11(b), the switching period is changed to t sw , which is synchronized to the commutation interrupt. Using (19), N cd is calculated as 4 and t sw is calculated as 65.135 μs. As a result, the torque ripples are reduced to a quarter of the reference torque.

V. EXPERIMENTAL RESULTS
To verify the proposed PWM methods, an experimental setup is established, as shown in Fig. 12. The 10-mm diameter slotless BLDC motor is controlled by the small-sized VSI for the partial prosthetic hand system. The 5-mm width small DSP (STM32G491REI6) is utilized to implement the proposed methods. Considering the dc-link voltage and the output capacitance of the power semiconductor, the maximum switching frequency of the system is derived as 120 kHz. The parameters of the slotless BLDC motor are listed in Table I.
The overall block diagram and flowchart of the proposed methods are shown in Fig. 13. The phase current and dc-link voltage are sampled by the analog-to-digital conversion (ADC) peripheral, and the rotor position is sampled by timer (TIM) peripheral in the DSP. The rotor position is used to calculate rotor speed by the Luenberger speed observer and reconstruct back EMF. Using the sampled parameters and the torque reference, the NSP commutation method and VSP conduction method are applied in each region, respectively. The switching between two regions is decided by the commutation interrupt and an algorithm in the NSP method. Then, the voltage reference and the PWM duty ratio are derived using the proposed methods and generated by the advanced control timer peripheral in the DSP. In addition, the VSP is implemented by modifying the register related to the peak of the carrier function. Fig. 14 shows the three-phase currents in the four cases of driving conditions. In Fig. 14(a), the conventional method is applied where the rising and falling times are different and the time delays in PWM exist. Accordingly, the current of the NC phase becomes uneven. In Fig. 14(b), both the NSP commutation method and the VSP conduction method are applied. Using (11), (12), and the parameters in Table I, the conditions of the commutation period are derived as 17.75 μs < t cm < 64.55 μs.
Therefore, the commutation constant is calculated as 3 and the commutation period is calculated as 25 μs. In addition, using (19), the VSP t sw is changed to 8.351 μs. As a result, both CSD and CED are eliminated and the commutation period is synchronized to three times of the switching period. Therefore, there is no fluctuation in NC phase current. The different speed condition is shown in Fig. 14(c), and the different torque condition is shown in Fig 14(d). The t cm and t sw are calculated in the same way. The ripple of the NC phase current is minimized, and the slopes of the OG and IC phase current are identical in all three cases of the proposed methods. This demonstrates the validity of the proposed methods even if the control parameters, such as the rotor speed and output torque, are changed.
In Fig. 15, the waveforms of the output torque at condition in Fig. 14(a)-(d) are compared and analyzed. While there are commutation torque ripples of 40.5% using the conventional method, the proposed method reduced the commutation torque ripples to 11.2%, which is 27.6% of the conventional ones. The torque ripples are also minimized in the case of different speed and output torque. Since the remaining torque ripples are generated due to the switching ripples of the phase current, it is considered that the commutation torque ripples are minimized in given driving conditions. Fig. 16 shows the dynamic response of the proposed methods in varying torque conditions. The proposed NSP commutation method and VSP conduction method are applied in each region. The torque reference is changed from 0.6T rated to 0.9T rated , while the speed is controlled steady to 28 kr/min. It is observed that there are not any current and torque ripples in the transient condition. Moreover, the commutation torque ripples are minimized in both dynamic and steady-state conditions. As a result, the dynamic output torque with fast response and fewer ripples can be obtained using the proposed methods.
The dynamic response of the proposed methods in varying speed conditions is shown in Fig. 17. The speed reference is changed from 14 to 28 kr/min, while the output torque is controlled steady to 0.6T rated . It is observed that there are not any current and torque ripples in the transient condition even if the speed accelerates in a short period. The output torque is increased in the transient condition, and the commutation torque ripples are effectively minimized in both dynamic and steady-state conditions. This indicates that the proposed methods are also effective in varying speed conditions. As a result, the proposed NSP commutation method and VSP conduction method have both good dynamic and steady-state performance. Fig. 16. Experimental results of three-phase current, output torque, and speed in varying torque conditions. The torque reference is changed from 0.6T rated to 0.9T rated . Fig. 17. Experimental results of three-phase current, output torque, and speed in varying speed conditions. The speed reference is changed from 14 to 28 kr/min.

VI. CONCLUSION
This article proposed the PWM methods for minimizing the commutation torque ripples in the low inductance BLDC motor drives. The analyses of CED and CSD represented that the time delays generate critical torque ripples. To eliminate CED, the NSP commutation method was proposed where the commutation period was controlled to be synchronized to the switching period. Then, to eliminate CSD, the VSP conduction method was proposed where the switching period was synchronized to the commutation interrupt. Consequently, both the commutation torque ripples and commutation period were minimized in the low inductance BLDC motor drives. Only a software algorithm and three-phase PWM VSI were required to implement the methods. The proposed methods were verified by the simulation and experimental results in both steady-state and dynamic driving conditions. The results showed that the commutation torque ripples were reduced by 27.6% in comparison to the conventional method. At the same time, the commutation period was minimized and synchronized to three times of the switching period. As a result, the proposed PWM methods provided the high efficiency and precise torque control in various slotless BLDC motor driving applications, such as partial prosthetic hand systems. Juwon Lee (Student Member, IEEE) was born in Seoul, South Korea, in 1997. He received the B.S. degree in 2020 from Seoul National University, Seoul, South Korea, where he is currently working toward the Ph.D. degree, all in electrical engineering.
His research interests include electric energy conversions, electric machine drives, and design and control of medical device systems.