Research on Suppressing Commutation Torque Ripple of BLDCM Based on Zeta Converter

Abstract

Torque ripple in a brushless DC motor (BLDCM) seriously restricts its application in high-performance fields. This paper proposes a commutation torque ripple suppression strategy based on a Zeta converter. The expected output voltage of a Zeta converter that suppresses the commutation torque ripple is obtained, according to the effect of the duty ratio of the Zeta converter on the turn-off phase freewheeling duration and the turn-on phase rising duration, during commutation. Based on the analysis of the dynamic response of the Zeta converter, the Zeta converter is adjusted to ensure that the Zeta converter reaches stability in sufficient time. During the commutation, the output voltage of the Zeta converter is connected to the main circuit to reduce the torque ripple during commutation, and the expected regulated duty cycle of the Zeta converter during the next commutation is calculated to adjust the output voltage of the Zeta converter. Based on this analysis, the experimental results verify the effectiveness of the proposed method.

1. Introduction

Brushless DC motors (BLDCMs) have received wide attention in regard to their many industrial applications due to its advantages, namely its high-power density, simple driving method, and easy maintenance [1,2]. However, due to the serious commutation torque ripple problem associated with the two-phase conduction mode, the stability of the BLDCM is reduced, which restricts its application in fields involving high precision [3,4,5]. During commutation of the BLDCM, the current cannot be abruptly changed due to the existence of inductance during motor winding, leading to the generation of commutation torque ripple [6,7,8,9]. On this basis, domestic and foreign scholars have performed lots of research on the commutation torque ripple of the BLDCM, from the perspective of circuit topology improvement and control strategy optimization.

In [10], the impact of stator iron loss and current-loop delay on torque control aimed at minimizing copper losses in a BLDCM are analyzed, and an improved control strategy is proposed, effectively reducing iron loss and torque ripple, while enhancing the overall efficiency of the motor. In [11], an efficient Maximum Power Point Tracking (MPPT) strategy using Ripple Correlation Control (RCC) is presented, which optimizes the power output from the PV system and significantly reduces torque ripple in a BLDCM-driven pumping system under varying solar irradiance conditions. In [12], an optimized switching technique focused on harmonic minimization is introduced, significantly improving the overall efficiency of the BLDC motor driving circuit and effectively reducing the torque ripple. In [13], an enhanced PWM-OFF-PWM technique is proposed, which modulates PWM signals during braking to smooth torque transitions, significantly reducing the commutation torque ripple and improving braking performance. Moreover, different voltage vectors are selected based on the output of the torque hysteresis comparator for different speed ranges. However, the three-phase conduction mode not only increases the switching loss of the inverter, but also causes serious error accumulation due to the set dead-zone time in the real system.

The buck–boost auxiliary circuit is designed in the front end of traditional inverters [14,15]. Based on this, the effects of different vector combinations on the commutation torque ripple and rotation speed are analyzed during commutation and conduction, respectively. The optimal vector is obtained to suppress the commutation torque ripple in two modulation cycles. In [16], the influence of three-level inverters on the motor torque is analyzed. The improved combination of the SEPIC converter and the three-level inverter is used to achieve commutation torque ripple suppression. However, the topology of this method is complex, which reduces the commutation torque ripple and increases the overall loss in the system. In [17], the auxiliary step-up front-end converter is designed based on a flyback transformer. On this basis, the relationship between the output voltage of the front end and the back EMF is adjusted to suppress the commutation torque ripple. However, this method uses a flyback transformer, which requires a higher performance capacitance and lacks analysis on the dynamic response of the auxiliary step-up front-end converter when the auxiliary circuit is commutated. In [18], a method utilizing an improved quasi-Z-source network is introduced, which optimizes the rise and fall times of phase currents during commutation, effectively reducing torque ripple in BLDC motors.

Unlike the buck–boost circuit, SEPIC converter, and flyback transformer, the unique inductance structure of the Zeta converter allows for continuous current output at a lower current ripple [19,20]. Although the Zeta converter consists of the same number of components as the Cuk converter [21,22], the Cuk converter is an inverting buck–boost circuit. Unlike the Cuk converter, the Zeta converter is a noninverting buck–boost circuit, which can eliminate the impact of negative voltage on the circuit, and it has a faster dynamic response in terms of the system [23,24]. Based on the advantages of the Zeta converter, combined with the application of other converters to BLDCM torque ripple suppression, this article analyzes the suppression of BLDCM commutation torque ripple based on the Zeta converter and provides the corresponding control strategies. Due to the existence of motor inductance, the turn-off phase current and turn-on phase current of the BLDCM cannot be abruptly changed during the commutation process, which results in the turn-off phase freewheeling duration not being equal to the turn-on phase rising duration, so that the noncommutated phase current changes, resulting in commutation torque ripple [25,26].

In this paper, first, based on the turn-off phase freewheeling duration and turn-on phase rising duration being equal, the adjusted duty ratio of the Zeta converter can be obtained. Then, the dynamic response of the Zeta converter and the commutation duration are analyzed. Considering that the expected output voltage of the Zeta converter without commutation torque pulsation remains the same during the two adjacent commutation cycles, according to the motor speed and load current at the previous commutation, the duty ratio of the Zeta converter under the stable operation of the motor can be adjusted. On this basis, the strategy for the adjustment of the noncommutation and commutation phases is constructed.

In Section 1, the article highlights the importance of BLDC motors in industrial applications and the challenge of commutation torque ripple, and discusses various strategies proposed to mitigate this issue, focusing on the advantages of using a Zeta converter. In Section 2, the analysis of commutation torque ripple reveals the sources of the ripple and explains how the Zeta converter can control the output voltage during commutation to reduce it. In Section 3, the proposed strategy for suppressing commutation torque ripple using the Zeta converter is detailed, including the design of a back-EMF observer to enhance system stability. And the dynamic response of the Zeta converter is analyzed, demonstrating its effectiveness in maintaining a stable output voltage during motor commutation. In Section 4, experimental results validate the proposed method using a 3.5 kW BLDC motor, showing a significant reduction in the torque ripple and improved performance compared to traditional methods. In Section 5, the conclusions affirm the effectiveness of the Zeta converter-based method in reducing commutation torque ripple and enhancing the stability and performance of BLDC motors.

2. Analysis of Commutation Torque Ripple in BLDCM

2.1. Equivalent Model of BLDCM with Zeta Converter

The stator winding of the BLDCM adopts a three-phase star connection method. Due to the existence of motor inductance and the impact of the motor speed, current, and conduction mode, the turn-off phase freewheeling duration is not equal to the turn-on phase rising duration, resulting in commutation torque ripple [23,24]. Therefore, the output voltage is controlled by adding the Zeta converter at the front end of the three-phase inverter, so that the conversion rates of the turn-off phase and turn-on phase currents are equal during commutation. Figure 1 displays the equivalent model of the driving circuit topology and the motor.

The three-phase stator windings of the motor are assumed to be symmetric. The nonlinear changes in the winding resistance and inductance during the operation of the motor are not taken into account. The voltage equation of the BLDCM is obtained as:

$$U_{\mathrm{xn}}=R_{\mathrm{x}}{i}_{\mathrm{x}}+L_{\mathrm{x}}{\displaystyle \frac{d{i}_{\mathrm{x}}}{dt}}+e_{\mathrm{x}}$$

(1)

where the phase current is i_x, the trapezoidal back EMF is e_x, the terminal-to-neutral point voltage is U_xn, the winding resistance is R_x = R, the winding inductance is L_x = L, and the phase is x = a, b, c. At the rotor mechanical angular velocity ω, the electromagnetic torque T_e is obtained as:

$$T_{\mathrm{e}}=(e_{\mathrm{a}}{i}_{\mathrm{a}}+e_{\mathrm{b}}{i}_{\mathrm{b}}+e_{\mathrm{c}}{i}_{\mathrm{c}})/\omega$$

(2)

2.2. Analysis of Commutation Torque Ripple Based on Zeta Converter

During motor operation, U_dc is the output voltage of the Zeta converter. It can be obtained as:

$$U_{\mathrm{dc}}={\displaystyle \frac{D}{1-D}}{U}_{\mathrm{in}}$$

(3)

where D is the duty ratio of the Zeta converter and U_in is the input voltage of the Zeta converter, which is the rated voltage of the motor. Taking the medium-speed stage of the BLDCM as an example, the impact of the change in the duty ratio D on the current and torque is analyzed by controlling the output voltage value of the Zeta converter during motor commutation.

When D = 50%, the turn-off phase freewheeling duration is equal to the turn-on phase rising duration, so that the noncommutation current and the electromagnetic torque remain constant, as shown in Figure 2a. When D < 50%, the turn-off phase freewheeling duration is less than the turn-on phase rising duration, so that the noncommutation current decreases, resulting in a decrease in the electromagnetic torque, as shown in Figure 2b. When D > 50%, the turn-off phase freewheeling duration is greater than the turn-on phase rising duration, so that the noncommutation current increases, resulting in an increase in the electromagnetic torque, as shown in Figure 2c. Analogously, the increase or decrease in electromagnetic torque is also influenced by the output voltage of the Zeta converter during low-speed and medium-speed commutation. Therefore, the condition for suppressing commutation torque ripple is that the turn-off phase freewheeling duration and turn-on phase rising duration are equal, and the commutation torque can be controlled by adjusting the output voltage of the Zeta converter.

3. Strategy of Suppressing Commutation Torque Ripple Based on Zeta Converter

3.1. Conditions for Suppressing Commutation Torque Ripple in BLDCM

Based on the general solution of the first-order nonhomogeneous differential equation, Equation (4) can be obtained from (1). Equation (4) is the relationship between the current during the commutation process and time.

$$i_x(t)=C{e}^{-tR/L}+\left(U_{\mathrm{xn}}-e_{\mathrm{x}}\right)/R$$

(4)

In the process of motor commutation, the turn-off phase freewheeling duration is t_off. The turn-on phase rising duration is t_on. The starting current value of the turn-off phase and the expected final value of the turn-on phase current are sI, where s is the sign function, s = 1 represents the upper-tube commutation, s = −1 represents the lower-tube commutation, and I is the absolute value of the starting current.

If during the commutation process, x represents the turn-off phase. When t = 0, then i_x(0) = sI. When t = t_off, then i_x(t_off) = 0. According to (4), the constant C₁ and the turn-off phase freewheeling duration t_off can be obtained as:

$$\left\{\begin{array}{l}{C}_1=sI-{\displaystyle \frac{U_{\mathrm{xn}}-e_{\mathrm{x}}}{R}}\\ t_{\mathrm{off}}={\displaystyle \frac{L}{R}}\ln \left(1-{\displaystyle \frac{sIR}{U_{\mathrm{xn}}-e_{\mathrm{x}}}}\right)\end{array}\right.$$

(5)

If during the commutation process, x represents the turn-on phase. When t = 0, then i_x(0) = 0. When t = t_on, then i_x(t_on) = sI. According to (4), the constant C₂ and the turn-on phase rising duration t_on can be obtained as:

$$\left\{\begin{array}{l}{C}_2=-{\displaystyle \frac{U_{\mathrm{xn}}-e_{\mathrm{x}}}{R}}\\ t_{\mathrm{on}}=-{\displaystyle \frac{L}{R}}\ln \left(1-{\displaystyle \frac{sIR}{U_{\mathrm{xn}}-e_{\mathrm{x}}}}\right)\end{array}\right.$$

(6)

The upper-tube commutation process from A⁺C⁻ to B⁺C⁻ is used as an example of the analysis. In Figure 3, VT₁ is turned off, VT₃ is turned on, VT₂ remains constant, and VD₄ freewheels. Ignoring the on-voltage drop of the switch tube and the freewheeling diode, the three-phase voltage meets U_cn − U_an = 0, U_bn − U_cn = sU_dc, and the three-phase current meets i_a + i_b + i_c = 0. According to (1), the voltage during the turn-off phase and the turn-on phase can be obtained as:

$$\left\{\begin{array}{l}{U}_{\mathrm{an}}=\left(e_{\mathrm{a}}+e_{\mathrm{b}}+e_{\mathrm{c}}-s{U}_{\mathrm{dc}}\right)/3\\ U_{\mathrm{bn}}=\left(e_{\mathrm{a}}+e_{\mathrm{b}}+e_{\mathrm{c}}+2s{U}_{\mathrm{dc}}\right)/3\end{array}\right.$$

(7)

When A⁺C⁻ to B⁺C⁻ commutation occurs, e_a = sE, e_b = sE, e_c = −sE, E is the amplitude of the back EMF, that is, E = kω, k is the back EMF coefficient, ω is the motor speed, and the initial value of the turn-off phase current is sI.

On this basis, substituting (7) into (5) and (6), the turn-off phase continuation freewheeling duration t_off and the turn-on phase rising duration t_on can be obtained as:

$$\left\{\begin{array}{l}{t}_{\mathrm{off}}={\displaystyle \frac{L}{R}}\ln \left(1+{\displaystyle \frac{3IR}{2E+U_{\mathrm{dc}}}}\right)\\ t_{\mathrm{on}}=-{\displaystyle \frac{L}{R}}\ln \left(1+{\displaystyle \frac{3IR}{2E-2U_{\mathrm{dc}}}}\right)\end{array}\right.$$

(8)

Similarly, the analysis of the lower-tube commutation also satisfies (8). According to (3) and (8), the corresponding 3D graph is shown in Figure 4.

When the speed and current are fixed, the U_dc can be obtained as (9), and the expected duty ratio D of the Zeta converter can be obtained as (10), by taking t_off = t_on as the condition of no commutation torque ripple, according to (3) and (8). The corresponding 3D graph is shown in Figure 5.

$$U_{\mathrm{dc}}=3IR+4E=3IR+4k\omega$$

(9)

$${\displaystyle \frac{D}{1-D}}={\displaystyle \frac{3IR+4E}{U_{\mathrm{in}}}}={\displaystyle \frac{3IR+4k\omega }{U_{\mathrm{in}}}}$$

(10)

3.2. Design of Back-EMF Observer for Brushless DC Motor

The structure of the observer could be seen in Figure 6. Based on the above analysis and considering the impact of the back EMF on the torque ripple suppression conditions of the brushless DC motor as stated in Equation (10), using the back-EMF constant to calculate the back EMF can lead to calculation errors due to the neglect of the system’s nonlinear effects, which in turn affects the results from Equation (10). Therefore, to enhance the suppression effect on the torque ripple in the brushless DC motor, this paper designs a back-EMF observer for the brushless DC motor and proves its stability.

Based on voltage Equation (1) of the mathematical model of a BLDC motor, this paper chooses the state vector x = [I_a, I_b, I_c, E_a, E_b, E_c]^T. The system’s state equation can be expressed as:

$$\dot{x}=Ax+Bu$$

(11)

where:

$$A=\left[\begin{array}{cccccc}-{\displaystyle \frac{R}{L}}& 0& 0& -{\displaystyle \frac{1}{L}}& 0& 0\\ 0& -{\displaystyle \frac{R}{L}}& 0& 0& -{\displaystyle \frac{1}{L}}& 0\\ 0& 0& -{\displaystyle \frac{R}{L}}& 0& 0& -{\displaystyle \frac{1}{L}}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$$

(12)

$$B=\left[\begin{array}{ccc}{\displaystyle \frac{1}{L}}& 0& 0\\ 0& {\displaystyle \frac{1}{L}}& 0\\ 0& 0& {\displaystyle \frac{1}{L}}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(13)

The observer takes the form:

$$\left\{\begin{array}{l}\dot{x}=A\widehat{x}+Bu+K(y-\widehat{y})\\ \widehat{y}=C\widehat{x}\end{array}\right.$$

(14)

where $\widehat{y}$ is the estimated output, $y$ is the actual measured output, and C is the output matrix:

$$C=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right]$$

(15)

Define the error $e=x-\widehat{x}$. The error dynamics equation is:

$$\dot{e}=\dot{x}-\dot{\widehat{x}}$$

(16)

Based on (14), the error is:

$$\dot{e}=(A-KC)e$$

(17)

To prove system stability, it needs to show that all the eigenvalues of the matrix A − KC lie in the left half-plane, by choosing a Lyapunov function V(e) = e^TPe, where P is a symmetric positive definite matrix. Solve the Lyapunov equation as follows:

$${(A-KC)}^{T}P+P(A-KC)=-Q$$

(18)

where Q is a symmetric positive definite matrix. If positive definite matrices P and Q can be found, the system is asymptotically stable.

Based on (14), the structure of the observer can be derived as shown below:

The specific steps for solving P, Q, and K are as follows: First, determine the system matrix A and the input matrix B, based on the motor parameters shown in

Table 1. Main parameters of the BLDCM and Zeta converter.

	Parameters	Symbol	Value
BLDCM	Rated voltage (V)	UN	200
	Rated power (kW)	PN	3.5
	Pole pairs	np	4
	Phase resistance (Ω)	R	0.684
	Phase inductance (mH)	L	1.234
	Back-EMF coefficient (V/(rad/s))	k	0.528
Zeta	Inductance of Zeta converter (mH)	LA and LB	1
	Capacitance of Zeta converter (μF)	CA and CB	1000

. Then, select the desired closed-loop poles λ_n (n = 1, 2, 3, 4, 5, 6) based on these motor parameters. The poles for the BLDC motor controller are calculated to ensure system stability. Following this, the poles for the observer are initially chosen to be 5–10 times the real part of the controller poles, which is a common criterion for ensuring a fast and accurate response of the observer without introducing excessive noise sensitivity. These initial selections are further refined through simulation and experimental adjustments to optimize the performance of the observer. Finally, the pole placement function ‘place(A, C, λ)’ in Python is used to solve the gain matrix K, and the Lyapunov solver function ‘solve_continuous_lyapunov(A − KC, −Q)’ is employed to solve P, where Q is chosen as the identity matrix.

3.3. Analysis of the Dynamic Response of the Zeta Converter

The Zeta converter converts the DC voltage into a high-frequency voltage through a high-frequency vibration circuit and outputs the required DC voltage through an inductor pulse transformation and capacitor filtering. The topology is shown in Figure 1. Due to the presence of inductance and capacitance in the circuit, it can be approximated as an RLC second-order system, and the equivalent second-order differential equation of the Zeta converter is (19), without considering the motor inductance.

$$RC{\displaystyle \frac{d^2{U}_{\mathrm{dc}}(t)}{d{t}^2}}+{\displaystyle \frac{d{U}_{\mathrm{dc}}(t)}{dt}}+{\displaystyle \frac{R}{L}}{U}_{\mathrm{dc}}(t)={\displaystyle \frac{R}{L}}{U}_{\mathrm{in}}(t)$$

(19)

where R represents the equivalent load of the Zeta converter, C represents the equivalent filter capacitor of the Zeta converter, L represents the equivalent inductance of the Zeta converter, U_dc(t) represents the Zeta converter output voltage, and U_in(t) represents the Zeta converter input voltage. The natural frequency ω_n and the damping ratio ξ can be obtained as:

$$\left\{\begin{array}{l}{\omega }_{\mathrm{n}}=1/\sqrt{LC}\\ \xi =\sqrt{L}/\left(2\sqrt{R^{2}C}\right)\end{array}\right.$$

(20)

The stability of the second-order system of the Zeta converter depends on ξ. Based on the analysis of the dynamic response of the second-order system in the underdamped state, the error between the actual response of the system and the steady-state output can be obtained by combining (19) and (20).

$$\mathrm{\Delta }\le {\mathrm{e}}^{-\xi {\omega }_{\mathrm{n}}t}/\sqrt{1-{\xi }^2}$$

(21)

Assuming ξ $\le$ 0.8, bringing ξ = 0.8 into the denominator of (21), and selecting the steady-state error $\mathrm{\Delta }$ = 0.05 or 0.02, the expression of the adjustment time t_s is obtained as:

$$\left\{\begin{array}{l}{t}_{\mathrm{s}}=3.5/\xi {\omega }_{\mathrm{n}}\begin{array}{c} \end{array}\mathrm{\Delta }=0.05\\ t_{\mathrm{s}}=4.5/\xi {\omega }_{\mathrm{n}}\begin{array}{c} \end{array}\mathrm{\Delta }=0.02\end{array}\right.$$

(22)

On this basis, the parameters of C and L can be selected by considering the power level of the actual motor. The dynamic response of the Zeta converter is analyzed according to Table 1. The adjustment characteristics concern the start of the Zeta converter and the step response of the expected output voltage from 200 V to 300 V.

As shown in Figure 7, the adjustment process of the Zeta converter takes 10 ms, while the commutation duration of the BLDCM is generally tens or hundreds of microseconds. Although the dynamic response of the Zeta converter can be improved by reducing its inductance and capacitance values, it also exacerbates the ripple of the output voltage and current, leading to the unstable output of the Zeta converter. Because the adjustment time of the Zeta converter is much longer than the commutation duration of the BLDCM, it is impossible to achieve real-time changes in the output voltage of the Zeta converter during the commutation duration of the motor. Considering that the expected output voltage of the Zeta converter without commutation torque pulsation remains the same in the two adjacent commutation cycles, according to the motor speed and load current at the previous commutation, the duty ratio of the Zeta converter under the stable operation of the motor can be adjusted by (10), and the expected output voltage of the Zeta converter can be adjusted in real time.

3.4. Implementation of Commutation Torque Ripple Suppression Strategy

When the motor runs in the noncommutation region, the S₁ is turned on and the S₂ is turned off, U_dc = U_in. During that time, the BLDCM drive system is controlled with traditional double-loop PI regulators (i.e., the speed loop and the current loop), and the speed and torque can achieve high stability, such as in Figure 8a. At the same time, the Zeta converter is disconnected from the inverter circuit and adjusted according to (10) to achieve stability.

When the motor runs in the commutation region, S₁ is turned off and S₂ is turned on. The Zeta converter is connected to the main circuit; U_dc is the output voltage of the Zeta converter, such as in Figure 8b. According to the motor speed and load current at the previous commutation, the duty ratio of the Zeta converter under the stable operation of the motor can be adjusted by (10). On this basis, a similar analysis can be performed according to (5)–(7), where d is the duty ratio of the inverter circuit. The expected duty ratio of the inverter circuit is obtained as:

$$d={\displaystyle \frac{3IR+4E}{U_{\mathrm{dc}}}}={\displaystyle \frac{3IR+4k\omega }{U_{\mathrm{dc}}}}$$

(23)

On the basis of the above analysis, the BLDCM control system is designed. Figure 9 shows the configuration of the proposed BLDCM control system. The working status of the motor is judged by the Hall signal. If the motor works in the noncommutation region, the system uses double-loop PI regulators to control the stable operation of the motor. If the motor works in the commutation region, on the basis of the Zeta converter output voltage adjustment after the previous commutation, the commutation time and the duty ratio of the inverter are calculated according to (10) and (15) to suppress the motor commutation torque ripple. At the end of commutation, the expected output voltage of the Zeta converter is adjusted according to (10) to ensure that the Zeta converter can output a stable expected voltage at the next commutation.

4. Simulation and Experimental Results

The experimental platform is set up based on a 3.5 kW BLDCM to verify the effectiveness of the proposed method, as shown in Figure 10. The system controller is DSP (TMS320F28335). The three-phase inverter is composed of IPM (PM50RL1A060). The Zeta converter is composed of IGBT (FF75R12RT4). The specific parameters of the BLDCM and the Zeta converter are shown in Table 1. The modulation method of the three-phase inverter is ON-PWM, and the switching frequency of the three-phase inverter is 20 kHz.

The torque ripple is defined as:

K_r = T_p-p/T_avg × 100%

(24)

where T_p-p is the peak–peak value of the torque and T_avg is the average torque.

As shown in Figure 11, the proposed method maintains a consistent rate of change during the turn-off phase and turn-on phase currents compared with traditional methods, while the noncommutation phase currents remain constant. The commutation torque ripple can be suppressed.

Figure 12 shows the experimental results when the motor runs at the torque reference value of 10 N·m and a speed of 1000 r/min. Figure 12a shows the experimental results for the traditional double-loop PI. The value of K_r is higher than 30%. Figure 12b shows the experimental results for the proposed method based on the Zeta converter. The value of K_r is less than 15%. Compared to traditional methods, the commutation time of the proposed method has also been significantly reduced, as shown in Figure 12.

Figure 13 shows the experimental results when the motor runs at the torque reference value of 20 N·m and a speed of 1500 r/min. As shown in Figure 13a, the value of K_r is higher than 45%. Figure 13b shows the experimental results presented in reference [15]. Figure 13c shows the experimental result for the proposed method based on the Zeta converter. The value of K_r is less than 20% in the proposed method. Compared with traditional methods and the method in [15], the proposed method maintains a consistent rate of change during the turn-off phase and turn-on phase currents, while the noncommutation phase currents remain constant. It can be seen that using the traditional method, the commutation time of the BLDCM is higher than 300 us, and after an improvement, the commutation time is lower than 200 us. Therefore, the BLDCM drive system optimized by the Zeta converter not only reduces the torque ripple, but also effectively shortens the commutation time.

The standard deviation to indicate a quantitative value for the torque ripple is as follows:

$${\sigma }_T=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(T_i-T_{avg}\right)}^2}}$$

(25)

Based on the stability and dynamic experimental results, the proposed method compared to the traditional method shows that the value of the torque ripple percentage falls within the range of 15% to 20%, and the standard deviation of the torque ripple is between 1.5 Nm and 2 Nm under full-load conditions, effectively reducing the torque ripple in the brushless DC motor. To analyze the suppression effect of torque pulsation, this paper provides a comparative table illustrating torque ripple suppression under different drive strategies and various operating conditions. This provides for a clearer observation of the effectiveness of the proposed method in suppressing torque ripple in BLDC motors, as shown in

Table 2. Comparative table on torque ripple suppression under different drive strategies.

Speed (r/min)	Torque (Nm)	Traditional Control Strategy		Proposed Method
		Torque Ripple (Kr)	Standard Deviation (σT)	Torque Ripple (Kr)	Standard Deviation (σT)
1000	10	30%	2.7 Nm	15%	1.5 Nm
1500	20	45%	3.9 Nm	20%	2 Nm

.

Figure 14 shows the output current and voltage waveforms of the Zeta converter under the rated conditions. From Figure 14, it can be seen that under the influence of the Zeta converter, the turn-on phase current can rapidly increase, effectively reducing the torque ripple.

Figure 15 shows the experimental results for the method proposed in this paper under accelerated operation and dynamic load. Setting the reference speed as the slope command, the speed is accelerated from 1000 r/min to 1500 r/min within 1 s, and the load torque is set to 20 N·m, as shown in Figure 15a. The load torque is changed from 10 N·m to 20 N·m, and the speed is 1500 r/min, as shown in Figure 15b.

The output voltage of the Zeta converter rises with increasing speed during the acceleration of the motor, so the reference speed can be tracked effectively. From the magnified result in Figure 15, it can be seen that during the acceleration of the motor, the noncommutation current ripple during commutation is not obvious, and the commutation torque ripple is suppressed. Thus, this experiment verifies the effectiveness of the proposed method during the dynamic process.

5. Conclusions

In this paper, the suppression of commutation torque ripple in BLDCMs, based on the Zeta converter, is proposed. The analysis covers the effect of the duty ratio of the Zeta converter on both the turn-off phase freewheeling duration and the turn-on phase rising duration during commutation, combined with an examination of the dynamic response of the Zeta converter. The proposed BLDCM control system is implemented to validate the effectiveness of the method. The experimental results demonstrate that the proposed method reduces the torque ripple percentage to less than 15% at a torque reference of 10 Nm and less than 20% at 20 Nm, compared to over 30% and 45%, respectively, using traditional methods. Additionally, the commutation time is significantly shortened, indicating the effective suppression of the torque ripple and an improvement in the overall stability and performance of the BLDC motor system. Further research on the Zeta converter will focus on further optimizing its parameters and exploring its application in other motor control scenarios to enhance the robustness and efficiency of BLDC motor systems.