Commutation Error Compensation Strategy for Sensorless Brushless DC Motors

Abstract

Sensorless brushless DC (BLDC) motor drive systems often suffer from inaccurate commutation signals, which result in current fluctuation and high conduction loss. To improve precision of commutation signals, this paper presents a novel commutation error compensation strategy for BLDC motors. First, the relationship between the line voltage difference integral in 60 electrical degree conduction interval and the commutation error is analyzed. Then, in terms of the relationship derived, a feedback compensation strategy based on the line voltage difference integral is proposed to regulate commutation signals by making three-phase back electromotive force (EMF) integral to zero, and the effect of the freewheeling process on the line voltage difference integral is considered. Moreover, an incremental PI controller is designed to achieve closed-loop compensation for the commutation error automatically. Finally, experiment results verify feasibility and effectiveness of the proposed strategy.

1. Introduction

Brushless DC (BLDC) motors are widely used in aerospace, electric vehicle, and household applications due to their inherent advantages, which include high-power density, high efficiency, and simple structure [1,2,3]. Position sensors, such as Hall-effect sensors, are often employed to acquire rotor position signals for proper commutation, however, misalignment in the mechanical installation of the sensors, extreme temperature conditions, or electromagnetic interference may induce errors in the rotor position information, which severely limit the application scope of BLDC motors [4]. To overcome these disadvantages, many sensorless BLDC motor techniques have been proposed in recent years. The zero-crossing points (ZCPs) of the back electromotive force (EMF) are easily detected and widely used in sensorless control. Among the various sensorless strategies currently available, these presented ZCP detection techniques show satisfactory results in specific application scopes; unfortunately, these strategies are still disturbed by various commutation errors [5,6,7]. The ZCPs of back-EMF are generally abstracted from the terminal voltage, but the terminal voltage is vulnerable to the high-frequency components due to the pulse-width modulation (PWM) switching. A low pass filter is usually applied to filter high-frequency switching noise of the terminal voltage, whereas the filters will introduce phase delay for ZCPs detection of back-EMF and the phase delay varies with the operating speed. The lag angle will increase as the motor speed rises. The distorted terminal voltage caused by the freewheeling process also will generate the commutation error. The disturbed terminal voltage may make position detection signal phase ahead, deviating from the ideal commutation instants. The inaccurate commutation signals will induce current pulsation and increase conduction loss.

In reference [8], a hysteresis comparator composed of discrete components is presented to compensate phase delay from the low-pass filter. In reference [9], the authors analyze the errors caused by signal extraction circuits and machine parameters in the sensorless control technique based on average line to line voltage. In reference [10], commutation errors due to the armature reaction and low-pass filters are analyzed, and an error compensation circuit is proposed to achieve proper commutation. In reference [11], the effects of the low-pass filter, noise, and nonideal properties of the estimated back-EMF on the commutation instants are considered. To adjust commutation instants for BLDC motors, a modified digital phase shifter is designed to implement compensation of commutation errors. In the research mentioned previously [8,9,10,11], some specific error factors are discussed and investigated, and the accuracy of the commutation signal is improved accordingly. To achieve more precise commutation signals of sensorless BLDC motors, the commutation error compensation strategies which consider more comprehensive error factors should be studied.

Many estimation and observer methods of commutation error compensation have been proposed to achieve high-precision commutation [12,13,14,15,16,17,18]. In reference [12], the torque constant is employed as the reference signal of the rotor position to obtain accurate commutation signals. In reference [13], the rotor position is acquired from the back-EMF difference, which is estimated from the disturbance observer structure. Discrete-time sliding mode observer [14] and iterative sliding mode observer [15] have been introduced to obtain precise commutation signals for sensorless permanent magnet synchronous motor control. Estimation strategies based on Kalman filters [16] and extended Kalman filters [17,18] have also been presented to estimate information of rotor position accurately. However, these algorithms are complicated and require a large amount of calculation.

Commutation error factors are reflected in the commutation signals, and thus voltages or currents under inaccurate commutation will contain the commutation error information. Many compensation strategies of commutation errors based on the phenomenon of voltages or currents produced under inaccurate commutation have been proposed. In reference [19], an intelligent self-tuning strategy is presented to finely adjust commutation instants, and regulation of the commutation instants is realized by minimizing the stator current. In reference [20], a self-compensation strategy of the commutation angle based on the DC-link current is proposed; in this technique, the commutation instants are corrected by regulating the estimated commutation time error to zero. However, this method is based on the linearized approximation between the DC-link current difference and commutation time error in low-inductance motors. In reference [21], commutation errors are compensated by keeping the freewheeling current of the unenergized phase symmetrical. However, when dealing with some PWM strategies featuring suppression of the freewheeling current [22], such as PWM_ON_PWM mode, the method is unsuitable. The research in [23] proposed a current index optimal approach to adjust commutation errors. However, the back-EMF is assumed to be a trapezoidal waveform with less than 120° flat-top width, which limits the applications of the proposed method. In reference [24], the symmetrical characteristics of the unenergized phase terminal voltage are presented to regulate commutation instants, but the effect of the freewheeling period is not considered.

This paper presents a novel compensation strategy for commutation errors of sensorless BLDC motors. In this paper, the relationship between the commutation error and the line voltage difference integral in 60 electrical degrees interval is analyzed, and a feedback compensation strategy based on the line voltage difference integral is presented to regulate commutation instants and the effect of the freewheeling period is considered. In addition, a PI controller is designed to achieve closed-loop compensation for commutation errors. The remainder of the paper is organized as follows. The proposed commutation error compensation strategy is proposed in Section 2. Section 3 provides experimental results to support the validity and effectiveness of the proposed method. The conclusions are presented in Section 4.

2. Proposed Commutation Error Compensation Strategy This paper presents a commutation error compensation strategy based on the line voltage difference integral. In this section, the relationship between the line voltage difference integral and the commutation error is theoretically analyzed. Then, a feedback compensation strategy with the line voltage difference integral is presented. Finally, an incremental PI controller is designed to regulate the commutation error. 2.1. Mathematical Model of BLDC Motor

The equivalent circuit of the BLDC motor drive system is shown in Figure 1. Assuming that the three-phase stator windings are symmetrical, the phase resistance and inductance are constant, and the armature reaction is negligible. Then, the voltage equations are described as

[_uaubuc]=[R000R000R][_iaibic]+[L000L000L]ddt[_iaibic]+[_eaebec]+[_uNuNuN]

where_ua,_ub, and_ucare the three-phase voltages;_ia,_ib, and_icare the three-phase currents;_ea,_eb, and_ecare the phase back-EMFs;Ris the stator resistance;Lis the phase inductance; and_uNis the neutral voltage.

The line voltage equations are expressed as

{_uab=_ua−_ub=R(_ia−_ib)+Ld(_ia−_ib)dt+_ea−_ebubc=_ub−_uc=R(_ib−_ic)+Ld(_ib−_ic)dt+_eb−_ecuca=_uc−_ua=R(_ic−_ia)+Ld(_ic−_ia)dt+_ec−_ea

In general, the back-EMF of BLDC motors is between those of a trapezoidal wave and a sine wave since it contains many high-order harmonics. Hence, the back-EMF of the BLDC motor can be expressed as

{_ea=_{∑n=0∞E2n+1}sin((2n+1)θ)=_ea1+_e3+_ea5+_ea7+_e9+…_eb=_{∑n=0∞E2n+1}sin((2n+1)(θ−2π/3))=_eb1+_e3+_eb5+_eb7+_e9+…_ec=_{∑n=0∞E2n+1}sin((2n+1)(θ+2π/3))=_ec1+_e3+_ec5+_ec7+_e9+…

where_E2n+1is the back-EMF coefficient; (2n + 1) is the order of harmonics;θis the rotor position;_eam,_ebm, and_ecmrepresent the mth harmonics of the three-phase back-EMF and the subscript m denotes harmonic order;_e3and_e9represent the third harmonics of the back-EMF.

Figure 2 shows phase diagram between back-EMFs and phase current, where_ea,_eb, and_ecdenote the back-EMFs, and_ia,_ib, and_icdenote phase current.αrepresents the commutation error, which is assumedα∈[−π/6,π/6]. The sign of the commutation errorαreflects the commutation information (i.e., delayed or advanced commutation). A negative commutation errorαdenotes advanced commutation, whereas a positive error represents a delayed one.

2.2. Line Voltage Difference Integral without Consideration of Freewheeling Period

For a BLDC motor, there are six operation modes and every operation modes last 60 electrical degrees. In the 60 electrical degree conduction interval, two periods are included: the freewheeling period and the normal conduction period. In the freewheeling period_tfw, the incoming phase current begins to rise, the un-commutated current remains conducting, and the outgoing phase current gradually decays due to the existence of phase inductance. When the outgoing phase current drops to zero, the freewheeling period is over and the motor starts operating in the normal conduction period.

Assume at some step that VT₁ and VT₆ are conducting, while VT₅ is floating. The line voltage difference(_ubc−_uca)is expressed as

_ubc−_uca=R(_ia+_ib−2_ic)+Ld(_ia+_ib−2_ic)/dt+_ea+_eb−2_ec

Considering a BLDC motor with a negligible freewheeling period_tfw, the current equation satisfies_ia=−_iband_ic=0during the entire 60-electrical degree period whent∈[π/6,π/2] , as shown in Figure 2. Substituting the current equation into (4), the line voltage difference(_ubc−_uca)is simplified as

_ubc−_uca=_ea+_eb−2_ec

By integrating (5), the line voltage difference integraldis achieved as follows:

d=_{∫tπ/6tπ/2}(_ubc−_uca)dt=_{∫tπ/6tπ/2}(_ea+_eb−2_ec)dt

Figure 3 shows the three-phase back-EMF under accurate, delayed, and advanced commutation; the shadowed areas in each subfigure represent the integral of each back-EMF. As seen in Figure 3a, the integral of the back-EMFs_eaand_ebyields equal amplitudes but opposite signs, i.e.,_{∫tπ/6tπ/2ea}dt=−_{∫tπ/6tπ/2eb}dt; hence, the sum of the integrated back-EMFs_eaand_ebis zero in the interval. Similarly, the integral of the back-EMF_ecis zero, i.e.,_{∫tπ/6tπ/2ec}dt=0, due to the AC symmetry of the back-EMF_ec. From this analysis, (6) satisfies the condition

d=_{∫tπ/6tπ/2}(_ubc−_uca)dt=_{∫tπ/6tπ/2}(_ea+_eb−2_ec)dt=0

The line voltage difference integraldunder accurate commutation yields a value of zero. When the commutation instants are delayed or advanced byα, the conduction period of VT₁ and VT₆ lasts for the intervalt∈[α/6+α,α/2+α], and the line voltage difference integraldis written as

d=_{∫tπ/6+αtπ/2+α}(_ubc−_uca)dt=_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθ=_dtrp+_doth

where_dtrp=_{∫π/6+απ/2+α}[_{∑n=1,4,7,…E2n+1}sin((2n+1)θ)+_{∑n=1,4,7,…E2n+1}sin((2n+1)(θ−2π/3)) −2_{∑n=1,4,7,…E2n+1}sin((2n+1)(θ+2π/3))]dθ,_doth=_{∫π/6+απ/2+α}[_{∑n=0,2,3,5,…E2n+1}sin((2n+1)θ)+_{∑n=0,2,3,5,…E2n+1}sin((2n+1)(θ−2π/3)) −2_{∑n=0,2,3,5,…E2n+1}sin((2n+1)(θ+2π/3))]dθ.

Since the third harmonics of the three three-phase back-EMFs are equal,_dtrp=0. Then, (8) is simplified as

_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθ=_doth

Solving and simplifying the definite integral gives

_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθ=3_{∫π/6+απ/2+α}[_{∑n=0,2,3,5,…E2n+1}sin((2n+1)(θ−π/3))]dθ =−3_{∑n=0,2,3,5,…}(_E2n+1/(2n+1))·cos((2n+1)(α+π/6)) +3_{∑n=0,2,3,5,…}(_E2n+1/(2n+1))·cos((2n+1)(α−π/6))

As the back-EMF harmonic coefficients satisfy the condition_E1/_E7≫1, the high-order terms of the back-EMF in (10) are negligible, and (10) can be simplified as

_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθ=−3_E1[cos(α+π/6)−cos(α−π/6)] −35_E5[cos(5α+5π/6)−cos(5α−5π/6)]

As seen in (11), the three-phase back-EMF integral_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθis a function of the commutation errorα. The relationship between the three-phase back-EMF integral and the commutation errorαwill be discussed in the following section.

Accurate commutation: when the commutation errorα=0 , the motor works under accurate commutation, as shown in Figure 3a, and the back-EMF integral_∫π/6π/2(_ea+_eb−2_ec)dθ=0.

Advanced commutation: when the commutation errorα∈[−π/6,0) , the motor works under advanced commutation, as shown in Figure 3b, and the three-phase back-EMF integral_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθ<0.

Delayed commutation: when the commutation errorα∈(0,π/6] , the motor works under delayed commutation, as shown in Figure 3c, and the three-phase back-EMF integral_{∫π/6+απ/2+α}(_ea+_eb−2_ec)dθ>0.

Similarly, when VT₃ and VT₄ are in conduction state, the phase current flows in opposite direction compared with the direction during the conduction period of VT₁ and VT_6. The three-phase back-EMF integral_{∫7π/63π/2}(_ea+_eb−2_ec)dθ=0under accurate commutation,_{∫7π/6+α3π/2+α}(_ea+_eb−2_ec)dθ<0under delayed commutation, and_{∫7π/6+α3π/2+α}(_ea+_eb−2_ec)dθ>0under advanced commutation.

From previous research on three-phase back-EMF integral and the line voltage difference integral expression in (7), when the freewheeling period_tfwis negligible, the line voltage difference integraldis proportional to the commutation errorαand can be used as an indicator of commutation errors.

The relationship between the conduction switch, the line voltage difference integral, and the commutation information is shown in Table 1; here,_tsand_teare the lower and upper limits of integral, respectively.

When the motor is in no-load state, the effect of the freewheeling period_tfw is negligible. Figure 4 shows the diagram of the line voltage difference integraldin the 60-electrical degree interval of A+B− and A−B+, where_TA+B−and_TA−B+represents the conduction period of phase A+B− and A−B+, respectively. The integrald in the A+B− conduction interval is zero under accurate commutation (Figure 4a), negative under advanced commutation (Figure 4b), and positive under delayed commutation (Figure 4c). In the A−B+ conduction interval, the integraldis opposite compared with that in the A+B− conduction interval. These results agree with the theoretical analysis.

2.3. Line Voltage Difference Integral Considering Freewheeling Period

During the freewheeling period_tfw, the line voltage difference is distorted by the DC-link voltage and clamped at the level of the DC-link voltage_udc. When line voltage difference integral is performed, the distorted line voltage difference in the interval_tfwis also integrated, which will cause the line voltage difference integral to deviate from the accurate line voltage difference integral. In this section, the line voltage difference integral of a BLDC motor considering the freewheeling period will be discussed.

Assume that the commutation error isαand the conduction interval of VT₁ and VT₆ lasts for the period whent∈[_tπ/6+α,_tπ/2+α] , as shown in Figure 2. In the conduction period, the line voltage difference integral^d∗can be expressed as

^d∗=_{∫tπ/6+αtπ/2+α}(_ubc−_uca)dt=_{∫tπ/6+αtπ/6+α+tfw}(_ubc−_uca)dt+_{∫tπ/6+α+tfwtπ/2+α}(_ubc−_uca)dt

Figure 5 shows the equivalent circuit when the commutation occurs from VT₅ to VT_1. In the commutation period_tfw, VT₅ is turned off, VT₁ is turned on, and VT₆ continues conducting. The three phase voltages are expressed as

{_ua=R_ia+Ld_iadt+_ea+_uN=_udcub=R_ib+Ld_ibdt+_eb+_uN=0_uc=R_ic+Ld_icdt+_ec+_uN=0

The line voltage difference meets the condition_ubc−_uca=_udc, and the line voltage difference integral during the freewheeling period_tfwis reformulated as

_{∫tπ/6+αtπ/6+α+tfw}(_ubc−_uca)dt=_{∫tπ/6+αtπ/6+α+tfwudc}dt

In (14), the line voltage difference(_uab−_ubc)is equal to the DC-link voltage_udcin the freewheeling period_tfw; hence, the line voltage difference integral shows a trend of linear growth in the freewheeling period_tfw.

Figure 6 shows the diagram of the line voltage difference(_ubc−_uca)and its integral^d∗ in the conduction period of phase A+B− and A−B+. The as seen in Figure 6, the line voltage difference(_ubc−_uca)is equal to the DC-link voltage_udcin the freewheeling period_tfwand the corresponding integral^d∗ linearly increases in the interval. Compared with that in Figure 4, since the disturbed voltage difference is integrated in the freewheeling period_tfw, the integral^d∗under different commutation produces an amplitude deviation. Especially under accurate commutation, for example, the integral^d∗deviates from zero, and becomes positive and negative in the conduction period of phase A+B− and A−B+, respectively.

The other voltage spikes in Figure 6 are analyzed as followed. When the commutation occurs, the outgoing phase current continues conducting through the anti-parallel diode, so the terminal voltage is clamped at the level of 0 V or_udc. When the motor commutates between different phases, the clamped voltage value of three-phase voltage changes, hence the line voltage difference(_ubc−_uca)will also change accordingly. The cause of the voltage spikes is the same under delayed, advanced, and accurate commutation. To facilitate the analysis, the voltage spike under the accurate commutation is taken as an example.

(1) When commutation occurs from A+B− to A+C−, it meets_ua=_udc,_ub=_udc, and_uc=0in the freewheeling period_tfw. Therefore,_ubc−_uca=2_udc, and it corresponds to the positive spike whose magnitude exceeds_udc.

(2) When commutation occurs from B+A− to C+A−, it meets_ua=0,_ub=0, and_uc=_udcin the freewheeling period_tfw. Therefore,_ubc−_uca=−2_udc, and it corresponds to the negative spike whose magnitude exceeds−_udc.

(3) When commutation occurs from A+C− to B+C−, it meets_ua=0,_ub=_udc, and_uc=0in the freewheeling period_tfw. Therefore,_ubc−_uca=_udc, and it corresponds to the positive spike whose magnitude is equal to_udc.

(4) When commutation occurs from C+A− to C+B−, it meets_ua=_udc,_ub=0, and_uc=_udcin the freewheeling period_tfw. Therefore,_ubc−_uca=−_udc, and it corresponds to the negative spike whose magnitude is equal to−_udc.

The voltage spikes of the line voltage difference(_ubc−_uca) in 360-degree electrical angle are listed in Table 2.

In the normal conduction period, whent∈[_tπ/6+α+_tfw,_tπ/2+α], the phase currents i_a and i_b conduct, while the outgoing phase current_icremains zero, i.e.,_ia=−_iband_ic=0. Substituting the current equation and (2) into the second item of (12), the line voltage difference integral_{∫tπ/6+α+tfwtπ/2+α}(_ubc−_uca)dtis rewritten as

_{∫tπ/6+α+tfwtπ/2+α}(_ubc−_uca)dt=_{∫tπ/6+α+tfwtπ/2+α}[R(_ia+_ib−2_ic)+Ld(_ia+_ib−2_ic)/dt+_ea+_eb−2_ec]dt =_{∫tπ/6+α+tfwtπ/2+α}(_ea+_eb−2_ec)dt

By decomposing (15) into a combination of two items, it is reformulated as

_{∫tπ/6+α+tfwtπ/2+α}(_ubc−_uca)dt=_{∫tπ/6+α+tfwtπ/2+α}(_ea+_eb−2_ec)dt=_{∫tπ/6+αtπ/2+α}(_ea+_eb−2_ec)dt−_{∫tπ/6+αtπ/6+α+tfw}(_ea+_eb−2_ec)dt

Substituting (14) and (16) into (12), and (12) is rewritten as

^d∗=_{∫tπ/6+αtπ/2+α}(_ubc−_uca)dt=_{∫tπ/6+αtπ/2+α}(_ea+_eb−2_ec)dt+_{∫tπ/6+αtπ/6+α+tfw}(_udc−_ea−_eb+2_ec)dt

The phase current satisfies_ia+_ib+_ic=0in the freewheeling period_tfw. Substituting the phase current relationship, (13) into the second item of (17), it is simplified as

_{∫tπ/6+αtπ/6+α+tfw}(_udc−_ea−_eb+2_ec)dt=−3_{∫tπ/6+αtπ/6+α+tfw}[R_ic+Ld_ic/dt]dt=−3R_{∫tπ/6+αtπ/6+α+tfwic}dt+3L_Ic

where_Icis the final value of the outgoing phase current before commutation.

During the commutation period_tfw, the outgoing phase current drops from the final value_Icto zero, the integral item_{∫tπ/6+αtπ/6+α+tfwic}dtcan be approximated by−_{Ic tfw}/2, and (18) is rewritten as

_{∫tπ/6+αtπ/6+α+tfw}(_udc−_ea−_eb+2_ec)dt=3R_{Ic tfw}/2+3L_Ic

From (13), the phase current_icin the freewheeling period_tfwis solved as

_ic(t)=−[β−(R_Ic+β)^e−Rt/L]R

whereβ=_udc+2_ec−_ea−_eb3.

From (20), the freewheeling period_tfwfor the outgoing phase current_icdropping from final value_Icto zero is solved as

_tfw=−LRlnββ+R_Ic

Substituting (21) into (19), and (19) is shown as

_{∫tπ/6+αtπ/6+α+tfw}(_udc−_ea−_eb+2_ec)dt=3L_Ic(1−lnββ+R_Ic)

In (22),_{∫tπ/6+αtπ/6+α+tfw}(_udc−_ea−_eb+2_ec)dtis related toβ. Next, the effect ofβwill be discussed.

In the freewheeling period whent∈[_tπ/6+α,_tπ/6+α+_tfw],(2_ec−_ea−_eb)is expressed as

2_ec−_ea−_eb=2_{∑n=0∞E2n+1}sin((2n+1)(θ+2π/3+α)) −_{∑n=0∞E2n+1}sin((2n+1)(θ−2π/3+α))−_{∑n=0∞E2n+1}sin((2n+1)θ+α)

Considering_E1≫_E7, the high-order terms of the back-EMF in (23) are negligible, and (23) can be simplified as

2_ec−_ea−_eb=2(_ec1+_e3+_ec5)−(_ea1+_e3+_ea5)−(_eb1+_e3+_eb5) =2_ec1+2_ec5−_ea1−_ea5−_eb1−_eb5 =2_E1sin(5π/6+α)+2_E5sin(π/6+5α)−_E1sin(π/6+α) −_E5sin(5π/6+5α)−_E1sin(α−π/2)−_E5sin(5α−π/2)

By calculating the trigonometric functions, (24) is simplified as

2_ec−_ea−_eb=32_E1cosα−332_E1sinα−32_E5cos(5α)−332_E5sin(5α)

Both sides of (25) are multiplied by1/_E5, and refined as

f(α)=2_ec−_ea−_ebE5=32·_E1E5·cosα−332·_E1E5·sinα−32cos(5α)−332sin(5α)

In general, the EMF harmonic coefficients of a BLDC motor with a trapezoidal back EMF [25]_E1/_E5∈[5,125]. The relationship betweenf(α),_E1/_E5, and the commutation errorαduring the freewheeling period_tfwcan be obtained in (26). When_E5>0,_E1/_E5∈[5,125] , and the corresponding 3-D graph is shown in Figure 7a. It is observed that with respect to_E1/_E5∈[5,125],f(α)is greater than zero when the commutation errorα∈[−π/6,π/6], so2_ec−_ea−_eb=f(α)·_E5>0. When_E5<0,_E1/_E5∈[−125,−5] , and the corresponding 3-D graph is shown in Figure 7b.f(α)is less than zero when the commutation errorα∈[−π/6,π/6], and2_ec−_ea−_eb=f(α)·_E5>0. Based on the relationship, it is derived thatβ=_udc+2_ec−_ea−_eb3>_udc3andR_Icβ<3R_Icudc. Both sides of3R_Icudcare multiplied by_Ic, and the item3R_Icudcis represented as3R_{Ic2udc Ic}=1.5_PcuPin, where_Pcudenotes copper loss and_Pcu=2_Ic2R._Pindenotes the input power of the motor and_Pin=_{udc Ic}. In general, the copper loss_Pcuis negligible, hence it satisfies3R_Icudc=3R_{Ic2udc Ic}=1.5_PcuPin≪1, which implieslnββ+R_Ic=ln11+R_Ic/β≈0Therefore, (19) is simplified as

_{∫tπ/6+αtπ/6+α+tfw}(_udc−_ea−_eb+2_ec)dt≈3L_Ic

Combining (27) and (17), the latter is reformulated as

^d∗=_{∫tπ/6+αtπ/2+α}(_ubc−_uca)dt=_{∫tπ/6+αtπ/2+α}(_ea+_eb−2_ec)dt+3L_Ic

According to the previous analysis, the general function of the line voltage difference integral^d∗is given as

^d∗=_∫tste(_uxz−_uzy)dt=_∫tste(_ex+_ey−2_ez)dt+3L_Iz

where the subscripts x and y represent the active phase, and z represents the inactive phase._tsand_teare the lower and upper limits of integral, respectively.

2.4. Commutation Error Compensation Strategy

In Section 2.2 and Section 2.3, the respective line voltage difference integralsdand^d∗without and with consideration of the freewheeling period_tfware discussed. Next, the commutation error compensation strategy is described based on the analysis of the line voltage difference integral.

From (29), the three-phase back-EMF integral is rewritten as

_∫tste(_ex+_ey−2_ez)dt=^d∗−3L_Iz

According to the research in Section 2.2, three-phase back-EMF integral can reflect commutation errors. In (30), three-phase back-EMF integral can be obtained from the combination of the integral^d∗and−3L_Izto determine commutation errors when the freewheeling period_tfwis considered.

In general, the commutation instants are obtained by delaying ZCPs by 30 electrical degrees in the sensorless technique and reactivating the motor windings according to the predefined commutation sequence. The predefined commutation sequences are shown in Table 3; here,_Hva,_Hvb, and_Hvcare virtual hall signals.

In Table 1, the sign of the three-phase back-EMF integral changes according to the direction of the phase current. The three-phase back-EMF integral_dccan be updated as

_dc=^{(−1)(_{Hva +Hvb +Hvc}+ρ)}·_∫tste(_ex+_ey−2_ez)dt

whereρdenotes the rotation direction of the motor rotor. Whenρ=0, the motor runs in the anti-clockwise rotation, and whenρ=1 , the motor rotates in the opposite one. When the sensorless BLDC motor runs, the windings are excited in according with the predefined commutation sequence listed in Table 3, and the conduction switches are obtained. Then the line voltage difference integral^d∗ is calculated with Table 1, and the outgoing phase current_Izbefore commutation is sampled. By using (30), the three-phase back-EMF integral_∫tste(_ex+_ey−2_ez)dtcan be given. Considering the rotation directionρof the motor rotor is known, the three-phase back-EMF integral_dcis converted to a positive value under delayed commutation and to a negative one under advanced commutation with (31), which also makes PI controller available.

Here, a PI regulator is also designed to compensate commutation errors, and the PI regulator is defined as

u(k)=_kp(e(k)−e(k−1))+_{ki Ti}e(k)+u(k−1)

where three-phase back-EMF integral errore(k)=_dref−_dc(k), and the reference value_dref=0._kpis the proportional coefficient, and_kiis the integral coefficient._Tiis the integral period and_Ti=0.0001 s.u(k)is the kth output value of the PI regulator,u(k−1)is the(k−1)th output value of the PI regulator. Here, k represents the number of iterations.

For the sensorless BLDC motor based on ZCP detection of the back-EMF, the accurate commutation instants are obtained by 30 electrical degrees delayed shift of ZCPs. Therefore, after the commutation error is determined, the commutation shift is updated as

θ(k)=θ(k−1)−u(k)

whereθ(0)=π/6.

Figure 8 shows the block diagram of a sensorless BLDC motor drive system and the proposed commutation error compensation strategy is shown in the shadowed area. The system contains a speed-loop controller, a current-loop controller, and a commutation error compensation link. In the commutation error compensation link, when commutation signals are detected, the outgoing phase current final value_Izis sampled and the line voltage difference is integrated until the commutation occurs again. Then, the line voltage difference integral^d∗is recorded, and the three-phase back-EMF integral_dcis calculated from the sampled current value_Izand the line voltage difference integral^d∗ . The entire compensation process is then resumed. A flowchart of the proposed method is shown in Figure 9.

Figure 10 shows the back-EMF waveforms by the constant speed test of 1200 rpm.

3. Experiment Results

An experimental prototype is set up to verify the feasibility and effectiveness of the proposed strategy. The experimental platform, which consists of the control system and the experimental motor, is shown in Figure 11. The BLDC motor is connected to the generator by a flexible coupling, and its rated parameters are shown in Table 4. The core processor is DSP (TMS320F28335) with a 150 MHz clock frequency, and the three-phase inverter is IPM PM50RL1A060 (Mitsubishi). The line voltage and phase current are sampled by three voltage sensors (LV25-P) and three current sensors (LTS15-NP), respectively. To remove the effect of switching noise, a low-pass FIR filter with the Hamming window function was designed. The cutoff frequency and the order of the filter were selected as 3 kHz and 50, respectively. The filter provides 65 dB of signal attenuation at the switching frequency. Many experiments were conducted to obtain the proportional coefficient_kpand the integral coefficient_kiof the PI regulator. Based on the test results, the proportional coefficient and integral coefficient are designed as_kp=8.16and_ki=0.023, respectively. The modulation method of the three-phase inverter is PWM-ON, and the switching frequency of three-phase inverter and the sampling frequency of the line voltage are 10 and 200 kHz, respectively. The experimental results were monitored and recorded by a digital oscilloscope DL750 (YOKOGAWA), and the line voltage difference integral was observed through a DAC.

Figure 12 shows the phase current_icand the line voltage difference integral^d∗under advanced, delayed, and accurate commutation. In the experiment, the motor runs at a speed of 500 rpm, the load torque is 7.5 Nm, and the commutation instants are set to be advanced and delayed by 10°. The conduction periods of A+B− and A−B+ are taken as examples, and the outgoing phase current final value_Icbefore commutation and the line voltage difference integral_∫tste(_uca−_uab)dt are marked in solid circles. In Figure 12a,b, there is an obvious current ripple in phase current_icand the line voltage difference integral^d∗ is different. Figure 12c shows that the current ripple is attenuated obviously and the integral^d∗also changes correspondingly when the commutation instants are accurate. In addition, as shown in the dashed circle, the line voltage difference integral^d∗takes on an approximately linear variation trend during the freewheeling period_tfw.The experimental results are consistent with the theoretical analysis.

The performance of the proposed strategy is tested in the steady and transient states. In the steady-state experiment, the motor runs at a fixed speed and load. In the transient experiment, the motor works at varying speeds and a fixed load. Figure 13 shows the phase currents and integral^d∗before and after compensation under different test conditions. It is observed that the phase current_icand the integral^d∗fluctuates obviously under delayed and advanced commutation, yet the current ripple is effectively reduced after the proposed compensation strategy is used, and the integral^d∗also changes correspondingly when the commutation instants are accurate. The experimental results show that the proposed strategy can accurately compensate commutation errors at different speeds and load torques.

Subsequently, the performance of the proposed strategy during the transient process is tested. In the experiment, the commutation instants are delayed by about 10°, and the motor is accelerated from 300 rpm to 1500 rpm at a load torque of 12 Nm. Figure 14 shows the phase current, the line voltage difference integral, and their enlarged vision with the proposed strategy. When the motor speed ramps up, the compensation strategy is initiated at the points S₁ and S₂, and the commutation error is almost removed at the points T₁ and T₂, respectively. It is seen that, after the compensation strategy is activated, the current fluctuation is effectively reduced. Therefore, the proposed compensation strategy exhibits satisfactory dynamic performance during variable-speed operation.

To test the convergence rate of the proposed strategy under the steady speed operation, the commutation error is set to about 10°, and the motor runs at a speed of 800 rpm and load torque of 12 Nm. Figure 15 shows the phase current, the line voltage difference integral^d∗, and their enlarged vision with PI controller. It is found that, before the point of S_a, the phase current_iccurls up and the integral^d∗is high, while the compensation strategy with PI controller is activated at the point S_a, the current ripple and the integral^d∗ gradually decreases. After 1.05 s, the current fluctuation is effectively suppressed, and the commutation error is almost eliminated. In addition, the convergence rate of the proposed method is tested at different speeds, and the test results are shown in Table 5, and the results show the proposed strategy converges quickly in a wide speed range.

Moreover, the power consumption is reduced after the commutation instants are compensated with the proposed strategy. In these power consumption tests, the commutation instants are set to be delayed and advanced by 10°and 12°, respectively, the speed range is 300–1500 rpm and the load torques_TL are about 11, 12, 13 and 15 Nm. Figure 16 shows the power consumption of the BLDC motor before and after compensation after steady operation for 40 min. The experimental results in Figure 16 show that the proposed strategy can reduce power consumption over wide speed ranges, especially in the high-speed ranges.

4. Conclusions This paper presents a novel compensation strategy for commutation error of sensorless BLDC motors. The method is suitable in both the steady-speed and variable-speed range. The relationship between the line voltage difference integral in 60 electrical degrees interval and the commutation error is established. A commutation error commutation strategy with respect of the line voltage difference integral is proposed to correct the inaccurate commutation instants by regulating the three-phase back-EMF integral to zero. In this method, the influence of the voltage pulses caused by the diode-freewheeling conduction on the line voltage difference integral is analyzed, making the proposed strategy applicable to BLDC motors considering the freewheeling period. Furthermore, a PI controller is also designed to realize closed-loop compensation for commutation errors. Finally, experimental results show the effectiveness and feasibility of the proposed strategy under different test conditions.

Figure 1. Equivalent circuit of a brushless DC (BLDC) motor drive system.

Figure 2. Phase diagram of the back electromotive forces (EMFs) and phase current.

Figure 3. Three-phase back-EMF. (a) Accurate commutation; (b) Advanced commutation; (c) Delayed commutation.

Figure 4. Line voltage difference(ubc-uca)and integraldin the conduction period of phase A+B- and A-B+. (a) Accurate commutation; (b) Advanced commutation; (c) Delayed commutation.

Figure 4. Line voltage difference(ubc-uca)and integraldin the conduction period of phase A+B- and A-B+. (a) Accurate commutation; (b) Advanced commutation; (c) Delayed commutation.

Figure 5. Equivalent circuit when commutation occurs from VT5 to VT1.

Figure 6. The line voltage difference(ubc-uca)and corresponding integrald∗in the conduction period of phase A+B- and A-B+. (a) Accurate commutation; (b) Advanced commutation; (c) Delayed commutation.

Figure 7. Relationships betweenf(α),E1/E5, and the commutation errorαduring the freewheeling periodtfw. (a)E5>0; (b)E5<0.

Figure 7. Relationships betweenf(α),E1/E5, and the commutation errorαduring the freewheeling periodtfw. (a)E5>0; (b)E5<0.

Figure 8. Sensorless BLDC motor drive system with the proposed strategy.

Figure 9. Flowchart of the commutation error compensation procedure.

Figure 10. Measured back-EMF waveforms of the BLDC motor at 1200 rpm.

Figure 11. Experimental setup of the BLDC motor drive system.

Figure 12. Phase current and the line voltage difference integral. (a) Advanced commutation; (b) Delayed commutation; (c) Exact commutation.

Figure 13. Current waveform of phase C before and after compensation. (a) Commutation instants are delayed by 10° at a speed of 1000 rpm and load torque of 12 Nm; (b) Commutation instants are delayed by 12° at a speed of 1500 rpm and load torque of 16 Nm; (c) Commutation instants are advanced by 12° at a speed of 850 rpm and load torque of 10 Nm; (d) Commutation instants are advanced by 14° at a speed of 1200 rpm and load torque of 14 Nm.

Figure 13. Current waveform of phase C before and after compensation. (a) Commutation instants are delayed by 10° at a speed of 1000 rpm and load torque of 12 Nm; (b) Commutation instants are delayed by 12° at a speed of 1500 rpm and load torque of 16 Nm; (c) Commutation instants are advanced by 12° at a speed of 850 rpm and load torque of 10 Nm; (d) Commutation instants are advanced by 14° at a speed of 1200 rpm and load torque of 14 Nm.

Figure 14. Transient performance and enlarged view with the proposed strategy when the motor accelerates from 300 to 1500 rpm.

Figure 15. Comparative results of the phase current, the integral of the line voltage difference, and their enlarged view at a speed of 800 rpm and load torque of 12 Nm.

Figure 16. Power consumption comparison before and after compensation. (a) Under delayed commutation; (b) Under advanced commutation.

Conduction Switch	d	Sign of d	Advanced/Delayed
VT1 -VT6	∫tste(ubc−uca)dt	>0	Delayed
		<0	Advanced
VT1 -VT2	∫tste(uab−ubc)dt	<0	Delayed
		>0	Advanced
VT3 -VT2	∫tste(uba−uac)dt	>0	Delayed
		<0	Advanced
VT3 -VT4	∫tste(ubc−uca)dt	<0	Delayed
		>0	Advanced
VT5 -VT4	∫tste(uab−ubc)dt	>0	Delayed
		<0	Advanced
VT5 -VT6	∫tste(uba−uac)dt	<0	Delayed
		>0	Advanced

Commutation Case	ua	ub	uc	ubc−uca
C+A− → C+B−	udc	0	udc	−udc
C+B− → A+B−	udc	0	0	udc
A+B− → A+C−	udc	udc	0	2udc
A+C− → B+C−	0	udc	0	udc
B+C− → B+A−	0	udc	udc	−udc
B+A− → C+A−	0	0	udc	−2udc

Hva, Hvb, Hvc	001	101	100	110	010	011
Direction	Clockwise
Conduction switch	VT3 -VT2	VT3 -VT4	VT5 -VT4	VT5 -VT6	VT1 -VT6	VT1 -VT2
Direction	Anti-clockwise
Conduction switch	VT5 -VT6	VT1 -VT6	VT1 -VT2	VT3 -VT2	VT3 -VT4	VT5 -VT4

Motor Parameter	Value	Unit
Rated voltage	200	V
Rated power	3.15	kW
Rated speed	1500	rpm
Rated torque	20	Nm
Pole pairs	4
Phase inductance	1.234	mH
Phase resistance	0.0654	Ω
Back-EMF coefficient	0.528	V/(rad/s)

Speed (rpm)	300	500	800	1200	1500
Convergence rate (s)	2.52	1.59	1.05	0.713	0.565

Author Contributions

This paper is the result of the hard work of all of the authors. X.Y. and J.Z. conceived and designed the proposed method. X.Y. and J.Z. conceived and analyzed the data. X.Y. provided funding support. J.Z. and G.L. conducted the experiment and verified the proposed method. J.Z., G.L., and H.L. wrote the paper, and H.L. and J.W. contributed to the review of the document. All authors gave advice for the manuscript.

Funding

This work was supported by the High-tech Ship Project of China under Grant No. GJYF-043/6.

Conflicts of Interest

The authors declare no conflict of interest.

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