Digital fixed-frequency hysteresis current control of a BLDC motor applied for aerospace electrically powered actuators

Jin FU,Yongling FU,Liming YU,Jinwen GUO,*,Rongrong YANG,Mingkng WANG

aSchool of Mechanical Engineering and Automation,Beihang University,Beijing 100083,China

bFlying College,Beihang University,Beijing 100083,China

1.Introduction

Safer,cheaper,and greener technologies are important initiatives for the future of air transport.In response to these needs,the aerospace industry is searching for an innovation(incremental or disruptive)in safety-critical actuation systems.1Recently,significant interest is given toward ‘more electric air-craft”even ‘a(chǎn)ll electric aircraft”.2,3The trend is to increase the usage of power-by-wire electrical actuators:electro-hydrostatic actuators and electro-mechanical actuators.4These actuators are projected to replace the conventional hydraulic servo actuators.From the power-to-weight ratio viewpoint,the high performance(significant efficiency and torque/power density)5and maturity improvements of the robust design for permanent magnet BrushLess Direct Current(BLDC)motor and their power drive electronics make electrically supplied actuators increasingly attractive.6,7However,alternative solutions need to be studied for BLDC motor control to ensure the required dynamic performance of an actuator under a wide range of in- flight maneuvers.The current loop is an innermost control loop in the control of a motor drive electronics,and its control strategy has a dramatic effect on the performance of a motor-driven device.8Thus,the study of a highperformance current control strategy has been an issue amid studies of motor control.9

In practical applications,the triangular-carrier-based Pulse Width Modulation(PWM)is adopted as a common method in current modulation,and a certain upper controller,such as a PID controller,a sliding mode controller,a predictive controller,and so on,is used to control the current quantitatively.10–13In these current control strategies,the dynamic response of the current controller can be effectively improved through increasing the frequencies of current sampling and PWM carrier.However,because the current is not controlled directly and an updating delay of the PWM carrier duty cycle exists,the actual winding current can easily exceed its upper limit confined by the power device capability due to an extremely high rising rate of current.Efforts have been done to solve this problem.For instance,Zhang et al.proposed a feed-forward control method combined with a dual current sampling and dual PWM duty ratio update scheme in Ref.14to decrease the time delay caused by a digital control system.In Ref.15,twofold of current sampling and duty cycle updating were performed in a single carrier period,which decreased the updating delay to half of that in traditional PWM.Nevertheless,the disadvantage of these PWM-based current control strategies has not been overcome fundamentally.

Hysteresis Current Control(HCC)16is an alternative to achieve high-performance current control.In HCC,the control target is the bus current instead of the bus voltage,which overcomes the inherent disadvantage of PWM-based current control strategies.In addition,the simple structure,rapid response speed,inherent overcurrent protection function,lower switching losses,and excellent control stability of this strategy constitute the merits of HCC.17,18When HCC is applied to a current control loop,stability can also be achieved,although the outer speed control loop has a high control gain.Hence,HCC is fairly suitable for applications with high dynamical response.However,in traditional HCC,the switching frequency of a power device(namely,the modulating frequency)varies with a change of the system working state,which causes a severe electromagnetic compatibility problem and may result in an excessive switching frequency that will damage the power device;this disadvantage hinders the spread and application of HCC.19,20

Extensive work has been conducted to overcome the disadvantages of HCC.The use of a flexible hysteresis band size is a common way to achieve a fixed modulating frequency.21–27.In these HCCs,the hysteresis band size is calculated in each switching period in real time according to transient system state variables.However,this calculation is computationally expensive and suffers from stability problems.25Meanwhile,other ways exist.In Ref.28,to realize a fixed-frequency modulation,the hysteresis band is removed,and the switch-on and switch-off times are determined using the predicted reference current,system behavior,and past time within a predefined switching period.Comparing the real-time current with the hysteresis bounds to determine the switching signal at a fixed frequency is another way to reach the target,29and if the hysteresis band size is set as zero,this current control strategy becomes similar to the bang-bang control.30Nevertheless,this strategy has the same problem concerning the sampling interval as that in PWM-based current control strategies.The triangular carrier-based fixed-frequency HCC strategy is a new way to obtain the fixed modulating frequency.31,32In this strategy,the triangular carrier technology and HCC are combined,and by tuning the amplitude of the carrier and the hysteresis band size,only a pair of switching signals is guaranteed to occur during a carrier period.However,strict constraints on the rising and descending rates of winding current are observed,and these constraints are usually ineffective in the unsteady state.

In this paper,a flexible-bound-size quasi- fixed-frequency HCC based on the triangular carrier-based fixed-frequency HCC is proposed to tackle the problem of traditional HCC.The digitalization of the proposed HCC is conducted to expand its feasible range to the entire running process.This paper is organized as follows.In the second section,the mathematical model of a BLDC motor using a PWM_ON modulation mode is constructed.In the third section,the triangular carrier-based fixed-frequency HCC is studied,and the new method is applied to analyze constraints to realize a fixed frequency modulation control.In the fourth section,based on the results obtained in the third section,a new flexiblebound-size fixed-frequency HCC feasible during the entire running process is proposed,and the process of obtaining the flexible bound size is analyzed.In the fifth section,simulations and experiments are conducted to verify the effectiveness of the newly proposed digital fixed-frequency HCC.Finally,conclusions are made in the last section.

2.Modelling of a BLDC motor

In this paper,a Y-shaped BLDC motor is used as the research target.This motor is driven by a three-phase inverter controlled using six steps for commutation,in which only two phases conduct current at any time;for each phase,the conducting interval is 120 electrical degrees.For the modulation of conducting current,a PWM_ON modulation mode is used to produce switching signals for power electronic devices.Specifically,assumptions are made as follows:

(1)The three-phase windings are perfectly symmetrical.

(2)The back-Electro Motive Force(back-EMF)is a trapezoidal wave of 120 electrical degrees.

(3)The stator core is unsaturated.

In addition,the armature reaction,eddy current,hysteresis loss,and cogging effect are neglected.The simplified circuit diagram of the BLDC motor and the three-arm power drive electronics(each arm consists a transistor and an antiparallel diode)are presented in Fig.1,where Usand Isare the supply voltage and current,respectively.

Then,the three phase winding voltages can be expressed as33

where u is the phase voltage;R is the equivalent phase winding resistance comprising the winding resistance and the equivalent resistance of the power components;i is the phase current;L is the equivalent inductance of phase winding and equal to the self-inductance of phase winding minus the mutual inductance;e is the back-EMF;the subscripts a,b,and c indicate the three phase windings respectively;UNis the center voltage of the three phase windings.

Under the assumptions mentioned above and without considering the commutation process,when the PWM signal is ON,that is,two phase windings are in the conducting state,the equivalent circuit of the motor can be simplified as in Fig.2.

Then,Eq.(1)can be reduced to

where U is the bus voltage,and E is the magnitude of the back-EMF.In this paper,the parameters used in the simulation(listed in Table 1)are consistent with those in the experiment.In addition,the modulation frequency of current control is 20 kHz.

3.Expansion of the triangular carrier-based fixed-frequency HCC

In the traditional HCC,the modulating frequency varies with the hysteresis band size ΔI,the rising rate of the winding current tanθ (where θ is the rising angle of the winding current),and the descending rate of the winding current tanα (where α is the descending angle of the winding current),which may generate an undesirable current harmonics and electromagnetic compatibility problem.The current ripple is presented as IRipa triangular carrier is introduced into the HCC to address this problem and obtain a fixed modulating frequency,31and the schematic of the mechanism of this improved HCC is shown in Fig.3.

In this HCC,the reference current IRAis obtained by superimposing a triangular-waveform current Itrwith a fixed frequency to the original reference current Iref.Then,the same procedures as in the traditional HCC are applied to determine the switching states of power devices.By carefully tuning the amplitude of Itr(marked as Atr)and the hysteresis band size ΔI,a fixed-frequency modulation is obtained,and the frequency is the same as that of Itr.In addition,IArepresents actual current,IUindicates the upper bound of the current,while ILindicates the lower bound of the current.

Fig.1 Equivalent circuit diagram of a BLDC motor system.

Fig.2 Simplified equivalent circuit diagram of a BLDC motor system in the two-phase conduction state.

Table 1 Parameters of target motor.

Fig.3 Mechanism of the triangular carrier-based hybrid HCC.

However,as stated in Ref.31,constraints exist for this hybrid fixed-frequency HCC to achieve a constant modulating frequency.In details,tanθ and tanα should be located in two separate parameter-dependent regions respectively.From Fig.4,the two pairs of bounds can be determined as

Fig.4 Fixed-frequency modulation in the triangular carrierbased fixed-frequency HCC.

where θmaxand θminare the upper and lower limits for the rising angle of current,respectively;αmaxand αminare the upper and lower limits for the descending angle of current,respectively;Ttrand Atrare the period and amplitude of the triangular waveform,respectively.

In fact,the bounds of tan θ and tanα given in Eq.(3)are too restricted with given ΔI and Atrbecause these bounds are determined separately.Furthermore,if it is only required that the modulating frequency is fixed during the steady state,the restrictions upon tan θ and tanα can be relaxed by considering the rising and descending processes of the current integratedly.Moreover,we call this hybrid fixed-frequency HCC strategy with relaxed restrictions the expanded quasi- fixed-frequency HCC.Considering that the constraints for the condition that tan θmin≤ tan θ ≤ tan θmaxhave been given in Eq.(3),to expand the bounds of tan θ with a given combination of ΔI and Atr,restricting tanα is an alternative to make the combination of ΔI and Atradaptive for the condition that tan θ ＞ tan θmax.

In Fig.5,given that tan θ ＞ tan θmax,when the motor works in the steady state,the curves of the actual current IAand ILintersect periodically,and both of the lines BC and EF are parallel to the time axis.In this situation,if a fixed modulating frequency is required,the descending rate of the current must be in the region(tan αmin2,tan αmax2).

By analyzing triangles ABC and DEF in Fig.5,we can get the constraint relationship between Atr,Ttr,and ΔI to obtain the fixed modulating frequency as follows:

These constraints for the condition that tan θmin≤ tan θ ≤tan θmaxgiven in Eq.(3)can also be relaxed when the rising and descending processes of the current are analyzed integratedly.Without loss of generality,take θ as shown in Fig.6.Apply the same analyzing procedure as that used for the condition that tan θ ＞ tan θmax,we get the constraint on tanα as follows:

Fig.5 Fixed-frequency modulation in the expanded quasi- fixed frequency HCC when tan θ ＞ tan θmax.

Fig.6 Fixed-frequency modulation in the expanded quasi- fixedfrequency HCC when tan θmin ≤ tan θ ≤ tan θmax.

Then,the constraints on tan α for tan θ ＞ tan θminby combining Eqs.(4)and(5)can be expressed as

Notably,when tan θ =tan θmax,we obtain

Meanwhile,both of the two expressions on the right side of Eq.(7)decrease monotonously with tan θ.Thus,F(tan θ)decreases continuously and monotonously with tan θ.

From Figs.4 and 6,we can see that,with given ΔI and Atr,the permitted range of tanα is expanded using the analyzing procedure proposed in this study compared with that in Ref.31 on the condition that tan θmin≤ tan θ ≤ tan θmax.Simultaneously,the geometric upper limit on tan θ disappears,and if Eqs.(6)and(7)are satisfied,tan θ can be any value higher than tan θminand within the electrical capability of the motor system.We can select the values of ΔI and Atrusing Eqs.(6)and(7).However,larger values of ΔI and Atrmay increase the steady-state error of the current;thus,smaller values of ΔI and Atrare preferred.

4.Design of the flexible-band-size fixed-frequency HCC

As is known,tan θ and tan α vary with a change of the rotating speed of the motor.Consequently,Eqs.(6)and(7)may fail to hold as the rotating speed changes with fixed ΔI and Atr;thus,the modulating frequency varies.Hence,a scheme to determine the combination of ΔI and Atrflexibly is required to accommodate the variations of tan θ and tan α to obtain a fixed modulation frequency.In this study,particularly,Atris fixed and a flexible ΔI is used to reach this target.

4.1.Methodology

Initially,we should know the expressions for tan θ and tan α.We can obtain the following by solving Eq.(2):

where I0is the initial value of the phase current.Similarly,when the PWM signal is OFF,the phase current can be expressed as

Differentiating Eqs.(9)and(10)with respect to t and considering that I0R and Rt/L can be taken as zero,we get

and

Considering that Ttrtanθ-2ΔI-4Atr＞0,?tanθ＞tanθmax,and Ttrtanθ-2ΔI-4Atr＞0 is not required for tan θmin≤ tan θ ≤ tan θmax,by substituting Eqs.(11)and(12)into Eqs.(6)and(7),respectively,we obtain the bounds for ΔI as follows:

Notably,the values of U and L are known,and E is determined by the rotating speed of the motor;then the values of tan θ and tan α can be obtained from Eqs.(11)and(12).In addition,Ttris predetermined,and the bounds of ΔI will be determined if Atris given.Moreover,a fixed-frequency modulation control can be obtained by selecting a suitable ΔI dynamically based on Eqs.(13)and(14)according to the transient E.In addition,only the condition in the steady state is considered here.Thus,we call this newly proposed strategy the ‘ flexible-bound-size quasi- fixed-frequency HCC”.

4.2.Design of the digital fixed-frequency HCC

The modulating frequency can remain constant using the newly proposed HCC in the steady state.Nevertheless,the frequency may change in the unsteady state and even exceed the permitted switching frequency of a power device.In addition,in applications with a fixed ΔI,the variable E also leads to a variety of modulating frequency.For example,as shown in Fig.7,EM represents the effective model with a fixed ΔI of 3 A and an Atrof 7 A,the feasible region of E is approximately between 10 V and 122 V.However,in regions from 0 V to 10 V and 122 V to 135 V,a fixed-frequency control cannot be obtained using the traditional triangular carrier-based fixedfrequency HCC.Fortunately,the application of digital control technology makes solving these problems easy.

Fig.7 Curves of tan θ,tan α,and their limits relative to E in the flexible-bound-size quasi- fixed-frequency HCC.

To realize the modulating process with a fixed switching frequency in all work states,digital rules are applied to the expanded quasi- fixed-frequency HCC as follows:

Rule 1.During a period of the triangular carrier waveform,only one switch-on signal is permitted.

Rule 2.At the peak of the triangular carrier waveform,if the actual current is lower than the corresponding upper limit of ΔI,the power devices switch to on-state.

Rule 3.At the trough of the triangular carrier waveform,if the actual current is higher than the corresponding lower limit of the hysteresis band,the power devices switch to off-state.

Rule 4.If the actual current remains exceeding the upper limit of the hysteresis band through a period of the triangular carrier waveform,or the motor works in the second or fourth quadrant,switch to the HPWM_LPWM modulating mode.

Fig.8 Examples of digital rules.

The explanations of Rule 1 and 2 are provided in Fig.8(a).As Rule 1 states,in the time interval[t0,t2)which represents a period of the triangular carrier waveform,power devices can switch to on-state only once;as Rule 2 states,at the peak time of the triangular carrier waveform t1,because the actual current is lower than the corresponding value in the IUwaveform at t1,a switch-on signal is given.Meanwhile,Rule 3 is explained in Fig.8(b),where a switch-off signal is obtained because the actual current is higher than the corresponding value in the ILwaveform at t0.

In fact,the applications of Rule 1 to 4 aim to guarantee a fixed modulating frequency when the constraints derived from the geometrical analysis are unsatisfied or the motor works in the unsteady state.Rule 1 guarantees that the switching frequency of the power devices cannot exceed the predetermined fixed modulating frequency,whereas Rule 2 and 3 prevent the switch from being missed during a period of the triangular carrier waveform due to a too-small tan α and a too-small tan θ,respectively.Rule 4 speeds up the descending process of the winding current to help the track of the actual current hit the lower limit curve of the hysteresis band.

4.3.Determination of ΔI

In an engineering scenario,determining ΔI should consider the resultant current error IErr,the current ripple IRip,and the difficulty and stability as well.Eqs.(13)and(14)provide a way to determine ΔI.

Fig.9 shows the limits for ΔI with respect to E when Atris 7A,whereas the zones filled with 45°dashed lines(Z1)and 135°dashed lines(Z2)comprise the feasible region for ΔI.In this figure,Z1is determined by Eq.(13),in which tan θ ＞ tan θmax;Z2is determined by Eq.(14),in which tan θmin≤ tan θ ≤ tan θmax.There are a lot of strategies to select ΔI from Z1and Z2,for example,we can simply set ΔI as the midpoint of the feasible range at the transient E,that is,ΔI=[min(ΔImin1,ΔImin2)+max((ΔImax1, ΔImax2)]/2.By using a suitable ΔI selected from Z1and Z2according to the transient E,a fixed-frequency switching control can be obtained throughout the whole scope of E,which greatly expands the effective range of the traditional triangular carrier-based fixed-frequency HCC.

In addition,considering that simplification is made in the deductions of Eqs.(13)and(14)and some other constraints are disregarded,the limits for ΔI given in Eqs.(13)and(14)are approximate results.Consequently,if the feasible range of ΔI is too narrow or the selected ΔI is too close to the boundary,the stability to realize a fixed-frequency modulation control will become worse,and even some tiny disturbances may lead to a failure of fixed-frequency modulation control.We call this phenomenon the boundary effect.Notably,when E becomes close to its limits,the boundary effect becomes worse.Digital Rule 2 and 3 help relieve the boundary effect because the feasible range of ΔI is dramatically enlarged,and each ΔI above the bottom boundaries of Z1and Z2is feasible.Moreover,a fixed-frequency modulation can be realized throughout the whole range of E even with a constant ΔI.

Fig.9 Curves of ΔI relative to E in the flexible-band-size quasifixed-frequency HCC.

Meanwhile,IErr,IRip,and the efficiency are important issues for ΔI determination.Here,IErris defined as the difference between Irefand the midpoint of the maximum and minimum of IA,while IRipis defined as the difference between the maximum and the minimum of IA.For the efficiency analysis,the copper loss is considered,which is the major effect of the motor losses and cannot be neglected,and it is defined as

where η is the motor efficiency,TIis the period to input Irefand equal to 1 ms,and Iavgis the average of the current during TIand equal to Irefplus IErr.Fig.10 shows the influences of ΔI on IErr,IRip,and η when Atris 7 A and E is 20 V.In Fig.10,every feasible ΔI at the left side of point B locates in Z1or Z2(in Fig.9),and the other feasible ΔIs locate above the upper bounds of Z1and Z2.In the former condition,IErris constant,while in the latter condition,IErrincreases linearly with an increase of ΔI because only IUis hit right now.In addition,the selection of ΔI has no impact on IRip.Consequently,according to Eq.(15),ΔI influences η only through IErr,and their relationship is shown in Fig.10.

4.4.Influence of Atron determining ΔI

In Section 4.3,Atris fixed as 7 A,but the selection of Atrhas a great influence on that of ΔI.Fig.11(a)and(b)provide the limits for ΔI relative to E in the flexible-bound-size quasifixed-frequency HCC when Atris 3 A and 12 A,respectively.Comparing Figs.11 and 9,it can be seen that the feasible range of ΔI at the transient E becomes wider with an increase of Atr,which means a larger Atrcan help relieve the boundary effect.Nevertheless,as shown in Fig.12 by using the digital fixedfrequency HCC where ΔI is 7 A and E is 20 V,a larger Atrbrings about a larger IErrand a lower η.Therefore,a smaller Atris preferable.Moreover,when we select Atr,some compromise has to be made,and the selection depends on the requirement of control performance.

4.5.Discussions about IErrand IRip

Fig.10 Curves of IErr,IRip,and η relative to ΔI in the digital if xed-frequency HCC.

Fig.11 Limits for ΔI relative to E with different Atr.

Fig.12 IErr,IRip,and η relative to Atr.

In this digital fixed-frequency HCC,due to the existence of Atrand the use of digital Rules 2 and 3,the maximum and minimum of IAare not symmetrical about IRef,and a none-zero IErris inevitably produced.Although IErrcannot be eliminated,it can be lowered by reducing ΔI or Atraccording to Figs.10 and 12.However,this will lower the stability and feasibility of fixed-frequency modulation.In practice,to obtain a low IErr,ΔI should be selected within Z1or Z2,and Atrshould be as small as possible on the condition that the stability requirement of fixed-frequency modulation is satisfied.

As for IRip,it is inevitable to result in a larger current ripple using the triangular carrier-based fixed-frequency HCC than that using the traditional HCC,and from Fig.3 as well as Eqs.(11)and(12),we can get the expression for IRipin the steady state as follows:

where T is the period of the current ripple.It can be seen that IRipis determined by the transient E and Ttr,and when E=U/4,the maximum magnitude is obtained.Fig.13 gives a visualized version of Eq.(16).Here,both ΔI and Atrare 7 A.According to Fig.13,if E changes suddenly,for example,when the BLDC motor is connected to a huge load abruptly,IErrwill change greatly,and IRipwill increase sharply.This situation will become severer if digital Rule 2 or 3 is triggered,because on that condition,only one bound is hit,and IErrchanges more quickly(corresponding to segments AB and CD in Fig.13)than that in the normal control state(corresponding to segment BC in Fig.13).Consequently,the resultant IRipcan be even higher than that in the traditional HCC using the same ΔI.Nevertheless,this situation happens rarely during the whole running process of the motor,and normally,E changes smoothly.In practice,the rise of IRipmainly results from the variety of Iref.Here,IRipcan be divided into two parts,one,denoted by IRipI,resulting from the variety of Irefand whose magnitude is the same as the change of Iref,and the other,denoted by IRipS,resulting from the current control strategy.If IRipIis removed,the magnitude of IRip(actually IRipS)is close to that in the steady state.In addition,considering that the frequency of IRipIis the same as that of Irefand generally smaller than 1 kHz,the duration of the unsteady state is normally shorter than 10%of the whole current control process.As a result,the energy loss increase owing to an increase of IRipin the unsteady state can be ignored.For example,when E is 10 V,Atris 7 A,ΔI is 7 A,and Irefincreases from 20 A to 40 A at tS,it takes 1.5Ttrs for the current control process to get steady.During the first modulation period after tS,IRipis about 25.20 A,and then IRipSis 5.20 A.After that,IRipis 4.63 A,and the change of Irefjust results in a 0.57 A rise of IRip.As for average efficiency,the value from tSto tSplus TIis 89.12%,the same as that without the unsteady modulation periods.

Compared with the traditional triangular carrier-based fixed-frequency HCC,IRipis lowered using the newly proposed HCC.In this digital fixed-frequency HCC,Rule 2 or 3 is triggered to obtain the required modulation frequency,and IRipis determined by Eq.(16).Nevertheless,in the traditional triangular carrier-based fixed-frequency HCC,the curves of IUand ILhave to be hit alternatively,and Ttrincreases multiply.Consequently,according to Eq.(16),the magnitude of IRipis times of that in the newly proposed digital fixed-frequency HCC.Moreover,according to Eq.(15),if their Iavg’s are the same,the newly proposed digital fixed-frequency HCC is more energy-efficient compared to the traditional triangular carrierbased fixed-frequency HCC.

5.Simulation and experimental results

Fig.13 IErr,IRip,and η relative to the E.

5.1.Simulative effectiveness validation

A simulative model to verify the effectiveness of the proposed digital fixed-frequency HCC is constructed based on the MATLAB/Simulink software package.Considering that the timescale at which E evolves is much larger than the switch period,we take E and ΔI as constant values in the interested time interval to reveal the temporal evolution of the current and the switching state.The related parameters of the target motor are listed in Table 1.Atris selected with the same value as that used in Fig.9,and both ΔI and Atrare 7 A,while different E’s are used to verify the effectiveness of the proposed the digital fixed-frequency HCC strategy in different conditions.Simulation results are presented in Figs.14–17.

Fig.14 shows an example in which Rule 2 is triggered.In this figure,the constraint for the modulating process at a fixed frequency cannot be satisfied because of a too high tan θ.In Fig.14,executing Rule 2 results in a large deviation of the mean value of the actual current from the original reference current,namely,a large steady-state error of current,and this steady-state error can be higher than Atr,though this error can be reduced by lowering ΔI or Atr.Furthermore,a low amplitude of the current ripple is obtained because the time interval given for the rising of the actual current is very short.

Fig.15 represents the scene where the normal modulating process of the expanded quasi- fixed-frequency HCC runs.Compared with the curves illustrated in Fig.14,the steadystate error of current is much lower,but the amplitude of the current ripple is much higher.Figs.16 and 17 depict examples of step and sinusoidal responses of the digital fixed frequency HCC,respectively.In Fig.17,when an amplitude of 10 A,a frequency of 1 kHz,and a mean value of 25 A(Iref)was applied,the frequency response of the current can reach as high as 1 kHz.

In Figs.14–17,using the digital fixed-frequency HCC can realize the fixed-frequency modulating process,although the constraints on tanθ and tanα derived from geometrical analysis cannot be satisfied.Furthermore,comparing the fixed frequency modulation control patterns realized in Figs.14–17 with those predicted in Fig.9 with the same parameters,we can find that they are coincident,which verifies the correctness of Eqs.(13)and(14).

5.2.Simulative comparison with other HCC strategies

The subsection above verifies the ability of the newly proposed HCC to realize the current control process with a fixed switching frequency.Moreover,owing to the digital rules,this newly proposed HCC has a great advantage over traditional HCCs,especially on conditions that tan θ or tan α is very large or very small when it is hard to realize a fix-frequency modulation.To demonstrate the advantage of the newly proposed HCC,Fig.18 gives simulative results when an amplitude of 5A,a frequency of 1 kHz,and a mean value of 25 A(Iref)was applied and using different HCCs with when Eqs.(6)and(7)cannot be satisfied due to a high tan θ.Here,both Atrand ΔI are 5 A and E is 6 V.

Fig.14 Simulative current response to a constant Irefof 30 A(E=10 V).

Fig.15 Simulative current response to a constant Irefof 40 A(E=60 V).

Fig.16 Simulative current response to a step Ireffrom 20 A to 40 A(E=10 V).

Fig. 17 Simulative current response to a sinusoidal Iref(E=30 V).

Obviously,we can see that a fix-frequency modulation cannot be realized using the traditional HCC or the triangular carrier-based fixed-frequency HCC;meanwhile,a fixed switching frequency is perfectly realized using the newly proposed HCC.Considering that the traditional HCC was not designed for fix-frequency modulation,failing to do that is inevitable.Regarding the reason for the failure of the triangular carrierbased fixed-frequency HCC,it is because tanα is too small and the actual current cannot drop quickly enough to reach the lower band during the left part of a period,and then the off state keeps for more than one period,which results in a varying switching frequency.On the other hand,when the newly proposed HCC is used,owing to the digital rules,the power device is forced on even though the actual current doesn’t hit the lower band to avoid missing the expected onstate switch,and a fixed-frequency modulation is realized.In addition,the ripple of the actual current with the proposed HCC is about 3 A,much smaller than those using the other two HCCs.

Fig.18 Simulative current responses to a sinusoidal Iref comparing different HCC.

Therefore,the switching frequency using the newly proposed HCC is higher than those using the other two HCCs,and the time for the current to rise and drop is shorter.However,IErris higher using the newly proposed HCC than those using the other two HCCs,because the actual current with the newly proposed HCC only hits the upper bound,while those with the other two HCCs hit both bounds.

5.3.Comparative experimental results

Fig.19 Description of the experimental platform.

An experimental platform of a BLDC motor driver and controller presented in Fig.19 is developed and used to verify the effectiveness of the proposed digital fixed-frequency HCC.In this experimental platform,the controller is designed basedon TMS320F28335from Texas Instruments and EPM1270from ALTERA.A SiC-MOSFET module CCS050M12CM2 from CREE is taken as the power device,and the modulating frequency is set as 20 kHz.The properties of the motor are the same as those listed in Table 1.Experimental results are shown in Figs.20–24.In Figs.20–23,both Atrand ΔI are 7 A,and In Fig.23 the Irefis with an amplitude of 10 A,frequency of 1 kHz,and a mean value of 25 A.While in Fig.24,the Irefis with an amplitude of 5 A,a frequency of 1 kHz,and a mean value of 25 A,both Atrand ΔI are 5 A,and E is 6 V.

Fig.20 Experimental current response to a constant Irefof 30 A(E=10 V).

Fig.21 Experimental current response to a constant Irefof 40 A(E=60 V).

Fig.22 Experimental current response to a step Ireffrom 20 A to 40 A(E=10 V).

Fig.23 Experimental current response to a sinusoidal Iref(E=30 V).

Fig.24 Experimental current responses to a sinusoidal Iref comparing different HCC(E=6 V).

Comparing Figs.20–23 with Figs.14–17 respectively,it can be seen that experimental results are consistent with those of simulations,which verifies the correctness of the simulation model.Furthermore,similar to the results of the simulations,modulation at a fixed switching frequency is realized in all of these experiments,and the capabilities to track the step and sinusoidal signals of this proposed current control strategy seem substantial.Likewise,results in Figs.24 and 18 are also consistent,which verifies the advantage of the newly proposed HCC.

6.Conclusions

In this paper,a digital fixed-frequency hysteresis current control method based on the PWM_ON modulating mode has been proposed.When a BLDC motor has an ultra-low inductance and a requirement of high-frequency response,this strategy is more suitable and can contribute to a performance improvement.The digital fixed-frequency HCC strategy,similar to the traditional HCC strategy,is easy to implement,with great control stability and inherent overcurrent protection function.Ultimately,this strategy can achieve full fixed frequency modulation.

(1)Through considering the rising rate tan θ and descending rate tanα of the winding current interestedly,the triangular carrier-based hybrid fixed-frequency HCC is expanded by relaxing the constraints on tan θ and tan α.

(2)Using the results about the constraints on tan θ and tanα,a flexible-band-size quasi- fixed-frequency HCC is designed,on which the determination method of the flexible band size ΔI and analysis of the influence of the amplitude of the triangular carrier are given.In this newly proposed HCC,a feasible ΔI can be obtained throughout the entire scope of the back-EMF E,that is,a fixed-frequency modulation control can be realized in the entire running process by carefully tuning ΔI,which greatly expands the feasible range for fixedfrequency modulation control compared with that of the traditional triangular carrier-based fixed-frequency HCC.This newly proposed HCC is called the flexibleband-size quasi- fixed-frequency HCC because its effectiveness is confined in the steady state.

(3)The digitization of the flexible-band-size quasi- fixedfrequency HCC is conducted.Considering that it may fail to realize a fixed-frequency modulation control in the unsteady state or when E becomes close to its limits,four digital rules are designed to guarantee a fixedfrequency modulation control throughout the entire running process.Then,simulation and experimental platforms are constructed to verify the effectiveness of this digital fixed-frequency HCC.Results prove the effectiveness of the proposed digital fixed-frequency HCC.In addition,simulative and experimental comparisons represent the advantage of the newly proposed digital HCC over the traditional HCC and the triangular carrier-based fixed-frequency HCC regarding the current ripple and the capacity of realizing a fix-frequency switch when tan θ or tan α is too high or too low.

Acknowledgements

This research was supported by the National Natural Science Foundation of China(Nos.51275021,61327807).

1.Bennett J,Mecrow BC,Atkinson DJ,Atkinson G.Safety-critical design of electromechanical actuation systems in commercial aircraft.IET Electron Power App 2011;5(1):37–47.

2.Rosero JA,Ortega JA,Aldabas E,Romeral L.Moving towards a more electric aircraft.IEEE Aero El Sys Mag 2010;22(3):3–9.

3.Noriko M,Michiya T,Hitoshi O.Moving to an all-electric aircraft system.IHI Eng Rev 2014;47(1):33–9.

4.MaréJ-C,Fu J.Review on signal-by-wire and power-by-wire actuation for more electric aircraft.Chin J Aeronaut 2017;30(3):857–70.

5.Liu H,Zhu ZQ,Mohamed E,Fu Y,Qi X.Flux-weakening control of nonsalient pole PMSM having large winding inductance,accounting for resistive voltage drop and inverter nonlinearities.IEEE Trans Power Electron 2012;27(2):942–52.

6.Cao W,Mecrow BC,Atkinson GJ,Bennett JW,Atkinson DJ.Overview of electric motor technologies used for more electric aircraft(MEA).IEEE Trans Ind Electron 2012;59(9):3523–31.

7.Boglietti A,Cavagnino A,Tenconi A,Vaschetto S.The safety critical electric machines and drives in the more electric aircraft:a survey.35th annual conference of IEEE industrial electronics;2009 Nov 3–5;Porto;Piscataway:IEEE Press;2009.p.2587–94.

8.Wang L,Xiao K,De Lillo L,Empringham L,Wheeler P.PI controller relay auto-tuning using delay and phase margin in PMSM drives.Chin J Aeronaut 2014;27(6):1527–37.

9.Guo H,Xu J,Kuang X.A novel fault tolerant permanent magnet synchronous motor with improved optimal torque control for aerospace application.Chin J Aeronaut 2015;28(2):535–44.

10.Wang L,Xiao K,De Lillo L,Empringham L,Wheeler P.PI controller relay auto-tuning using delay and phase margin in PMSM drives.Chin J Aeronaut 2014;27(12):1527–37.

11.Chen J,Tang PC.A sliding mode current control scheme for pwm brushless dc motor drives.IEEE Trans Power Electron 1999;14(3):541–51.

12.Barrero F,Prieto J,Levi E,Gregor R.An enhanced predictive current control method for asymmetrical six-phase motor drives.IEEE Trans Ind Electron 2011;58(8):3242–52.

13.Sathyan A,Milivojevic N,Lee YJ,Krishnamurthy M,Emadi A.An FPGA-based novel digital PWM control scheme for BLDC motor drives.IEEE Trans Ind Electron 2009;56(8):3040–9.

14.Zhang X,Chen J,Ma Y,Wang Y,Xu D.Bandwidth expansion method for circulating current control in parallel three-phase pwm converter connection system.IEEE Trans Power Electron 2014;29(12):6847–56.

15.Wang H,Yang M,Niu L,Xu D.Current-loop bandwidth expansion strategy for permanent magnet synchronous motor drives.2010 5th IEEE conference on industrial electronics and applications;2010 June 15–17;Taiwan.Piscataway:IEEE Press;2010.p.1340–5.

16.Dawande MS,Kanetkar VR,Dubey GK.Three-phase switch mode rectifier with hysteresis current control.IEEE Trans Power Electron 1996;11(3):466–71.

17.Mohseni M,Islam SM.A new vector-based hysteresis current control scheme for three-phase PWM voltage-source inverters.IEEE Trans Power Electron 2010;25(9):2299–309.

18.Vahedi H,Sheikholeslami A,Bina MT,Vahedi M.Review and simulation of fixed and adaptive hysteresis current control considering switching losses and high-frequency harmonics.Adv Power Electron 2011.https://doi.org/10.1155/2011/397872.

19.Brod DM,Novotny DW.Current control of VSI-PWM inverters.IEEE Trans Ind Appl 1985;21(3):562–70.

20.Buso S,Fasolo S,Malesani L,Mattavelli P.A dead-beat adaptive hysteresis current control.IEEE TransInd Appl2000;36(4):1174–80.

21.Bose BK.An adaptive hysteresis-band current control technique of a voltage-fed PWM inverter for machine drive system.IEEE Trans Ind Electron 1990;37(5):402–8.

22.Malesani L,Mattavelli P,Tomasin P.Improved constant frequency hysteresis current control of VSI inverters with simple feed forward bandwidth prediction.IEEE Trans Ind Appl 1997;33(5),2633–40.

23.Malesani L,Tenti P.A novel hysteresis control method for current-controlled voltage-source pwm inverters with constant modulation frequency.IEEE Trans Ind Appl 1990;26(1):88–92.

24.Pereira RR,Silva CHD,Cavalcanti LEM,Silva LEBD.A simple full digital adaptive current hysteresis control with constant modulation frequency for active power filters.2007 IEEE 42nd IAS annual meeting conference record of the industry applications;2007 Sept 23–27;New Orleans.Piscataway:IEEE Press;2007.p.1644–8.

25.Tekwani PN,Kanchan RS,Gopakumar K.Novel current error space phasor based hysteresis controller using parabolic bands for control of switching frequency variations.IEEE Trans Ind Electron 2007;54(5):2648–56.

26.Rahman KM,Khan MR,Choudhury MA,Rahman MA.Variable-band hysteresis current controllers for PWM voltagesource inverters.IEEE Trans Power Electron 1997;12(6):964–70.

27.Liu F,Maswood AI.A novel variable hysteresis band current control of three-phase three-level unity PF rectifier with constant switching frequency.IEEE Trans Power Electron 2006;21(6):1727–34.

28.Ho CNM,Cheung VS,Chung HSH.Constant-frequency hysteresis current control of grid-connected VSI without bandwidth control.IEEE Trans Power Electron 2009;24(11):2484–95.

29.Stefanutti W,Mattavelli P.Fully digital hysteresis modulation with switching-time prediction.IEEE Trans Ind Appl 2006;42(3):763–9.

30.Wu F,Feng F,Luo L,Duan J.Sampling period online adjustingbased hysteresis current control without band with constant switching frequency.IEEE Trans Ind Electron 2015;62(1):270–7.

31.Rahman AM,Radwan TS,Osheiba AM,Lashine AE.Analysis of current controllers for voltage-source inverter.IEEE Trans Ind Electron 1997;44(4):477–85.

32.Kadjoudj M,Benbouzid MEH,Ghennai C,Diallo D.A robust hybrid current control for permanent-magnet synchronous motor drive.IEEE Trans Energy Conver 2001;19(1):109–15.

33.He Y,Zheng S,Fang J.Start-up current adaptive control for sensorless high-speed brushless DC motors based on inverse system method and internal mode controller.Chin J Aeronaut 2017;30(1):358–67.