A 16

Position Sensorless Direct Torque Control of BLDC Motor by Using Modifier 1

G. R. Arab Markadeh1, S. I. Mousavi2, E. Daryabeigi2 Faculty of Engineering Dept., Shahrekord University, Shahrekord, Iran, [email protected] 2 Azad University, Najafabad Branch, Young Reseacher Club, Isfahan, Iran,

Abstract— Direct torque control (DTC) can directly control the inverter states in order to reduce the torque error within the prefixed band limit. DTC of brushless dc drive with trapezoidal back-EMF is presented in this paper. Using the rotor flux vector position in α − β axis stationary reference frame and torque error, the proper switching pattern can be selected to control the generated torque and reducing commutation torque ripple. Sliding mode observer, which is robust to parameter uncertainties can be used to estimate the back-EMF and the generated torque. The rotor flux vector position estimation and its novel modifier can be achieved using a rotor flux observer. Simulation and experimental results confirm that the proposed method is good idea to introduce reliable sensorless scheme. Index Terms— Brushless dc (BLDC) drives, direct torque control (DTC), permanent-magnet motor, sensorless, rotor angle modifier.

I. INTRODUCTION Permanent magnet brushless machines are extensively used for servo derives, ship propulsion systems and traction drives. The back-EMF waveform of brushless AC (BLAC) drive is sinusoidal and is supplied with SPWM voltage source inverter, while the Brushless DC (BLDC) stator phase currents are rectangular and its back-EMF is trapezoidal due to concentrated winding. BLDC motor has better efficiency, higher torque density, lower cost and simpler structure, comparing to BLAC motors. Furthermore, while BLAC drive requires an accurate encoder sensor, BLDC drive needs to discrete position sensor such as Hall Effect device. The main drawback of BLDC is higher torque ripple due to machine structure and feeding system. Machine structure causes to cogging torque and waveform imperfection; and those coming from the power electronic feeding system are switching techniques and phase current commutation. Two decades later, DTC method is developed and presented for induction machine drives [4]. Since the beginning, the new technique was characterized by simplicity, good performance and robustness [5]. o DTC of BLDC with 120 conduction angle is proposed in [7, 8]. The stator flux linkage and torque commands were obtained from hysteresis controllers by comparing the estimated electromagnetic torque and stator flux linkage with their demanded values and the switching pattern of the inverter was obtained according to the stator flux vector position, torque and stator flux errors. In that control method voltage vector is selected from a look-up table witch is depends on rotor mechanical position and irrespective to the stator phase

current or back-EMF which produces higher torque ripple and switching frequency. In this paper, the DTC method is used which is introduced before in [7]. However, it is tried to solve two fundamental disadvantages. The first one is the backEMF in [7] is measured by look up table and position sensor, in fact the mentioned scheme assumed that the relationship between back-EMF and electrical velocity is linear by a constant factor. Therefore, the estimated torque depends on position sensor sensitivity. Accordingly, sliding mode observer is robust to parameter uncertainty therefore can be used to estimated the back-EMF accuracy. The second one is using sensor position, which is sensitive to mechanical disturbance. However, in the proposed method in this article, instead of rotor mechanical position, the rotor flux vector position is estimated by using the rotor flux observer. The essential advantage of the rotor flux observer is in its rotor angle modifier. In this novel method, the state equation of the BLDC motor is utilized to achieve a relationship between the angle of the stator current vector and the back-EMF vector angle. By using of this method, the error angle estimation is approximated to minimum value. Therefore, this scheme cause to reduce the torque ripple in commutation regions. In the proposed control method in this research work, the proper voltage vector is selected from look up table using the rotor flux vector position and torque error. This voltage vector leads the torque error to the predefined hysteresis. II. BLDC MODEL Brushless DC motor state space equations with 120° conduction angle and trapezoidal back-EMF waveform, can be expressed as ias = ∫

2 Vab + Vbc − 2 eas + ebs + ecs − 3 Rias d t (1a) 3 Ls

ibs = ∫

− Vab + Vbc + eas + 2ebs + ecs − 3 Ribs d t (1b) 3 Ls ics = −ias − ibs

(1c)

and the electromagnetic generated torque is i e +i e +i e Te = p a as b bs c cs ωe

(2)

were ias , ibs , ics ,Vab ,Vbc are the stator currents and line voltages, respectively; eas , ebs , ecs are back-EMF voltages

of each phase and p is the number of pole pairs, θr is the rotor electrical angle, Tl is the load torque, B is the friction coefficient, J is the moment of inertia, R is stator resistance and Ls = L − M , where L and M are the self inductance and mutual inductance of stator phase, respectively. III. DIRECT TORQUE CONTROL OF BLDC MOTOR Using transformation, the state space equation of BLDC motor in α − β axis stationary reference frame can be written as: V sα = R i sα + L

d i sβ d i sα + eα ; Vsβ = R isβ + L + eβ (3) dt dt

and the electromagnetic generated torque can be expressed as [2], [4]: Te =

d ϕ rβ eβ ⎞ 3 p ⎛ eα ⎞ 3 p ⎛ dϕ rα ⎜ ⎜ i sα + isβ ⎟⎟ = i sα + isβ ⎟⎟ ⎜ ⎜ ωr 2 ⎝ dθ r dθ r ⎠ 2 ⎝ ωr ⎠

(4) where φra and φrb are the α − β axis rotor flux vector components.

isα and is β are the stator current vector

components in α − β reference frame. The equations between the stator flux and rotor flux are: ϕ sd = Ld isd + ϕed ; ϕ sq = Lq isq + ϕ eq

(5)

where isd and isq are the stator current vector components in d-q reference frame, Ld and Lq the d-q axis stator inductances, φrd, φrq, φsd and φsq are the d-q axis rotor and stator flux linkages, respectively. The differential forms of rotor flux components respect to θr can be derived from the ratio of the back-EMF to the electrical angular velocity, ωr, i.e. dϕ rα 1 dϕ rα eα ; = = dθ r ωr dt ωr

dϕ rβ dθ r

=

1 dϕ rβ eβ = ω r dt ωr

(6)

2

2

∠

⎛ eβ dϕr e =∠ = tan −1⎜⎜ ωr dθ r ⎝ eα

eα2 + eβ2

⎞ ⎟ = ∠e = θe ⎟ ⎠

(7a)

(7b)

Where φr and e are the rotor flux and the back-EMF vectors, respectively and θe is the back-EMF vector angular phase. The phase difference between stator current vector and back-EMF vector is defined as θerr = θis − θe

isα = is cos(θ e + θ err ); isβ = is sin (θ e + θ err ) (9)

and d ϕ rβ

dϕ rα dϕ r = cos(θ e ); dθ r dθ r

dθ r

=

dϕ r sin (θ eα ) (10) dθ r

Substituting (10) and (9) in (5) we obtain; Te =

3p is dϕ r / dθ r cos θ err 2

(12)

It can be seen that the phase difference of each stator current components with its back-EMF causes to torque pulsation in commutation region. In the other word if the stator current with back-EMF is in phase, then the torque pulsation decreased and higher density of torque can be achieved. Obviously, it is known that the dϕ r / dθ r term is relative to rotor flux or permanent magnet profile and is independent of the stator currents. From (12), it can be shown that generated electromagnetic torque depends on the stator currents, dϕ r / dθ r value and the phase difference between back-EMF (or consequently rotor flux) vector and stator current vector. In previous research works, it is proved that due to the sharp dips in the stator flux space vector at every commutation region, there is no easy way to control the stator flux magnitude [7]. In the other word against of induction motor the stator flux hasn’t directly effect on generated torque while from (12) the rotor flux position has extremely effects on torque and it is better to use the rotor flux vector position to control the electromagnetic torque. To control the generated torque in two-phase conduction a switching pattern is selected according to torque error respect to its reference value. Table I indicates the proper switching pattern in each sector which is determined based on rotor flux vector position. IV. ROTOR FLUX OBSERVER

where ωr = dθ r / dt Then, the amplitude and the angle of (dϕ r / dθ r ) vector can be written as: ⎛ dϕ ⎞ ⎛ dϕrβ ⎞ dϕr 3 1 ⎟ = = = ⎜⎜ rα ⎟⎟ + ⎜⎜ ⎟ ω dθr ωr d d θ θ r ⎠ r ⎝ ⎝ r ⎠

where θis is the stator current vector angular phase. The stator current vector components in α − β axis can be rewritten as

(8)

In (6), rotor flux vector components can be calculated using rotor flux observer ϕ rα = − Ls isα + ∫ (Vα − R isα ) dt

(13a)

ϕ rβ = − Ls isβ + ∫ Vβ − R isβ d t

(13b)

(

)

Switching effects in stator voltage can be filtered with integrator operator in (13). The rotor flux position is related to the back-EMF position directly, witch the backEMF vector position has 90o leads to the rotor flux angle, therefore θbemf can be calculated based on rotor flux angle θbemf = 90º+θr. On the other hand from (13) the stator flux position is not related to back-EMF vector position directly. Rotor flux position which is needed to DTC technique can be obtained as

(

θ r = tan −1 ϕ rβ / ϕ rα

)

(14)

TABLE I DTC VECTOR SELECTION FOR BLDC MOTOR

ϕs 0

Tl

θr o o 330 − 30

o o 30 − 90

o o 90 − 150

o o 150 − 210

o o 210 − 270

1

V 2(001001)

V 3(011000)

V 4(010010)

V 5(000110)

V 6(100100)

o o 270 − 330 V 1(100001)

-1

V 5(000110)

V 6(100100)

V 1(100001)

V 2(001001)

V 3(011000)

V 4(010010)

Figure 2. Sliding mode observer block diagram.

Substituting (16) in (17), we obtain

V. ROTOR FLUX ANGLE ESTIMATOR MODIFIER Using the BLDC motor rotor flux angle calculation by utilizing (13) is fluctuated with some problems. So the pure integral operator in (13) must be replaced with low pass filter. In addition, a high pass filter must filter the dc offset of measured current. The mentioned filters cause angle between the actual and estimated rotor flux angle and affects to the calculated active torques. In order to reduce the torque ripple in commutation regions, as show in Fig.1 the current and back-EMF vectors must be synchronized as well as possible. Therefore, the estimated rotor angle because of filtering and calculation time lag to actual one. Hence, the modified rotor angle can be calculated as follow: θ'r = θr − θerr

(15)

where θ'r is the modified rotor flux angle and used to voltage vector selection in DTC and

⎡1 θ err = tan −1 ⎢ ⎢⎣ Te

dϕ rβ ⎞⎤ ⎛ dϕ rα ⎟⎥ ⎜ i i ⋅ − ⋅ s β s α ⎟ ⎜ dθ dθ e e ⎠⎥⎦ ⎝

(18)

Rotor speed for electromagnetic torque calculation can be obtained, using a differential operator. Rotor flux vector components have smooth shape and there is not any spike in them then a simple differential can be used to obtain the rotor speed. VI. SLIDING MODE OBSERVER Sliding mode observer can be used to estimate nonsinusoidal back-EMF of BLDC motor. Sliding mode observer is proposed as: ) − Rs ˆ eα Vα ⎧ˆ ˆ ⎪isα = L ⋅ isα − L + L + K s1 ⋅ Sgn isα s s s ⎪ ) eβ Vβ − Rs ˆ ⎪ˆ ˆ (19) ⎨ isβ = L ⋅ isβ − L + L + K s1 ⋅ Sgn isβ s s s ⎪ ) eα = K s 2 ⋅ Sgn iˆsα ⎪ ) ⎪ eβ = K s 2 ⋅ Sgn iˆsβ ⎩

( )

( )

( ) ( )

) ) where, iˆsα , iˆsβ , eα , eβ are the estimation of α-β axes stator currents and back-EMF respectively and stator currents estimation error define as ) ~ isα = isα + isα (20a) ) ~ isβ = isβ + isβ

Fig. 1 stator current and back-EMF vectors

tan (θis ) =

(

)

isβ isα

tan θbemf =

eβ eα

(16a)

(16b)

It will be consider that tan θerr =

tan θis − tan θbemf 1 + tan θis ⋅ tan θbemf

(17)

(20b)

If sampling period is significantly less than electrical and mechanical time constants then back-EMF value can be assumed to remain constant during each sampling period and, d eβ d eα = 0; =0 dt dt By subtracting (19) from (3), the estimation error of stator currents and back-EMF can be expressed as in (21):

− Rs ~ e~ V ~ ~ isα = ⋅ isα − α + α + K s1 ⋅ Sgn isα Ls Ls Ls e~β Vβ − Rs ~ ~ ~ ⋅ isβ − + + K s1 ⋅ Sgn isβ isβ = Ls Ls Ls

( )

(21a)

( )

(21b)

( ) ( )

~ e~α = − K s 2 ⋅ Sgn isα ~ e~β = − K s 2 ⋅ Sgn isβ

EMF estimation without need to look up table and position sensor. In Fig. 6, simulation results for rotor position estimator and its error respect to position sensor values are shown. The estimated rotor position error is less than 5 degrees

(21c) (21d)

Defining positive definite Lyapunov function V(x) as: V=

(

1 ~ 2 ~ 2 isα + isβ 2

)

(22) Figure 3. Sensorless DTC of BLDC block diagram.

Derivation of V(x) with respect to time concludes:

(

)

(

R ~ 1 ~ ~ 2 ~ ~ 2 ~ V& = − s isα 2 + isβ 2 − eα isα + eβ isβ Ls Ls ~ 2 ~ 2 − K s1 isα − K s1 isβ < 0

) (23a) (23b)

ks1 can be selected such that V&

The lower bound of

became negative definite and then the estimated stator current components converge to their measured values.

(

)

R ~ ~ ~2 K s1 > − s − e~α isα + e~β isβ / Ls is Ls

TABLE II PARAMETERS OF

(24)

BLDC MOTOR

Rs (ohm)

0.62

Ls (mH)

1

Flux induced by magnets (Wb)

0.07627

Rated torque

6.1

Poles

16

DC link voltage (v)

36

Rated speed (rad/s)

31

It is sufficient that:

TABLE III

(

)

~ ~ ~2 K s1 > max e~α isα + e~β isβ / Ls is

THE SLIDING MODE OBSERVER PARAMETERS

(25)

k s1

300

ks 2

25

kp

0.4

kI

1

Bound of the Current Sat(.) function

1 10

To decrease the chattering effects near the sliding surfaces the Sgn( ) function is replaced with saturation function as: ~ ⎛i Sat ⎜⎜ s ⎝q

⎧1 ⎞ ⎪~ ⎟ = ⎨ is ⎟ ⎠ ⎪− 1 ⎩

~ is ≥ q ~ is < q ~ is ≤ q

(26)

in order to decrease the pure integrator influences in back-EMF estimation , it is recommended that a PI structure to be used as follows : ~ e~α = − K t e~α + K s 2 Sgn isα (27a)

( ) ~ e~β = − K t e~β + K s 2 Sgn (isβ )

Bound of the back-EMF Sat(.) function

VIII. EXPERIMENTAL RESULTS The eZDSP based on TMS320F2812 DSP is used to implement the discussed control method. To reach a higher dynamic in control method and estimate the parameters accurately, the sapling time have to select as more as possible.

(27b)

40

35

VII. SIMULATION RESULTS

25

rad/s

The overall block diagram of the proposed control scheme which is shown in Fig. 3 , is simulated with Matlab/ Simulink software. The motor parameters are listed in Table II, as well as the sliding mode observer coefficients are shown in Table III. Fig. 4 shows the speed estimator performance. Fig. 5 shows the sliding mode back-EMF observer results. It is obviously shown that the estimated back-EMF is accurate and the sliding mode observer is a good option for back-

30

20

15

10

5

0

0.05

0.1

0.15

0.2

0.25

sec

Figure 4. Simulation results for rotor speed: the red waveform simulated: the blue waveform - estimated.

20

15

10

V

5

0

-5

-10

-15

-20 0.25

0.26

0.27

0.28

0.29

0.3 sec

0.31

0.32

0.33

0.34

0.35

(a) 20

15

10

V

5

(a)

0

3.5

-5

-10

3

-15

2.5 -20 0.25

0.26

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

2

e

T (N.m)

sec

(b) Fig.5 Back-EMF vector components in stationary reference frame : (a)simulated (b)estimated

1.5

1

0.5

8

0 0.4

0.41

0.42

0.43 time(s)

0.44

0.45

(b) Fig.7 Experimental result for BLDC motor (300 r/min) (a)phase current (b)estimated electromagnetic torque

ra d

6 4 2 0 0.45

7

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

time(sec) (a)

6 5

A n g le (ra d )

0

4

-0.05

ra d

3

-0.1

2 1

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

time(sec) (b) Fig.6 (a)estimated rotor positions is in blue and simulated rotor positions is in red(b) the angle error between estimated and simulated rotor position

Fig. 7 illustrates the experimental results of phase-a current and electromagnetic torque. The torque reference is 2.85 and the torque control is performed as shown in fig.(3). the hysteresis bandwidth is .01 N.m, the DC-link voltage is set to 35 V. As shown in Fig. 7a the current waveform verify the estimated rotor angle, because the error in rotor angle estimation causes to make spike in motor currents which is not in these current waveforms. The ripple of electromagnetic torque is less than 27%. Fig. 8 is the angle between of the modified rotor angle and measured electrical rotor angle in experimental implementation. Depends on elapsed calculation time and ADC accuracy, the sampling period is determined as 32 μs. to measure the phase current value, two (LEM LA25-NP) current sensors are used and the line-to-line voltage is calculated by measure the DC-link voltage through one (LV25-P) voltage sensor versus state of inverter switches witch produced in DSP320f2812.

0 0.4

0.402 0.405 0.408 0.411 0.414 0.417 0.420 0.423 0.426 0.429

Time(s) Fig. 8. Experimental Estimated modified rotor flux-linkage in red and measured electrical rotor angle in black

To yield more accurate measurement, range of current and voltage sensors are selected based on the maximum phase current and DC-link voltage, respectively. The other point which should be noted is that a deadtime is employed after each switch state variation to avoid make overlap in one leg of the inverter which causes to make high spike in phase current waveform. VIII. CONCLUSION In this paper, a sensorless direct torque control method for a brushless DC motor is developed. Using the rotor flux vector position, rotor angle modifier and torque error, the proper switching pattern can be selected from a look up table to limit the generated torque error respect to its reference in a predefined hysteresis band. Rotor flux vector angle is estimated by using rotor flux observer and back-EMF is calculated with a simple robust sliding mode observer. Simulation and experimental results show good performance for torque control and rotor position estimation.

REFERENCES [1]

D. Casadei, F. Profumo, G. Serra and A. Tani, “FOC and DTC: two viable schemes for induction motors torque control,” IEEE Trans. Power Electron., vol. 17, no. 5, pp. 779-787, Sep. 2002.

[2]

Y. Liu, Z.Q. Zhu, D. Howe “Direct torque control of brushless DC drives with reduced torque ripple”, IEEE Trans. Ind. App. vol.41, no.2, pp 599-608. March/April 2005

[3]

Y. Liu, Z.Q. Zhu, D. Howe, “Commutation torque ripple minimization in direct torque controlled PM brushless dc,” IEEE Proc. 14244-0365-0/06, pp. 1642-1648, 2006.

[4]

S.K. Chung, H.S. Kim, C.G. Kim, and M.J. Youn, “A new instanttaneous torque control of PM synchronous motor for highperformance direct-drive applications,” IEEE Trans. Power Electron., vol. 13, no. 3,pp. 388–400, May 1998.

[5]

B.H. Ng, M.F. Rahman, and T.S. Low, “An investigation into the effects of machine parameters on torque pulsation in a brushless dc drive,” in Proc. IEEE IECON’88, pp. 749-754, 1988.

[6]

Y. Liu, Z.Q. Zhu, and D. Howe”, “Instantaneous Torque Estimation in Sensorless Direct-Torque-Controlled Brushless DC Motors”, IEEE Trans. Ind. Appl. Vol.. 42, No. 5, Sept./ Oct. 2006.

[7]

S.B. Ozturk and H.A. Toliyat” Direct Torque Control of Brushless DC Motor with Non-sinusoidal Back-EMF,” IEEE Proc. 2007.

[8]

T. Kim, B.K. Lee, and M. Ehsani," Sensorless Control of the BLDC Motors From Near Zero to High Speed," IEEE Proc.2001

\