Direct Torque Control of Matrix Converter Fed BLDC Motor Ranganth Muthu1, M. Senthil Kumaran2, L.A. Abishek Rajaraman3, P. Ganesh4, P.Geeth Prajwal Reddy5 Electrical and Electronics Engineering, SSN College of Engineering, Anna University, Chennai, India {ranganathm1, senthilkumaranm2}@ssn.edu.in,
[email protected],
[email protected],
[email protected]
Abstract— This paper proposes a two-phase conduction Direct Torque Control (DTC) scheme for three-phase Brushless DC (BLDC) Motor in the constant torque region. The use of Matrix Converter (MC) produces rectangular output currents, while maintaining sinusoidal input currents close to unity power factor. The implemented scheme reduces the torque dip in the BLDC motor. The merits of the matrix converter fed BLDC motor is shown by comparing the results with a conventional AC–DC–AC converter fed BLDC, both implementing two–phase conduction DTC. A model based on Matlab/Simulink is implemented to verify the above-proposed theory. Keywords – BLDC, MC, DTC.
I. INTRODUCTION Brushless DC (BLDC) machines are becoming more popular in the industry due to their inherent advantages, such as superior speed torque characteristics, high power density, low maintenance, long operating life, high efficiency and noiseless operation [1]. As the name implies, BLDC motors do not use brushes for commutation [1],[2]. The commutation is performed electronically by using an array of switching devices based on the rotor position information. Normally, the rotor position information is obtained from position transducers such as Hall effect sensors, shaft mounted encoders or by analyzing the circuit parameters such as the stator third harmonic current waveforms. BLDC motors find major applications in computers and hard drives, industrial robotics, motion control and actuator systems and also in some heating and ventilation systems. BLDC motors are traditionally driven by the pulse width modulated (PWM) voltage source inverters (VSI). However, they have disadvantages such as the high source current harmonics caused by diode rectifier feeding the voltage source inverter, and requirement of bulky energy storage component such as the DC bus capacitor. Matrix converters are considered as a possible alternative to PWM–VSI [3]. Simple and compact power circuit, variable voltage and variable frequency output, near unity power factor operation and polyquadrant operation are the advantages of MC. The DTC of the BLDC motor is implemented using a two–phase conduction scheme. The stator flux linkage is intentionally kept constant, thereby eliminating the need for flux control, thus simplifying the control of the drive to torque alone.
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II.
BRUSHLESS DC MOTOR
A. Operation of BLDC motor The stator consists of a concentrated three–phase star windings similar to that of the synchronous machine. The rotor is made of a permanent magnet material, Neodymium Ferric Boron (NdFeB) and the magnets are usually surface mounted [4]. Unlike conventional motors, BLDC motors require rotor position sensing, which is done using the Hall effect sensors mounted on the stator surface [5]. When Hall effect sensors are placed 120° electrical apart on the inner surface of the stator and are subjected to the rotor magnetic field, a voltage is induced, which provides the controller with the present position of the rotor. From this information, the next position of the rotor can be estimated and the appropriate winding energization sequence realized in the converter. Table I shows the output of the Hall effect sensor and the appropriate phase excitation based on the rotor position. TABLE I. HALL EFFECT SENSOR OUTPUT AND PHASE EXCITATION Hall effect sensor
Phase
Rotor Position
HA
HB
HC
A
B
C
0° - 60° 60° - 120° 120° - 180° 180° - 240° 240° - 300° 300° - 360°
1 1 0 0 0 1
0 1 1 1 0 0
0 0 0 1 1 1
+1 +1 0 -1 -1 0
0 -1 -1 0 +1 +1
-1 0 +1 +1 0 -1
III.
MATRIX CONVERTER
The matrix converter is an array of bidirectional IGBTs that interconnects directly the three-phase AC supply to a threephase load, without using any DC link or large energy storage elements [3]. The Indirect Modulation Principle decouples the matrix converter into a current source rectifier on the source side [6] and a voltage source inverter on the load side as shown in Fig. 2. Thus the switching algorithm for the two stages are individually obtained as explained below. A. Switching Strategy for Rectifier stage Space Vector Modulation (SVM) is implemented on the rectifier side of the MC. The current and voltage transfer functions of the rectifier side are given by (1) and (2)
VDC+ IDC+
Ia
Va
SaA Ib
Vb
Ic
Vc
SaB
SaC S1
SbA
SbC
SbB
S5
S7
S9 S11 VA
Vb
VB
VDC
VC
Vc
ScA
ScB
ScC
IA
IB
IC
VA
VB
S2
𝐼! 𝑆! 𝐼! = 𝑆! 𝐼! 𝑆! 𝑉!"! 𝑆 = ! 𝑉!"! 𝑆!
𝑆! 𝐼 𝑆! × !"! 𝐼!"! 𝑆! 𝑆! 𝑆!
𝑉! 𝑆! × 𝑉! 𝑆! 𝑉!
(1)
S8 S10 S12
IDCVDC-
Inverter part
𝑇! 𝜋 = 𝑚! . 𝑠𝑖𝑛 − 𝜃! 𝑇! 3 𝑇! 𝑑! = = 𝑚! . 𝑠𝑖𝑛 𝜃! 𝑇! 𝑇!" 𝑑!" = = 1 − 𝑑! − 𝑑! 𝑇!
𝑑ϒ =
(2)
(3)
From the configuration of the converter, as shown in Fig. 2, nine switching states are possible. These switching states are enlisted in Table II.
Using equation (3), seven discrete space vectors can be obtained from the nine switching states. These vectors, when plotted on a complex plane, form a hexagon, as shown in Fig. 3. Reference vector IREF, can be synthesized within the hexagon sector by the vector sum of the components of the sector and the zero vectors, as shown in Fig. 4. For a small time interval TS, the reference vector is represented as the sum of the current-time products of the adjacent vectors, as shown in (4). (4)
Where, dδ and dγ are the duty cycles of the current vectors Iδ and Iγ respectively. The duration of the active vectors determines the direction of IREF while the zero vector interval is used to adjust the amplitude of IREF. The duty cycles are computed using (5) - (7)
(5) (6) (7)
where θC indicates the angle of the reference current vector within the hexagon sector and mc is the modulation index of the rectifier stage. Tδ and Tγ are the time periods for which Iδ and Iγ current vectors are applied. Toc and doc are the time period and duty cycle respectively of the zero current vector. TABLE II. RECTIFIER STAGE SWITCHING STATE MATRIX AND SWITCHING VECTORS Type
Vector I1 [ab]
I1[ab] indicates that the input phase ‘a’ is connected to the positive rail of the virtual DC link VDC+ and the input phase ‘b’ is connected to the negative rail VDC-.
𝐼!"# = 𝑑! . 𝐼! + 𝑑! . 𝐼!
S6
Fig. 2. Indirect Modulation Principle Equivalent Circuit
where Ia, Ib, Ic and Va, Vb, Vc are the 3Φ input currents and voltages respectively. IA, IB, IC are the 3Φ output currents. The input currents are expressed as a space vector, IIN using the transformation given in (3). !! !! 2 𝐼! + 𝐼! . 𝑒 ! ! + 𝐼! . 𝑒 ! ! 3
S4
Rectifier part
VC
Fig. 1. Matrix Converter Topology
𝐼!" =
S3
Va
I2 [ac] I3 [bc] Active I4 [ba] I5 [ca] I6 [cb] Zero
IV.
I0 [aa] [bb][cc]
1 1
0 0
𝑺𝟏 𝑺𝟐 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0
𝑺𝟑 𝑺𝟒 0 1 0 0 1 0 1 0 0 0 0 1 1 1
𝑺𝟓 𝑺𝟔 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0
𝑰𝑰𝑵 2 3 2 3 2 3 2 3 2 3 2 3 0 0
1 1
𝐼!" 𝐼!" 𝐼!" 𝐼!" 𝐼!" 𝐼!"
∠𝑰𝑰𝑵 𝜋 6 𝜋 6 𝜋 2 5𝜋 6 7𝜋 6 3𝜋 2
−
0
-
DIRECT TORQUE CONTROL OF THE BLDC MOTOR USING VOLTAGE SOURCE INVERTER
Unlike the implementation of the conventional DTC algorithm in AC machines, the requirement of Flux estimation is eliminated in the two-phase conduction scheme. This is done by intentionally keeping the stator flux linkage constant.
The variation of Lds and Lqs with θe can be neglected with the use of non-salient pole machines, where Lds=Lqs=Ls [8] or with the use of high coercive PM material [4], [5]. As a result, the electromagnetic torque equation for BLDC motor is given by (9).
Im I3[bc] I4[ba]
3
2
I2[ac]
5
I5[ca]
Re
1
4 6
𝑇!" =
I1[ab]
Fig. 3. Matrix Converter Topology
Im Iδ
Re
IREF
d γI γ
(10a)
𝜙!" = 𝜙!" 𝑠𝑖𝑛 𝜃! + 𝜙!" 𝑐𝑜𝑠 𝜃!
(10b)
Fig. 4. Reference Current Vector Synthesis
However, even without the flux estimation, DTC can be implemented successfully in BLDC, thereby simplifying the control of the motor. Furthermore, zero voltage space vector is not used, as they tend to decrease electromagnetic torque and also increase switching losses [7]. A. Proposed DTC Technique The general electromagnetic torque (Tem) equation of a BLDC motor, taken in dq reference frame, which is synchronous with the stator flux and includes the mutual coupling between d and q axis winding, is given by (8) 𝑑𝐿!" 𝑑𝜙!" 𝑖 + − 𝜙!" 𝑖!" 𝑑𝜃! !" 𝑑𝜃! 𝑑𝐿!" 𝑑𝜙!" + 𝑖 + − 𝜙!" 𝑖!" 𝑑𝜃! !" 𝑑𝜃!
𝑖!" = 𝑖!" 𝑐𝑜𝑠 𝜃! − 𝑖!" 𝑠𝑖𝑛 𝜃!
(11a)
𝑖!" = 𝑖!" 𝑠𝑖𝑛 𝜃! + 𝑖!" 𝑐𝑜𝑠 𝜃!
(11b)
Thus, the electromagnetic torque equation, in the stationary frame of reference, using (10), (11) in (9) becomes (12)
Iγ
3𝑃 4
𝜙!" = 𝜙!" 𝑐𝑜𝑠 𝜃! − 𝜙!" 𝑠𝑖𝑛 𝜃!
Similarly, the dq axis stator currents can also be expressed in the αβ-axis as shown in (11)
dδIδ
𝑇!" =
(9)
For the proposed DTC scheme for BLDC motor, the electromagnetic torque equation is expressed in the stationary frame of reference (αβ-axis) instead of the dq frame. The dq axis rotor flux linkages can be expressed in the αβ frame as shown in (10)
I6[cb]
θc
𝑑𝜙!" 3𝑃 𝑑𝜙!" − 𝜙!" 𝑖!" + − 𝜙!" 𝑖!" 4 𝑑𝜃! 𝑑𝜃!
𝑑𝜙!" 𝑒! 3 𝑃 𝑑𝜙!" 3𝑃 𝑒! 𝑇!" = ∙ 𝑖 + 𝑖 = 𝑖 + 𝑖 2 2 𝑑𝜃! !" 𝑑𝜃! !" 4 𝜔! !" 𝜔! !"
where ωe is the electrical rotor speed and eα and eβ are the αβ frame motor back-EMFs. In general, the back-EMF of a motor can be expressed as shown in (13)
𝜙!" = 𝐿!" 𝑖!" + 𝜙!" ϕ!" = L!" i!" + ϕ!" P is the number of poles of the stator, θe is the electrical rotor angle, isd and isq are the d axis and q axis stator currents. Lds and Lqs are the d and q axes stator inductances and ϕrd, ϕrq, ϕsd, ϕsq are the d and q axes rotor and stator flux linkages respectively.
𝑒! = 𝑘! 𝜃! . 𝜔!
(13a)
e! = 𝑘! θ! . ω!
(13b)
where, k ! θ! and k ! θ! are the back-EMF constants with respect to the rotor position. By substituting (13a) and 13(b) in (12), we get (14).
(8)
where,
(12)
𝑇!" =
3𝑃 𝑘! 𝜃! 𝑖!" + 𝑘! 𝜃! 𝑖!" 4
(14)
Since (14) does not involve the rotor speed in the denominator, the torque can be estimated even at zero and near zero speeds. B. DTC Operation Conventional two-phase conduction scheme causes the locus of the stator flux linkage to have a hexagonal trajectory [7], neglecting the open-phase back-EMF and freewheeling diode effects, as shown in Fig. 5 in dotted lines. However, in this scheme, it is observed that sharp dips in stator flux linkage occur every 60° electrical angle [1] due to the freewheeling
diodes, shown as Fig. 5 in solid lines, deviating from the desired circular flux trajectory as in the case of the PMSM drive [9]. By knowing the exact shape of the flux linkage, we can control its amplitude, but this method is tedious in the constant torque region. Therefore, the flux error in the voltage vector selection lookup table [7] is always taken as zero and only the torque error is used, as in Table III.
computed and multiplied to obtain the 3×3 switching matrix for the MC. This is shown in (16). 𝑆!" 𝑆!" 𝑆!" 𝑆! 𝑆! 𝑆 𝑆! 𝑆! (16) 𝑆!" 𝑆!" 𝑆!" = 𝑆! 𝑆!" × ! 𝑆! 𝑆! 𝑆! 𝑆!" 𝑆!" 𝑆!" 𝑆!! 𝑆!" The 3×3 matrix is applied to the corresponding switches of the MC shown in Fig. 1. VI.
SIMULATION
A. Triggering Pulses The layout of the Triggering pulse generator for the MC is shown in Fig. 7. The rectifier SVM algorithm is used to trigger the rectifier stage of the matrix converter. The DTC algorithm triggers the inverter stage.
2𝐻𝜑
Fig. 5. Representation of two–phase actual (solid line) and desired (dotted circle) stator flux linkage vectors in the stationary αβ–axes reference frame TABLE III. TWO-PHASE VOLTAGE VECTOR SELECTION FOR THE BLDC MOTOR τ +1 -1
θ1 V1 100001 V5 000110
θ2 V2 001001 V6 100100
Sector Number θ3 θ4 V3 V4 011000 010010 V1 V2 100001 001001
θ5 V5 000110 V3 011000
θ6 V6 100100 V4 010010
C. Implementation of DTC The phase voltages Van, Vbn, Vcn, supplied to the stator of the BLDC motor are determined by the status of the six switches of the inverter, represented by S7, S8, S9, S10, S11, S12. A total of six non–zero space vectors are possible. With two-phase conduction, at any instant, both switches in the non-conducting phase leg are always off. The six non–zero vectors, V1, V2, V3, V4, V5, V6, shown in Fig. 5, are 60° electrically apart from each other and 30° electrical phase shifted from the corresponding three-phase voltage vectors of the SVM voltage hexagon [6].
The DTC algorithm starts with the hysteresis controller subsystem. Fig. 8 shows the layout of this subsystem. The ‘w’ signal is the actual speed of the rotor and the reference speed is set to 100 rad/sec. This speed error is fed to a PI controller. This gives the reference torque, which is compared with the actual electromagnetic torque ‘t’ of the BLDC motor and this error ‘ε’ is the input of the hysteresis controller. The output, ‘hysop’, of the hysteresis controller is given by (17). An output of 1 will cause the forward torque to be applied and -1 will apply the reverse torque. ℎ𝑦𝑠𝑜𝑝 𝜀 =
1, −1,
𝜀 > 0.01 𝜀 < −0.01
(17)
With the rotor position, estimated from the Hall effect sensors, the sector number, as per Fig. 5, is obtained. Based on hysteresis controller output and sector number, the voltage vector as per Table III is applied. Finally, the switching matrix generated from the rectifier SVM algorithm and the DTC algorithm of the BLDC are multiplied in the Mat_Mul subsytem, as shown in (16), and the triggering pulses for the MC are obtained. The layout of the forward torque subsystem is shown in Fig. 9.
With reference to Fig. 2, the switching states applied to the VSI stage of the MC, is represented as a 3×2 matrix, given by (15). 𝑆! 𝑆! 𝑆!!
𝑆! 0 𝑆!" => 𝑉! = 1 𝑆!" 0
0 0 1
(15)
Fig. 7. General layout of triggering pulses of the DTC for the MC fed BLDC drive.
V. SWITCHING ALGORITHM FOR THE MATRIX CONVERTER As explained in the previous sections, the rectifier stage switching matrix and the inverter stage switching matrix are
Fig. 8. Hysteresis Current Controller
Fig. 12. FFT of the source phase current drawn by the MC
Fig. 9. Layout of forward torque subsystem
B. Matrix Converter The bidirectional switch used in the MC is simulated by antiparallel connection of the two reverse blocking IGBTs, as shown in Fig. 10.
The rotor speed is shown in Fig. 14. After the initial overshoot, the speed settles at the set speed of 100 rad/s. Once the load is applied, the speed initially dips and then returns to the set reference value. Fig. 15 shows the electromagnetic and load torques of the BLDC motor. Due to the use of DTC, the torque of the BLDC motor is restricted to a narrow band, except in the instances where current commutation occurs.
Fig. 11 shows the arrangement of switches in the MC and the 3-Φ AC voltage source feeding the MC. ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’, ‘7’, ‘8’, ‘9’ denote the triggering pulses to the MC. ‘Phase A’, ‘Phase B’, ‘Phase C’ represent the 3-Φ AC voltage outputs of the MC. VII.
SIMULATION RESULTS
From the simulation, it is observed that the MC draws sinusoidal input currents from the AC voltage source. Fast Fourier Transform analysis is performed on the source phase current and the result is shown in Fig. 12. The Total Harmonic Distortion (THD) was found to be 0.88 %.
Fig. 13. Stator current and back EMF of the BLDC motor
Fig. 13 shows the stator current and the back EMF waveforms of the BLDC motor. The current waveform settles to zero after the initial transient stage and the motor is run at no load before a load torque of 2 Nm is applied at t = 0.16s. Smooth rectangular output current waveform is obtained with a small dip in between, due to commutation at every 60°. Fig. 14. Rotor Speed of the BLDC motor
Fig. 10. Bidirectional switch
Fig. 11. Arrangement of switches in the MC
TABLE IV. SIMULATION PARAMETERS Parameters Values Source 230 V AC Frequency 50 Hz Motor Rating Values Stator Inductance per phase (L) 0.03126 H Stator Resistance per Phase (R) 2.8 Ω Torque Constant (Kt) 1.23 Nm/A Back EMF Constant(kb) 1.23 V/rad/s Pole pairs (P) 4 Moment of Inertia (J) 0.098 kgm2 Coefficient of friction (B) 0.0078 N/rad/s Proportional Controller 0.04 Integral Controller 0.9 Hysteresis Controller Limits 0.01 Nm Bidirectional IGBT rating 1200 V, 200 A, Gate Voltage: 20V
The ripple in the torque produced by BLDC motor using the two–phase conduction DTC when fed by the MC and by the AC–DC–AC converter are shown in Figs. 15 and 16 respectively.
requirement for digital implementation is also reduced. The simplified two-phase DTC algorithm coupled with the advantages of MC and BLDC motor makes an ideal drive system. Industrial automation, machine tool applications, elevators are some areas of application of this drive system.
Fig. 15. Torque ripple in matrix converter fed BLDC drive using DTC Fig. 18. Simulink model for Power Factor Measurement
Fig. 16. Torque ripple in AC–DC–AC converter fed BLDC drive using DTC
Power factor is computed by using the Simulink model shown in Fig. 18. Time integral is performed till the current and voltage individually become zero. Thus, the current and voltage time offsets are obtained, the difference between the two provides the time offset difference between the two waveforms. From the time offset difference, the phase difference between the voltage and current waveforms is computed. The cosine of the phase difference gives the power factor of the system.
Fig. 19. FFT of the source phase current drawn by the AC–DC–AC Converter
REFERENCES [1] [2]
[3]
[4]
[5]
Fig. 17. Source Phase Voltage and Current waveforms
VIII. CONCLUSION The use of the MC over conventional AC-DC-AC converter not only reduces the torque ripple, as shown in Fig. 15 and 16, but also results in a low THD of 0.88% over 3.70% in ACDC-AC converters. Power factor of 0.963 was measured with the MC fed BLDC motor, whereas the use of AC-DC-AC converter measured 0.213. Thus the power factor at the source was significantly improved with use of the MC. Requirement of only two control variables makes the dynamic response of torque using DTC very fast, and memory
[6] [7]
[8]
[9]
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