28
CHAPTER 2
MODELLING AND STUDY OF THE
BLDC MOTOR DRIVE
2.1
INTRODUCTION
This chapter discusses the simplified, and a fast running simulation
model of BLDC motor drive using MATLAB/SIMULINK. This model
provides a mechanism for monitoring and controlling the voltage, current,
speed and torque response of motor. This simplified model can be used to
predict the performance of BLDC motor during transient and steady state
conditions. The complete simulation model of BLDC motor drive has several
sub-blocks. The inverter and switching function are implemented as separate
S-function builder blocks. The developed model can serve as a design tool for
engineers, researchers, and whosoever concern working on BLDC drive to
reduce the computation time in developing and simulating a complex model.
2.2
MODELLING OF BLDC MOTOR
A three phase star connected BLDC motor driven by a three phase
inverter with six step commutation is considered in this analysis. Figure.2.1
shows the simplified BLDC motor drive scheme. The BLDC drive scheme
consists of a three phase inverter, BLDC motor and position sensor. Figure
2.2 shows the ideal current, back emf and rotor position signal for a three
phase BLDC motor. Each interval starts when rotor and stator field lines are
120° apart and ends when they are 60° apart.
29
a
b
c
Figure 2.1 Simplified BLDC motor drive
Figure 2.2
Ideal current, back emf and hall signals of three phase
BLDC motor
The conducting interval of each phase is 120 electrical degrees. The
commutation instant is determined from the hall sensors mounted on the shaft.
30
A BLDC motor normally operated in a two phase ON mode i.e. at any instant
of time only two phases conduct current, leaving the third phase open which
is available for measuring the back emf.
The entire simulation model is developed on Matlab/Simulink
environment. The analysis is based on the following assumptions: Stator
resistances of all the windings are equal, self and mutual inductances are
constant, the motor is not saturated, iron losses are negligible and
semiconductor switches are ideal.
The voltage equations of the BLDC motor shown in Figure (2.1)
are given in Equations (2.1) to (2.3).
vab
vbc
vca
R ia ib L
d
ia ib ea eb
dt
d
R ib ic L ib ic eb ec
dt
d
R ic ia L ic ia ec ea
dt
where,
R
= stator resistance per phase
L
= stator inductance per phase
ia , ib , ic
= instantaneous stator phase currents
vab , vbc , vca = instantaneous stator line voltages
ea , eb , ec
= instantaneous phase back emf.
The current relationship is given by,
(2.1)
(2.2)
(2.3)
31
ia ib ic 0
(2.4)
Equation (2.4) is rewritten as,
ic ia ib (2.5)
Using Equation (2.5) the line voltage equations are rearranged as
vab
Ria ib L
vbc
Ria 2ib L
d
ia ib ea eb
dt
d
ia 2ib eb ec
dt
(2.6)
(2.7)
Back emf depends on the flux of the permanent magnet rotor and
the speed of the rotor, and is given in Equation (2.8)
ªe a º
«e »
« b»
«¬ec »¼
ª
« F (T )
e
«
k eZm « §
2S
F ¨T e «
3
2
©
«
4S
§
« F ¨T e 3
«¬ ©
º
»
»
·»
¹̧»
»
·»
¹̧»¼
(2.8)
The generated electromagnetic torque is given by Equation (2.9)
2S · Kt §
4S · º
K §
ªK
Te « t FTe ia t F¨Te ib F¨Te ic
2 ©
3 ¹̧
2 ©
3 ¹̧ »¼
¬2
(2.9)
The dynamics of the motor and load are expressed as given in
Equations (2.10) to (2.14)
32
Te
d
Z m TL
dt
d
K f Zm J Z m dt
K f Zm J
Te TL
(2.10)
(2.11)
d
Zm Te TL K f Z m
dt
k
1
Z m' f Z m >Te TL @
J
J
(2.12)
J
(2.13)
T m' Zm
(2.14)
where,
2.3
Te
= Electrical angle, degrees
Zm
= rotor speed, rad/sec
ke
= back-emf constant, volts/rad/sec
J
= moment of inertia, kg/m2
kf
= friction constant, N-m/rad/sec
TL
= load torque, N-m
kt
= torque constant, N-m/A
p
Tm
2
MODELLING OF BLDC DRIVE SYSTEM
Inverter and its switching sequence are modeled using S-function.
Table 2.1 gives information on the switching sequence, current direction and
the position signals used in triggering the switch in a phase A or B or C which
is more positive with respect to the neutral point.
33
Table 2.1 Switching sequence
Switching
interval
Degree
Position
Sequence
Phase current
Switch
sensor
closed
number
H1 H2 H3
A
B
C
(0-60)
0
1
0
0
T1
T4
+Ia
-Ib
off
(60-120)
1
1
1
0
T1
T6
+Ia
Off
-Ic
(120-180)
2
0
1
0
T3
T6
off
+Ib
-Ic
(180-240)
3
0
1
1
T3
T2
-Ia
+Ib
off
(240-300)
4
0
0
1
T5
T2
-Ia
Off
+Ic
(300-360)
5
1
0
1
T5
T4
off
-Ib
+Ic
There are six states in a cycle of operation of BLDC motor. The
values of voltages of BLDC drive are given in Equations (2.15) to (2.32) for
the state 1 to state 6. Figures 2.3 to 2.8 show the circuit configuration for
complimentary PWM from state 1 to state 6.
State 1 (0-60ƕ): The phases A and B are conducting and phase C is
freewheeling through diode D6. Figure 2.3 shows the circuit configuration of
drive for complimentary PWM for state 1.
(a) ON state and commutation
Figure 2.3
(b) Freewheeling
Circuit diagram of drive during commutation, ON state and
free wheeling for state 1
34
During ON state (ic
vab eab
vbc ebc
0)
Vs ea eb
(2.15)
1
Vs ea eb 2
During commutation ( ic z 0 )
vab eab
Vs ea eb
vbc ebc
eb ec
(2.16)
During freewheeling ( ia z 0, ic z 0 )
Vab eab
Vs ea eb
vbc ebc
Vs eb ec
(2.17)
State 2 (60-120ƕ): The phases A and C are conducting and phase B is
freewheeling through diode D3. Figure 2.4 Shows the circuit configuration
of inverter for complimentary PWM, for state 2.
(a) ON state and commutation
Figure 2.4
(b) Free wheeling
Circuit diagram of drive during commutation, ON state and
freewheeling for state 2
35
During ON state (ib 0)
vab eab
vbc ebc
1
Vs ea ec 2
1
Vs ea ec 2
(2.18)
During commutation (ib z 0)
vab eab
vbc ebc
ea eb
(2.19)
Vs eb ec
During freewheeling (ic z 0, ib z 0)
Vab eab
vbc ebc
Vs ea eb
(2.20)
eb ec
State 3 (120-180ƕ): The phases B and C are conducting and phase A is
freewheeling through diode D2. Figure 2.5 shows the circuit configuration of
inverter for complimentary PWM, for state 3.
a, ON state and commutation
Figure 2.5
b, Free wheeling
Circuit diagram of drive during commutation, ON state and
freewheeling for state 3
36
During ON state (ia 0)
vab eab
vbc ebc
1
Vs eb ec 2
(2.21)
V s eb ec
During commutation (ia z 0)
vab eab
vbc ebc
Vs ea eb
(2.22)
Vs eb ec
During freewheeling (ib z 0, ia z 0)
Vab e ab
ea e b
Vbc ebc
V s e b e c
(2.23)
State 4 (180-240ƕ): The phases A and B are conducting and phase C is
freewheeling through diode D5. Figure 2.6 shows the circuit configuration of
inverter for complimentary PWM for state 4.
a, ON state and commutation
Figure 2.6
b, Free wheeling
Circuit diagram of drive during commutation, ON state and
freewheeling for state 4
37
During ON state (ic
vab eab
vbc ebc
0)
Vs ea eb
(2.24)
1
Vs ea eb 2
During commutation (ic z 0)
vab eab
vbc ebc
Vs ea eb
(2.25)
eb ec
During freewheeling (ia z 0, ic z 0)
Vab eab
Vs ea eb
Vbc ebc
Vs eb ec
(2.26)
State 5 (240-300ƕ): The phases A and C are conducting and phase B is
freewheeling through diode D4. Figure 2.7 shows the circuit configuration of
inverter for complimentary PWM, for state 5.
(a) ON state and commutation
Figure 2.7
(b) Free wheeling
Circuit diagram of drive during commutation, ON state and
freewheeling for state 5
38
During ON state (ib
vab eab
vbc ebc
0)
1
Vs ea ec 2
(2.27)
1
Vs ea ec 2
During commutation (ib z 0)
vab eab
vbc ebc
ea eb
(2.28)
Vs eb ec
During freewheeling (ic z 0, ib z 0)
Vab eab
Vs ea eb
Vbc ebc
eb ec
(2.29)
State 6 (300-360ƕ): The phases B and C are conducting and phase A is
freewheeling through diode D1. Figure 2.8 shows the circuit configuration of
inverter for complimentary PWM, for state 6.
a, ON state and commutation
Figure 2.8
b, Free wheeling
Circuit diagram of drive during commutation, ON state and
freewheeling for state 6
39
During ON state ( ia 0)
vab eab
vbc ebc
1
Vs eb ec 2
Vs eb ec
(2.30)
During commutation (ia z 0)
vab eab
vbc ebc
Vs ea eb
Vs eb ec
(2.31)
During freewheeling (ib z 0, ia z 0)
Vab eab
Vbc ebc
ea eb
Vs eb ec
(2.32)
where,
vab , vbc and vca are the instantaneous stator line voltages
ia ,ib and ic are the instantaneous stator phase current.
Vs is the dc source voltage
2.4
SIMULATION MODEL OF BLDC MOTOR DRIVE
The simulation model of BLDC motor drive consists of four blocks
namely, the BLDC motor block, switching sequence block, Inverter block and
the generation of Hall signals block. Equations (2.1) to (2.14) are used to
develop the simulation model of motor using Matlab/Simulink.
40
2.4.1
BLDC motor block
The BLDC motor block consisting of four sub-blocks as labeled in
the Figure 2.9 representing the Equations (2.1) to (2.8).
phi'r
1
Step
v ab
vab
Te
is
is
we
we
phi'r
the
the
eabc
Tm
m_mech
Te
2
Back EMF and Flux Linkage
we,the,te
Hthe
Hthe
habc
Electromagnetic Torque
2
v bc
3
vbc
is
bldc_mech
To Workspace
4
eabc
1
habc
Hall sensor1
simout1
bldc current equation
Figure 2.9 Simulation model of BLDC Motor
Sub-block 1 as shown in Figure 2.10 represents the BLDC motor
Equations (2.5) and (2.6) which takes the voltages vs, vab and vbc as inputs and
gives the phase current, is as output. The sub- block 2 implements Equation
(2.9) using the current as input along with the flux linkage to produce the
electromagnetic torque (Te) as the output. The sub-block 3 represents
Equations (2.10) to (2.14) and is modeled as the mechanical block which
takes the electromagnetic torque (Te) along with the load torque (TL) as input
and gives speed and position as outputs. The sub-block 4 takes speed and
position as input and gives back emf and flux linkages as output as given in
Figure 2.11.
41
1
vab
1/L
In
Gain2
Out
integrator
Gain3
Divide1
R
2
Constant1
Divide
-3
2
1/L
In
1
is
Constant
Out
vbc
Gain1
integrator1
Gain4
R
Figure 2.10 Model for voltage and current equations
Phi' r
2
p
cos(u[1])
Flux/trap
the
1
1
phi'r
p
we
Phi' r
cos(u[1]-2*pi/3)
Flux/trap
Phi' r
cos(u[1]+2*pi/3)
Flux/trap
2
eabc
Figure 2.11 Simulation model for Generation of back emf and flux
linkage
2.4.2
Inverter and Switching logic
The inverter and switching logic are implemented as an S-function
that takes DC source voltage and the firing sequence as the inputs as shown in
Figures 2.12 and 2.13. The output voltage of the inverter depends on DC
source voltage, rotor position, phase current and also the value of the back
42
emf. In this modelling, hall sensor is used to sense the position of the rotor.
The hall sensor mounted on the stator generates a three digit number that
keeps changing for every 60 degree electrical rotation. For every 60 degree of
electrical rotation, one hall sensor changes its state. Hence, it takes 6 states to
complete one 360 degree of electrical rotation. This is shown in Table 2.1.
pwm1
2
pwmin
Duty Cycle(%)
pwm2
ha
pwm3
switch_sequencer
1
Gates_pwm
pwm4
Habc
hb
1
pwm5
hc
pwm6
S-Function Builder2
Figure 2.12 Simulation model for Generation of switching sequence
ha
habc
1
hb
hc
vs
2
vs
vab
ea
1
vab
eb
5
eabc
ec
ia
is
4
inverter_nochop_double
iz
ib
zero detecter
ic
is
pwmd1
pwmd2
vbc
pwmd3
2
vbc
3
pwmduty
pwmd4
pwmd5
pwmd6
S-Function Builder1
Figure 2.13 Simulation model of Inverter using S-function builder
43
2.4.3
Generation of hall signals
The hall signals give the position and the speed information of the
rotor. Turn on and turn off of the power switches of the inverter are
synchronized with the position signal. Based on the position information, the
controller generates the PWM signal to turn on and turn off of the switches to
control the speed of the motor. When a switch in a phase is turned off, the
outgoing phase current freewheels through the diode where as the incoming
phase current increases from zero to the new steady state. The position of the
rotor decides the incoming and outgoing phases. The speed of the motor is
controlled by measuring the actual speed of the motor and comparing it with
the set speed to generate an error signal. This error signal is amplified with a
PI controller and it is used to vary the duty cycle, thereby varying the speed.
Figure 2.14 shows the hall sensor block, where the position signals are
generated by using simple relational and logical operators, and Figure 2.15
shows the generated hall sensor signals. These signals are then used in the
speed calculation. The specifications of the motor used for the simulation are
given in Table 2.2.
sin
1
Hthe
T rigonometric
Function
cos
T rigonometric
Functi on1
atan2
-K-
T rigonometric
Function2
Gai n
-60
Constant
120
>=
Relational
Operator
AND
Convert
<=
Logical
Operator
Data Type Conversion
OR
Convert
Relational
Operator1
Constant1
60
Constant2
-120
Constant3
-180
Constant4
0
Constant5
>=
Relational
Operator2
<=
Logical
Data T ype Conversion1
Operator2
1
habc
Relational
Operator3
>=
AND
Relational
Operator4
<=
Convert
Logical Data T ype Conversion2
Operator1
Relational
Operator5
Figure 2.14 Simulation model for generation of signals from hall sensors
44
Figure 2.15 Generated hall sensors signals
Table 2.2 Motor Parameters
MOTOR PARAMETERS
VALUES
Voltage
40 Volt
Current
20 Amps
Per phase resistance (R)
0.6 ohm
Per phase Inductance (L)
0.42mH
Moment of inertia (J)
0.0002 Kg-m2
Back emf Constant (Kb)
0.1 V/rad/sec
Torque Constant(Kt)
0.1 N-m/A
Number of poles
4
Speed
1200 rpm
45
2.5
OPEN LOOP CONTROL OF BLDC MOTOR DRIVE
In open loop control, the duty cycle is directly calculated from the
set speed as there is no feedback. Figure 2.16 shows the developed simplified
open loop simulation model of the BLDC motor drive to study the
characteristics of the system.
5
7
vab
gate_signal
habc
40
Habc
v ab
Habc
Terminal Voltage
we,the,te
pwmduty
3
is
Duty Cy cle(%)
is
duty cycle
1
we,the,te
Gates_pwm
.5
2
habc
v ab
vs
v bc
iabc
v bc
4
eabc
switch sequence
eabc
Inverter
6
BLDC Motor
eabc
vbc
F28335 eZdsp
Figure 2.16 Simplified simulation model of BLDC motor for open loop
operation
The performance during starting, running and speed reversal of the
BLDC motor drive under open loop operation are discussed in this section.
2.5.1
Starting operation
The BLDC drive system is started with a constant load of 0.5 N-m
from stand still condition. Figure 2.17 shows the starting response of the
phase current, back emf, speed and electromagnetic torque of the motor. The
speed response reaches the steady state value at time t=0.5 sec. The stator
46
phase winding draws a maximum current of 17 amps during starting and
reduces to 4 amps during steady state. The electromagnetic torque increases to
a maximum value of 12.2 N-m and then drops to 2.1 N-m at steady state.
(a) phase current
(b) back emf
Figure 2.17 (Continued)
47
(c) Speed waveform
(d) Electromagnetic torque
Figure 2.17 Starting responses of BLDC motor drive
2.5.2
Running operation
The response of the drive under load perturbation is shown in
Figure 2.18. When the motor is running at steady state for a constant load of
0.5 N-m, an additional load of 4 N-m was applied at t=2 seconds and was
removed at t=2.5 seconds. The change in load causes the speed to drop from
700 rpm to 600 rpm over a period of 0.025 second. The current reaches the
maximum value of 15 amps with in one cycle and the developed torque
48
increases to 4.5 N-m which is equal to the applied torque. It is seen that the
back emf decreases from 15 volt to 10 volt, as the speed falls to 600 rpm with
increase in load torque of 4N-m.
(a) Phase current
(b) back emf waveform
Figure 2.18 (Continued)
49
(c) Speed response
(d) Electromagnetic torque
Figure 2.18 Responses of BLDC motor during load perturbations
2.6
HARDWARE IMPLEMENTATION
Figure 2.19 shows the hardware setup for the validation of the
simplified modelling of BLDC motor drive. The BLDC motor scheme has
four major elements: A dSPACE DSP board, BLDC motor, processor board
and PC. The dSPACE DS2204 DSP board forms the main core of the closed
loop control system.
50
Figure 2.19 Hardware setup for BLDC motor drive
Apart from the duties of controlling the operator interface, it
performs the acquisition of the feedback signal, computes the error signal,
delivers the error signal to the control algorithm and executes the control
algorithm to determine the switching signal.
The control algorithm is built within Simulink environment
combined with the real time interface provided by dSPACE and is
implemented by the main processor of DS2204 board. The processor board is
interfaced with the DS2204 DSP board. The processor board is programmed
to monitor and measure the position and speed information and feed this
information back to the control algorithm.
A 0.5 hp, 2000 rev/min BLDC machine manufactured by Hurst
motors has been used. The BLDC motor is equipped with hall sensors and
incremental optical encoder. The sensors detect the rotor position and give the
information to the switching logic to excite the proper phase of the stator. The
51
optical position encoder with resolution of 500 pulse/ revolution is used to
give the speed and position information for the closed loop operation.
The measured line voltage Vab waveform is shown in Figure 2.20.
The measured phase current and speed waveforms on no load and loaded
condition is shown in Figures 2.21 and 2.22. The simulation and hardware
results are closely matched and thus validating the simplified modelling.
Figure 2.20 Measured line voltage Vab waveform
(a) Phase current
Figure 2.21 (Continued)
52
(b) Speed waveform
Figure 2.21 Measured phase current and speed waveform on no-load
(a) Phase current
Figure 2.22 (Continued)
53
(b) Speed waveform
Figure 2.22 Measured phase current and speed waveform on load
2.7
CLOSED LOOP OPERATION OF BLDC MOTOR DRIVE
In the closed loop operation, the set speed and the actual speed are
compared to generate an error signal. This error signal is fed to the PI
controller to control the duty cycle for varying the dc voltage applied to the
motor. The simulation model of BLDC motor drive for the closed loop
operation is shown in Figure 2.23.
54
<signal1>
-KGain
5
7
vab
gate_signal
habc
40
Habc
Habc
v ab
we,the,te
Terminal Voltage
gate_signal
pwmduty
is
Speed input
is
v bc
eabc
Inverter
6
<signal1>
1
we,the,te
3
iabc
v bc
eabc
Controller
2
habc
v ab
vs
BLDC Motor
4
eabc
vbc
Figure 2.23
Simplified simulation model of BLDC drive for closed loop
operation
2.7.1
Design of PI controller
The structure of PI controller selected to the BLDC motor drive is
shown in Figure 2.24.
Figure 2.24 Structure of a PI controller
55
Equation (2.33) gives the expression of the PI controller output.
The output of controller is sum of proportional gain, K p times the magnitude
of error and the Ki times the integral of error.
K p e(t ) K i ³ e(t ) dt
t
u (t )
(2.33)
0
where,
u(t) = output of the controller
e(t) = error difference between the set point and the actual
output
K p = Proportional gain
Ki = Integral gain
Table 2.3 gives a brief note on the effect of PI controller parameters
for a step response of speed.
Table 2.3
Effect of PI controller parameters for a step response of the
speed
Parameter Rise time Overshoot Settling time
Increase in Decrease Increase
Steady state
stability
error
Small change Decrease
Degrade
Increase
Degrade
Kp
Increase in Decrease Increase
Ki
Decrease
significantly
56
Due to its simplicity, PI controllers are used in more than 90% of
closed-loop control systems in industrial processes.
The PI controller was designed by trial and error method. The best
controller gains are found to be K p = 0.01 and Ki =0.000013. The responses
of the drive under different operating conditions are discussed in the
succeeding sections.
2.7.2
Load perturbation
The closed loop response of BLDC motor drive under load
perturbation is shown in Figure 2.25. The motor is run at 600 rpm under
steady state. A step load of 4 N-m is applied at t= 2 second and removed at t=
2.2 second. The responses of rotor speed, phase current and electromagnetic
torque are shown in Figure 2.25. The stator phase current reaches to the
steady state value of 12 amps within half a cycle. The dip in speed is around
100 rpm and the controller takes 0.25 second to reach the set speed. The
electromagnetic torque rises to a maximum value of 3.7 N-m and remains till
the load is changed.
(a) waveform of phase current
Figure 2.25 (Continued)
57
(b) Waveform of back emf
(c) Speed response
(d) Electromagnetic torque response
Figure 2.25 Responses of BLDC motor during load perturbations
58
2.7.3
Change in set speed
Figure 2.26 shows the response of the motor speed when the set
speed is changed for a constant load torque of 0.5 N-m. The motor starts at
t=1 second and reaches the set speed of 300 rpm in 0.25 seconds. The set
speed was again changed to 600 rpm at t=2 seconds. The controller tracks the
new set speed within 0.5 seconds. This clearly shows that the controller
behaves smoothly without any oscillations and steady state error. The set
speed is superimposed on the actual speed in order to compare its
performance.
Figure 2.26 Speed response of BLDC motor during change in set speed
2.7.4
Speed reversal
Figure 2.27 shows the response of the controller during speed
reversal. The motor operating with a constant load and with a set speed of 600
rpm in clock wise direction is suddenly made to rotate in the counter clock
wise direction at t= 2.5 sec. The controller smoothly shifts the rotor speed in
the opposite direction and it takes 0.5 sec to attain the set speed. This shows
that the controller works well even under speed reversal.
59
Figure 2.27 Speed response of BLDC motor during speed reversal
2.8
CONCLUSION
The simplified modelling and analysis of BLDC drive with hall
sensor using Matlab/Simulink has been discussed. The simulation model of
BLDC motor was developed as several sub blocks, which also includes the
inverter and switching logic control. This facilitates the analysis of transient
and steady state behavior of the drive in detail. The operation of the drive
during on state, commutation and freewheeling are discussed in detail with
the necessary equations. The responses of the BLDC motor drive during
starting, load perturbations, change in set speed and speed reversal were
monitored and compared with the ideal waveforms and also with the
waveforms available in the literature. The performance of BLDC drive system
was also compared with a hardware setup to validate the simulation model.
The simulation and hardware results are closely matched and hence this
model can be easily extended for the sensorless control of the BLDC motor
drive with an Artificial intelligence controller.