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A. SVPWM Technique
Normally, as Fig.1 shows three-phase PMSM is driven by
voltage source.
As it is known to all, PMSM becomes more and more
popular in the field of the production of electric vehicle
nowadays. However, the control performance at low speed of
the motor is still influenced by ripple torque because of
harmonic wave. Space Vector Pulse Width Modulation
(SVPWM) is available to improve the quality of the stator
currents and reduce the ripple torque remarkably [9].
1) Fundamental Space Voltage Vectors: The reference
stator voltage space vector for PMSM can be defined as:
2 2
3 3 2 ( )
3
j j
U U Ue Ue s ab c
π π −
= +⋅ +⋅ (1)
Where, Ua, Ub, Uc are motor phase voltages.
In order to simplify the computational process, we plan to
introduce - axes shown as Fig.2. So Eq. (1) can be written
as Eq. (2).
U U jU s = + s s α β (2)
Let’s assume the state of the voltage source inverter is
‘001’. In this case VF1, VF4 and VF6 would be turned on. So
that the current would flow into the motor through armature
‘A’ and stream out of the motor from armature ‘B’ and
armature ‘C’. And then, we can suppose that the voltage value
of the three-phase of the motor equals 2/3U,-1/3U,-1/3U
respectively. According to Eq. (1) and Eq. (2), it can be
obtained that:
0
2 0
3 U Uj = + (3)
Singer sewing machine showroom in chennai Therefore, other Fundamental Space Voltage Vectors can be
calculated as the same way. And the states of ‘000’ and ‘111’
are defined as the Zero Vector.
2) Synthesis of Space Voltage Vectors: Suppose Ux and
Ux+60 are two adjacent Fundamental Space Voltage Vectors
shown as Fig.4. And then, we can get that˖
In the expression above, t1 and t2 stand for the action time of
the two adjacent Fundamental Space Voltage Vectors; while t0
and t7 represent the action time of the Zero Vector.
Fig. 4 Synthesis of Space Voltage Vectors.
With sine theorem, it can be found that
1 2
60
sin(60 ) sin sin120
x x
PWM PWM s
t t U U
TT U
θ θ
+
= = −
(5)
Combined with Eq. (4), we can work out that
1
2
60
2 sin(60 ) 3
2 sin
3
s
PWM
x
s
PWM
x
U t T
U
U t T
U
θ
θ
+
= −
=
(6)
Where, means the angle between US and Ux. If t1+t2>TPWM,
t1=t1TPWM / (t1+t2); and t2=t2 TPWM / (t1+t2).
And with Eq. (4), t0 and t7 can be defined as
0 7 12 ( )/2 PWM t t T tt = = −− (7)
3) Realization of SVPWM: Presently, SVPWM can be
realized through the regulation of voltage source inverter. The
procedure of SVPWM algorithm has always been divided into
three steps including sector judgement, calculation of the
action time and distribution of the duty factor.
Step 1: Sector Judgement. We find u1, u2 and u3 as three
unit normal vectors as presented in Fig.5. And with the -
axes transform, it can be formed as
1
2
3
1 0
1 3
2 2
1 3
2 2
u u u u u
α
β
= −
− −
(8)
Thus, the formula that used to sector judgement can be
described as:
3 21 Sector sign u sign u sign u =++ 4 ()2 () () (9)
Step 2: Calculation of the Action Time. The action time
for VF1 to VF6 located in the source voltage inverter was
deduced in Ref. [9] and [10]. So it can be computed that:
1
2
2 sin(60 ) 3
2 sin
3
s
PWM
s
PWM
U t T
U
U t T
U
θ
θ
= −
=
(10)
1
2
2 sin
3
2 sin(60 ) 3
s
PWM
s
PWM
U t T
U
U t T
U
θ
θ
=
= −
(11)
If the motor is operating in Sector I, II and IV, the action
time t1 and t2 can be calculated as Eq. (10), else, the t1 and t2
should be expressed as Eq. (11).
In Eq. (10) and Eq. (11), U is Fundamental Space
Voltage Vector. https://vssewingmachine.in/ In view of the fact that the range of is
located on [0, 60], we must finish the transformation between
(the angle between Us and Ux) and (the angle between Us
and axes ). And with the purpose of forming a
revolving magnetic field as a circle inscribed in the regular
hexagon shown as Fig.3, Us can be formulated as:
3
2
U UU s = ⋅ PID (12)
Where, UPID is the output value of PID regulator, and 0
UPID 1.
Step 3: Distribution of Duty Factor. The formula adopted
to calculate the duty factor is shown as
12 0
2 0
0
0.5
0.5
0.5
a
b
c
t tt t
tt t
t t
=++ ⋅
=+ ⋅
= ⋅
(13)
And ta
, tb, tc
represent the action time of VF1, VF3 and VF5 of
the source voltage inverter.
B. Measurement of Motor Speed and Motor Position
As discussed above, it is known that achieving SVPWM
needs the accuracy information about the rotation angle of the
motor.
1) Speed Measurement: Speed measurement is supposed
as the first step for position determination. In this case we
make use of three HALL sensors which are fixed on the
surface of the motor with 120 mechanical degree’s phase
delay. It is capable of generating such waves demonstrated in
Fig.7 when the motor is running. And from Fig.7, we can also
find that each HALL state indicates an operating sector of the
motor.
The IC interrupt of MCU will be responded at once only
when HALL state has been changing. As the result, the width
of HALL signal could be measured accessibility. In IC
interrupt subprogram, the switching time of the HALL state
will be recorded by timer3 of MCU. And thus, the time span
between two adjacent HALL states can be acquired easily as:
( ) ( 1) T timer k timer k HALL = −− (14)
According to Eq. (14), the speed of the motor can be
illustrated as:
deg
deg
60
ree
HALL
ree
HALL
Speed T
HALL
P
=
=
(15)
Where, HALLdegree means mechanical angle of the sector for
the motor, and P stands for pole-pairs of the motor. Here P=3.
And with the purpose of enhancing the precision of the
calculated speed, average digital filter should be introduced as
demonstrated in Ref. [3].
2) Position Determination: Position determination will
be processed minutely in PWM interrupt subprogram. Thus, in
the initialization part of the main program, PWM interrupt is
activated and frequency of PWM is installed as 10K Hz. So
that, the system will response PWM interrupt in every 0.1ms,
and we can get the angle between Us and axes as:
360 int( ) 360
PWM
PWM
p T Speed p T Speed ⋅ ⋅ Θ= ⋅ ⋅ − ⋅ (16)
Consequently, in accordance with the table II, the value of
can be worked out at last.
C. Nonlinear PI Control Algorithm
Being simple, robust, effective and applicable to a broad
class of systems, PID (Proportiona1-Integral-Derivative)
controllers have been the most widely used and well known
controllers in the industry for over 50 years [11]. And more
than 90% of industrial controllers are based around PID
algorithm, particularly at low levels [12]. However, there exist
several contradictions between response time and overshoot of
the system when referred to traditional linear PID algorithm
[13]. In order to cope with the problem above, a kind of
nonlinear PID algorithm has been studied for years.
Usually, derivative unit may cause system oscillation,
and PI control algorithm is chosen for speed regulation of the
system so as to eliminate such phenomenon. The parameters
for proportional and integral can be defined as:
( ) [1 sec ( ( ))] K p pp p k a b hcek =+ − (17)
( ) sec ( ( )) Ki i ii k a hb c e k = ⋅ (18)
Thus, the expression for the nonlinear PI algorithm can be
written as the flowing type:
( ) ( )( ( ) ( 1)) ( ) ( )
( ) ( 1) ( )
PI P i
PI PI PI
uk K k ek ek K kek
u k u k uk
Δ = − −+
= − +Δ
(19)
In the Eq. (16), e(k) is the speed error, uPI(k) is the output, and
uPI(k) is the output increment.
The 6 parameters listed in Eq. (17) and Eq. (18) for
nonlinear PI algorithm can be set as ap=0.8, bp=0.25, cp
=1,
ai
=0.15, bi
=0.025, ci
=1. On the basis of Eq. (17) to Eq. (19),
the simulation model of the algorithm can be created as Fig.8
by MATLAB.
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