power system control

1. energies
Article
Determining the Position of the Brushless DC
Motor Rotor
Krzysztof Kolano
Faculty of Electric Drives and Machines, Lublin University of Technology, 20-618 Lublin, Poland;
k.kolano@pollub.pl
Received: 11 March 2020; Accepted: 30 March 2020; Published: 1 April 2020
Abstract: In brushless direct current (or BLDC) motors with more than one pole pair, the status of
standard shaft position sensors assumes the same distribution several times for its full mechanical
rotation. As a result, a simple analysis of the signals reflecting their state does not allow any
determination of the mechanical position of the shaft of such a machine. This paper presents a
new method for determining the mechanical position of a BLDC motor rotor with a number of
pole pairs greater than one. In contrast to the methods used so far, it allows us to determine the
mechanical position using only the standard position sensors in which most BLDC motors are
equipped. The paper describes a method of determining the mechanical position of the rotor by
analyzing the distribution of errors resulting from the accuracy proposed by the BLDC motor’s Hall
sensor system. Imprecise indications of the rotor position, resulting from the limited accuracy of the
production process, offer a possibility of an indirect determination of the rotor’s angular position of
such a machine.
Keywords: rotor position; BLDC motor; sensor misalignment
1. Introduction
Compared to conventional DC brush motors, BLDC brushless motors using shaft position
sensors are becoming increasingly popular in many applications, due to their relatively low cost,
high performance and high reliability. The operating issues associated with the control of such
machines have been extensively studied under the assumption of the correct operation of the shaft
position sensor system, namely Hall sensors, and under the assumption of the symmetry of the signals
generated by them [1–4].
Some industrial applications require that the position of the rotor is clearly defined and the
information about the rotor is used by the control system during the operation cycle of the device.
An example of this is the machine spindle, which must be properly positioned in relation to the cutter
hopper, in order for the tools to be automatically picked and replaced.
A common solution to this problem of determining the rotor position is to equip the drive system
with additional elements, namely, shaft position sensors, whose resolution is matched to the desired
accuracy of its positioning. All kinds of encoders of the motor shaft’s absolute position are used for
this purpose [5]: absolute position encoders, and incremental encoders with an additional index “I”
signal generated once per mechanical rotation.
The presence of Hall sensors, mounted in the motor, led the author to use them as an element
defining the mechanical angle of the motor shaft. Determining the position of the BLDC motor rotor
with an accuracy of 60 mechanical degrees for motors with one pair of poles is possible according to
the formula:
θe = p · θm (1)
Energies 2020, 13, 1607; doi:10.3390/en13071607 www.mdpi.com/journal/energies

2. Energies 2020, 13, 1607 2 of 9
(where p equals the number of pole pairs), and it is determined directly by the state of the shaft position
sensors, as the change in the electrical angle δe is equal to the change in the mechanical angle δm of the
motor shaft position – Figure 1a.
Energies 2020, 13, x FOR PEER REVIEW 2 of 9
(where 𝑝equals the number of pole pairs), and it is determined directly by the state of the shaft
position sensors, as the change in the electrical angle 𝛿 is equal to the change in the mechanical
angle 𝛿 of the motor shaft position – Figure 1a.
However, in machines with the number of p1 pole pairs (p=4 for most BLDC motors that are
present on the market), equipped only with standard Hall sensors, it is not possible to determine the
mechanical position of the shaft solely on the basis of the analysis of their state, because the Hall
sequence repeats p times per one mechanical revolution, as shown in Figure 1b.
(a) (b)
Figure 1. Relationships between the values of mechanical angle 𝜃 , electrical angle 𝜃 and the state
of Hall sensors for brushless direct current (or BLDC) motors for: (a) 1; (b) 4 pairs of poles.
Until now, the solution to this problem has been to install additional elements determining the
position of the motor shaft. Unfortunately, this makes it necessary to mechanically modify the
typical BLDC motors available on the market.
This article proposes a method that allows us to determine the mechanical position of the rotor
with an accuracy equal to 60/p mechanical degrees for BLDC motors with the number of pole pairs
greater than one, without the need for additional sensors. This accuracy value is sufficient for rotor
positioning in many industrial applications.
2. Errors in the Positioning of Motor Sensors
The disadvantages resulting from the real, different from ideal, arrangement of shaft position
sensors are known, and described in detail in the literature [6–13]. This problem is caused by the
relatively low accuracy of the motor shaft position determination system during mass production.
Researchersʹ efforts focus on compensating for sensor placement errors by assuming perfect
symmetry of the rotating magnetic element (Figure 1 𝛿 =0). This makes it possible to treat the
multipolar machine as a machine with one pair of poles [14–16]. Defining the errors in the
measurement of the speed and position of the motor shaft, resulting only from the incorrect
arrangement of sensors, is a large simplification in the analysis of the operation of the real system for
determining the position of the BLDC motor rotor. If the error concerned only the sensor location
accuracy (𝛿 – Figure 2), the speed measurement errors could be uniquely determined for each
combination of Hall sensor states. In fact, these errors result to the same extent from the accuracy of
sensor placement (𝛿 ), as from the accuracy of making the magnetic ring rotating coaxially with the
shaft (𝛿 ); see Figure2. The angular velocity 𝛿 of the shaft, measured by the microprocessor
system as a value inversely proportional to the time 𝑡 of changing the state of the position sensor
values, amounts to:
𝜔
𝜋 𝛿 𝛿
𝑝 ∙ 𝑡
(2)
Figure 1. Relationships between the values of mechanical angle θm, electrical angle θe and the state of
Hall sensors for brushless direct current (or BLDC) motors for: (a) 1; (b) 4 pairs of poles.
However, in machines with the number of p 1 pole pairs (p = 4 for most BLDC motors that
are present on the market), equipped only with standard Hall sensors, it is not possible to determine
the mechanical position of the shaft solely on the basis of the analysis of their state, because the Hall
sequence repeats p times per one mechanical revolution, as shown in Figure 1b.
Until now, the solution to this problem has been to install additional elements determining the
position of the motor shaft. Unfortunately, this makes it necessary to mechanically modify the typical
BLDC motors available on the market.
This article proposes a method that allows us to determine the mechanical position of the rotor
with an accuracy equal to 60/p mechanical degrees for BLDC motors with the number of pole pairs
greater than one, without the need for additional sensors. This accuracy value is sufficient for rotor
positioning in many industrial applications.
2. Errors in the Positioning of Motor Sensors
The disadvantages resulting from the real, different from ideal, arrangement of shaft position
sensors are known, and described in detail in the literature [6–13]. This problem is caused by the
relatively low accuracy of the motor shaft position determination system during mass production.
Researchers’ efforts focus on compensating for sensor placement errors by assuming perfect symmetry
of the rotating magnetic element (Figure 1 δmr = 0). This makes it possible to treat the multipolar
machine as a machine with one pair of poles [14–16]. Defining the errors in the measurement of the
speed and position of the motor shaft, resulting only from the incorrect arrangement of sensors, is a
large simplification in the analysis of the operation of the real system for determining the position of the
BLDC motor rotor. If the error concerned only the sensor location accuracy (δms– Figure 2), the speed
measurement errors could be uniquely determined for each combination of Hall sensor states. In fact,
these errors result to the same extent from the accuracy of sensor placement (δms), as from the accuracy
of making the magnetic ring rotating coaxially with the shaft (δmr); see Figure 2. The angular velocity
δr of the shaft, measured by the microprocessor system as a value inversely proportional to the time ts
of changing the state of the position sensor values, amounts to:
ωr =
π ± δms ± δmr
p · ts
(2)

3. Energies 2020, 13, 1607 3 of 9
Energies 2020, 13, x FOR PEER REVIEW 3 of 9
If non‐zero errors 𝛿 or 𝛿 are assumed, the speed values for the individual sectors may
74
vary despite the shaft rotating at a constant speed. This difference can be determined by comparing
75
the measured value with a reference value, measured e.g., by an external measuring system.
76
H2
H3
H1
ms
mr
Magnetic ring
Optimal axis
Real axis
Optimal axis
Real axis
77
Figure 2. Visualisation of sensor (𝛿𝑚𝑠) and magnet (𝛿𝑚𝑟) misalignment in a BLDC shaft sensing
78
system.
79
In extreme cases, assuming that both the sensor system and the magnetic ring are made with
80
limited accuracy, an error in shaft speed measurement can be calculated for each sector of the BLDC
81
motor (Figure 3), e.g., for a motor with four pole pairs, the number of sectors is 24 per full
82
mechanical rotation of the shaft (Figure 3b).
83
(a) (b)
1
0
1
1
0
0
1
1
0
0
1
0
0
1
1
0
0
1
1 pole pair ms≠0 and mr≠0
Hall
sensors
Elec. Angle

e
Mech. angle

m
360o
360o
60o
180o
120o
0o
0o
240o
300o
Error
e1
e2
e3
e4
e6
e5
1
2
3
4
6
5
Sector
number
12
1
0
1
1
0
0
1
1
0
0
1
0
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
4 pole pairs ms≠0 and mr≠0
Hall
sensors
360o
360o
90o
270o
180o
0o
0o
e6
e4
e2
e1
e3
e5
Error
e12
e24
Elec. Angle

e
Mech. angle

m
18
6
4
2
1
3
5
24
Sector
number
e18
Figure 3. Relationships between the values of mechanical angle 𝜃 , electrical angle 𝜃 and the state
84
of Hall sensors for brushless direct current (or BLDC) motors for: (a) 1; (b) 4 pairs of poles for 𝛿𝑚𝑠≠0
85
and 𝛿𝑚𝑟≠0.
86
For a motor with one pair of poles six unique error values can be assigned to a specific state of
87
Hall sensors (Figure 3a), while for a motor with the number of pole pairs greater than one, these
88
errors cannot be assigned to a specific state of sensors (Figure 3b).
89
Errors 𝛿 and 𝛿 result in switching the keys of the motor stator winding controller in a
90
suboptimal position of the rotor, which in turn leads to the electromagnetic torque pulsation of the
91
motor, increases the amplitude of its current (Figure 4) and the noise level of the motor operation.
92
Boosting the amplitude of the current results in an increase in electrical losses and electromagnetic
93
Figure 2. Visualisation of sensor (δms) and magnet (δmr ) misalignment in a BLDC shaft sensing system.
If non-zero errors δms or δmr are assumed, the speed values for the individual sectors may vary
despite the shaft rotating at a constant speed. This difference can be determined by comparing the
measured value with a reference value, measured e.g., by an external measuring system.
In extreme cases, assuming that both the sensor system and the magnetic ring are made with
limited accuracy, an error in shaft speed measurement can be calculated for each sector of the BLDC
motor (Figure 3), e.g., for a motor with four pole pairs, the number of sectors is 24 per full mechanical
rotation of the shaft (Figure 3b).
Energies 2020, 13, x FOR PEER REVIEW 3 of 9
If non‐zero errors 𝛿 or 𝛿 are assumed, the speed values for the individual sectors may
74
vary despite the shaft rotating at a constant speed. This difference can be determined by comparing
75
the measured value with a reference value, measured e.g., by an external measuring system.
76
H2
H3
H1
ms
mr
Magnetic ring
Optimal axis
Real axis
Optimal axis
Real axis
77
Figure 2. Visualisation of sensor (𝛿𝑚𝑠) and magnet (𝛿𝑚𝑟) misalignment in a BLDC shaft sensing
78
system.
79
In extreme cases, assuming that both the sensor system and the magnetic ring are made with
80
limited accuracy, an error in shaft speed measurement can be calculated for each sector of the BLDC
81
motor (Figure 3), e.g., for a motor with four pole pairs, the number of sectors is 24 per full
82
mechanical rotation of the shaft (Figure 3b).
83
(a) (b)
1
0
1
1
0
0
1
1
0
0
1
0
0
1
1
0
0
1
1 pole pair ms≠0 and mr≠0
Hall
sensors
Elec. Angle

e
Mech. angle

m
360o
360o
60o
180o
120o
0o
0o
240o
300o
Error
e1
e2
e3
e4
e6
e5
1
2
3
4
6
5
Sector
number
12
1
0
1
1
0
0
1
1
0
0
1
0
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
4 pole pairs ms≠0 and mr≠0
Hall
sensors
360o
360o
90o
270o
180o
0o
0o
e6
e4
e2
e1
e3
e5
Error
e12
e24
Elec. Angle

e
Mech. angle

m
18
6
4
2
1
3
5
24
Sector
number
e18
Figure 3. Relationships between the values of mechanical angle 𝜃 , electrical angle 𝜃 and the state
84
of Hall sensors for brushless direct current (or BLDC) motors for: (a) 1; (b) 4 pairs of poles for 𝛿𝑚𝑠≠0
85
and 𝛿𝑚𝑟≠0.
86
For a motor with one pair of poles six unique error values can be assigned to a specific state of
87
Hall sensors (Figure 3a), while for a motor with the number of pole pairs greater than one, these
88
errors cannot be assigned to a specific state of sensors (Figure 3b).
89
Errors 𝛿 and 𝛿 result in switching the keys of the motor stator winding controller in a
90
suboptimal position of the rotor, which in turn leads to the electromagnetic torque pulsation of the
91
motor, increases the amplitude of its current (Figure 4) and the noise level of the motor operation.
92
Boosting the amplitude of the current results in an increase in electrical losses and electromagnetic
93
Figure 3. Relationships between the values of mechanical angle θm, electrical angle θe and the state of
Hall sensors for brushless direct current (or BLDC) motors for: (a) 1; (b) 4 pairs of poles for δms , 0 and
δmr , 0.
For a motor with one pair of poles six unique error values can be assigned to a specific state of
Hall sensors (Figure 3a), while for a motor with the number of pole pairs greater than one, these errors
cannot be assigned to a specific state of sensors (Figure 3b).
Errors δms and δmr result in switching the keys of the motor stator winding controller in a
suboptimal position of the rotor, which in turn leads to the electromagnetic torque pulsation of the
motor, increases the amplitude of its current (Figure 4) and the noise level of the motor operation.
Boosting the amplitude of the current results in an increase in electrical losses and electromagnetic
interference, and forces the constructors of the controller to use in it transistors with a higher rated
current, so that they do not suffer thermal and dynamic damage during long-term operation [17].

4. Energies 2020, 13, 1607 4 of 9
Energies 2020, 13, x FOR PEER REVIEW 4 of 9
interference, and forces the constructors of the controller to use in it transistors with a higher rated
94
current, so that they do not suffer thermal and dynamic damage during long-term operation [17].
95
96
Figure 4. Measurement of phase currents of a BLDC motor (four pole pairs, nominal speed = 3000
97
rpm/min, nominal power = 380 W) during full mechanical rotation.
98
3. Laboratory Measurements Taken on Real Motors
99
In order to confirm the universality of this phenomenon in motors available on the market, it
100
was decided to purchase four types of BLDC motors from four different manufacturers, and to
101
analyze the performance of their shaft positioning systems. For this purpose, an experiment was
102
developed to compare the actual 𝜔 speed measured by the incremental encoder, with the speed
103
measured using the signal generated by the motor’s Hall sensors in an open loop drive system. The
104
measuring functions were executed by the dSpace GmbH (Paderborn, Germany) dSpace
105
MicroLabBox system supported by the Tektronix (Beaverton, Oregon United States) Digital
106
Phosphor Oscilloscope 5054-B (Figure 5). As a measure of the error in determining the speed in
107
individual sectors 𝑒𝑠𝑒𝑐, a percentage ratio of the 𝜔𝑠𝑒𝑐 calculated speed to the actual 𝜔 measured
108
by the reference element, i.e. the incremental encoder, was proposed. Additionally, the values of
109
phase currents of the BLDC motor were measured during the tests.
110
111
Figure 5. Research set-up.
112
One of the aims of the experiment was to determine the influence of sensor distribution error on
113
the pulsation of the actual rotational speed. It was predicted that significant oscillations of the motor
114
phase current amplitude, caused by the premature or delayed activation of the controller transistors,
115
may significantly increase this pulsation. During laboratory tests, it was found that in a wide range
116
Figure 4. Measurement of phase currents of a BLDC motor (four pole pairs, nominal speed = 3000
rpm/min, nominal power = 380 W) during full mechanical rotation.
3. Laboratory Measurements Taken on Real Motors
In order to confirm the universality of this phenomenon in motors available on the market, it was
decided to purchase four types of BLDC motors from four different manufacturers, and to analyze the
performance of their shaft positioning systems. For this purpose, an experiment was developed to
compare the actual ωref speed measured by the incremental encoder, with the speed measured using
the signal generated by the motor’s Hall sensors in an open loop drive system. The measuring functions
were executed by the dSpace GmbH (Paderborn, Germany) dSpace MicroLabBox system supported by
the Tektronix (Beaverton, Oregon United States) Digital Phosphor Oscilloscope 5054-B (Figure 5). As a
measure of the error in determining the speed in individual sectors esec, a percentage ratio of the ωsec
calculated speed to the actual ωref measured by the reference element, i.e. the incremental encoder,
was proposed. Additionally, the values of phase currents of the BLDC motor were measured during
the tests.
Energies 2020, 13, x FOR PEER REVIEW 4 of 9
interference, and forces the constructors of the controller to use in it transistors with a higher rated
94
current, so that they do not suffer thermal and dynamic damage during long‐term operation [17].
95
96
Figure 4. Measurement of phase currents of a BLDC motor (four pole pairs, nominal speed = 3000
97
rpm/min, nominal power = 380 W) during full mechanical rotation.
98
3. Laboratory Measurements Taken on Real Motors
99
In order to confirm the universality of this phenomenon in motors available on the market, it
100
was decided to purchase four types of BLDC motors from four different manufacturers, and to
101
analyze the performance of their shaft positioning systems. For this purpose, an experiment was
102
developed to compare the actual 𝜔 speed measured by the incremental encoder, with the speed
103
measured using the signal generated by the motor’s Hall sensors in an open loop drive system. The
104
measuring functions were executed by the dSpace GmbH (Paderborn, Germany) dSpace
105
MicroLabBox system supported by the Tektronix (Beaverton, Oregon United States) Digital
106
Phosphor Oscilloscope 5054‐B (Figure 5). As a measure of the error in determining the speed in
107
individual sectors 𝑒𝑠𝑒𝑐, a percentage ratio of the 𝜔𝑠𝑒𝑐 calculated speed to the actual 𝜔 measured
108
by the reference element, i.e. the incremental encoder, was proposed. Additionally, the values of
109
phase currents of the BLDC motor were measured during the tests.
110
111
Figure 5. Research set‐up.
112
One of the aims of the experiment was to determine the influence of sensor distribution error on
113
the pulsation of the actual rotational speed. It was predicted that significant oscillations of the motor
114
phase current amplitude, caused by the premature or delayed activation of the controller transistors,
115
may significantly increase this pulsation. During laboratory tests, it was found that in a wide range
116
Figure 5. Research set-up.
One of the aims of the experiment was to determine the influence of sensor distribution error on
the pulsation of the actual rotational speed. It was predicted that significant oscillations of the motor
phase current amplitude, caused by the premature or delayed activation of the controller transistors,
may significantly increase this pulsation. During laboratory tests, it was found that in a wide range of
speeds (from 10% to 100% of the nn rated speed), and for wide ranges of load torque changes (from idle
to rated load), the actual speed pulsation is imperceptible (Figure 6), despite significant (up to 80%;
Figure 4) phase current fluctuations of the motor.

5. Energies 2020, 13, 1607 5 of 9
Energies 2020, 13, x FOR PEER REVIEW 5 of 9
of speeds (from 10% to 100% of the 𝑛 rated speed), and for wide ranges of load torque changes
117
(from idle to rated load), the actual speed pulsation is imperceptible (Figure 6), despite significant
118
(up to 80%; Figure 4) phase current fluctuations of the motor.
119
120
Figure 6. Actual and measured speed of a BLDC motor with factory nonsymmetrical Hall sensor
121
array, for a BLDC motor with four pairs of poles.
122
The measurements showed that the accuracy of speed determination in the experiment,
123
assumed at 0.5%, was not exceeded. This means that, regardless of errors in the BLDC motor
124
position measurement system, we can assume that its average steady state rotational speed can be
125
considered constant, regardless of the phase current amplitude fluctuations of the motor. This is a
126
very important conclusion, allowing the thesis that, in the steady state the average actual speed for
127
individual motor sectors is equal to the average speed measured during the full mechanical rotation
128
of the shaft. The actual average speed of a full revolution can be measured correctly by any single
129
Hall sensor. This results in the possibility of not using an additional reference speed measurement
130
system (i.e. an additional encoder) in favor of using averaged full revolution speed, which is
131
independent of the sensor placement error [12].
132
Figure 7 shows the calculated speed error values for the individual 𝑒𝑠𝑒𝑐 sectors for four
133
different BLDC motors with powers ranging from 60 W to 1500 W. All motors had four pairs of poles
134
each, for which 24 sectors of speed measurement can be defined per one mechanical revolution. Each
135
of them corresponds to a mechanical rotation angle equal to π/(3∙p), which gives an angle value for
136
each sector equal to 15 mechanical degrees.
137
138
Figure 7. Rotational speed computation error „𝑒 ” in particular sectors of the tested motors.
139
Figure 6. Actual and measured speed of a BLDC motor with factory nonsymmetrical Hall sensor array,
for a BLDC motor with four pairs of poles.
The measurements showed that the accuracy of speed determination in the experiment, assumed at
0.5%, was not exceeded. This means that, regardless of errors in the BLDC motor position measurement
system, we can assume that its average steady state rotational speed can be considered constant,
regardless of the phase current amplitude fluctuations of the motor. This is a very important conclusion,
allowing the thesis that, in the steady state the average actual speed for individual motor sectors is
equal to the average speed measured during the full mechanical rotation of the shaft. The actual
average speed of a full revolution can be measured correctly by any single Hall sensor. This results
in the possibility of not using an additional reference speed measurement system (i.e. an additional
encoder) in favor of using averaged full revolution speed, which is independent of the sensor placement
error [12].
Figure 7 shows the calculated speed error values for the individual esec sectors for four different
BLDC motors with powers ranging from 60 W to 1500 W. All motors had four pairs of poles each, for
which 24 sectors of speed measurement can be defined per one mechanical revolution. Each of them
corresponds to a mechanical rotation angle equal to π/(3·p), which gives an angle value for each sector
equal to 15 mechanical degrees.
Energies 2020, 13, x FOR PEER REVIEW 5 of 9
of speeds (from 10% to 100% of the 𝑛 rated speed), and for wide ranges of load torque changes
117
(from idle to rated load), the actual speed pulsation is imperceptible (Figure 6), despite significant
118
(up to 80%; Figure 4) phase current fluctuations of the motor.
119
120
Figure 6. Actual and measured speed of a BLDC motor with factory nonsymmetrical Hall sensor
121
array, for a BLDC motor with four pairs of poles.
122
The measurements showed that the accuracy of speed determination in the experiment,
123
assumed at 0.5%, was not exceeded. This means that, regardless of errors in the BLDC motor
124
position measurement system, we can assume that its average steady state rotational speed can be
125
considered constant, regardless of the phase current amplitude fluctuations of the motor. This is a
126
very important conclusion, allowing the thesis that, in the steady state the average actual speed for
127
individual motor sectors is equal to the average speed measured during the full mechanical rotation
128
of the shaft. The actual average speed of a full revolution can be measured correctly by any single
129
Hall sensor. This results in the possibility of not using an additional reference speed measurement
130
system (i.e. an additional encoder) in favor of using averaged full revolution speed, which is
131
independent of the sensor placement error [12].
132
Figure 7 shows the calculated speed error values for the individual 𝑒𝑠𝑒𝑐 sectors for four
133
different BLDC motors with powers ranging from 60 W to 1500 W. All motors had four pairs of poles
134
each, for which 24 sectors of speed measurement can be defined per one mechanical revolution. Each
135
of them corresponds to a mechanical rotation angle equal to π/(3∙p), which gives an angle value for
136
each sector equal to 15 mechanical degrees.
137
138
Figure 7. Rotational speed computation error „𝑒 ” in particular sectors of the tested motors.
139 Figure 7. Rotational speed computation error “esec” in particular sectors of the tested motors.
It can be seen that the values of these errors are significant, and strongly influence the accuracy of
motor speed calculation. It is worth emphasizing that all of the tested motors were characterized by a
significant error in the arrangement of ee sensors, which ranged from a few to over 25◦ of the electric
angle, with respect to the optimal location according to the formula ee = esec·60◦.

6. Energies 2020, 13, 1607 6 of 9
4. Determination of the Absolute Position of the BLDC Motor Rotor
In the course of the work described above, errors in determining the rotational speed for individual
sectors of the motors, tested at different rotational speeds, were determined. It was shown that for a
wide range of rotational speeds, these errors are almost constant, and what is worth underlining, unique
for the specific motor used in the test. This is illustrated in Figure 8 for the motor marked as “D”.
Energies 2020, 13, x FOR PEER REVIEW 6 of 9
characterized by a significant error in the arrangement of 𝑒𝑒𝑒𝑒 sensors, which ranged from a few to
over 25° of the electric angle, with respect to the optimal location according to the formula
𝑒𝑒𝑒𝑒=𝑒𝑒𝑠𝑠𝑠𝑠𝑠𝑠·60°.
4. Determination of the Absolute Position of the BLDC Motor Rotor
In the course of the work described above, errors in determining the rotational speed for
individual sectors of the motors, tested at different rotational speeds, were determined. It was
shown that for a wide range of rotational speeds, these errors are almost constant, and what is worth
underlining, unique for the specific motor used in the test. This is illustrated in Figure 8 for the
motor marked as “D”.
Figure 8. Rotational speed computation errors determined at different rotational speeds of motor D
(four pole pairs, nominal speed = 3000 rpm/min, nominal power = 380W).
Figure 9 shows the error distribution of the speed determination for motor D. Using a
standard Hall sensor system causes that during the full rotation of the shaft, the sequence of sensor
states is repeated depending on the number of pole pairs of the motor (here, four times).
Figure 9. Distribution of errors in determining the 𝑒𝑒𝑠𝑠𝑠𝑠𝑠𝑠 speed of the BLDC motor, for motor D,
determined at 2000 rpm.
It can be seen that the value of the speed determination errors for individual sectors varies
greatly, creating a sequence unique for a specific motor. This makes it possible to identify the
position of the rotor using information about the states of the sensors, with the support of an
algorithm that looks for a specific pattern of error distribution. Determination of the rotor position
5 10 15 20
Calculated
speed
error
%
-30
-20
-10
0
10
20
30
40
50
500 rpm
1000 rpm
1500 rpm
2000 rpm
2500 rpm
3000 rpm
Sector number
Sector number
1 7 13 19
Speed
calculation
error,
%
-40
-30
-20
-10
0
10
20
30
40
50
60
H1 = 0
H2 = 0
H3 = 1
H1 = 0
H2 = 0
H3 = 1
H1 = 0
H2 = 0
H3 = 1
H1 = 0
H2 = 0
H3 = 1
Figure 8. Rotational speed computation errors determined at different rotational speeds of motor D
(four pole pairs, nominal speed = 3000 rpm/min, nominal power = 380W).
Figure 9 shows the error distribution of the speed determination for motor D. Using a standard
Hall sensor system causes that during the full rotation of the shaft, the sequence of sensor states is
repeated depending on the number of pole pairs of the motor (here, four times).
Energies 2020, 13, x FOR PEER REVIEW 6 of 9
It can be seen that the values of these errors are significant, and strongly influence the
140
accuracy of motor speed calculation. It is worth emphasizing that all of the tested motors were
141
characterized by a significant error in the arrangement of 𝑒 sensors, which ranged from a few to
142
over 25° of the electric angle, with respect to the optimal location according to the formula
143
𝑒 =𝑒 ·60°.
144
4. Determination of the Absolute Position of the BLDC Motor Rotor
145
In the course of the work described above, errors in determining the rotational speed for
146
individual sectors of the motors, tested at different rotational speeds, were determined. It was
147
shown that for a wide range of rotational speeds, these errors are almost constant, and what is worth
148
underlining, unique for the specific motor used in the test. This is illustrated in Figure 8 for the
149
motor marked as “D”.
150
151
Figure 8. Rotational speed computation errors determined at different rotational speeds of motor D
152
(four pole pairs, nominal speed = 3000 rpm/min, nominal power = 380W).
153
Figure 9 shows the error distribution of the speed determination for motor D. Using a
154
standard Hall sensor system causes that during the full rotation of the shaft, the sequence of sensor
155
states is repeated depending on the number of pole pairs of the motor (here, four times).
156
157
Figure 9. Distribution of errors in determining the 𝑒 speed of the BLDC motor, for motor D,
158
determined at 2000 rpm.
159
It can be seen that the value of the speed determination errors for individual sectors varies
160
greatly, creating a sequence unique for a specific motor. This makes it possible to identify the
161
position of the rotor using information about the states of the sensors, with the support of an
162
Figure 9. Distribution of errors in determining the esec speed of the BLDC motor, for motor “D”,
determined at 2000 rpm.
It can be seen that the value of the speed determination errors for individual sectors varies greatly,
creating a sequence unique for a specific motor. This makes it possible to identify the position of the
rotor using information about the states of the sensors, with the support of an algorithm that looks
for a specific pattern of error distribution. Determination of the rotor position can be done in relation
to the reference value of the eref error distribution, stored in the electrically erasable programmable
read-only - EEPROM memory of the motor control device.

7. Energies 2020, 13, 1607 7 of 9
The sequence of errors is repeated with the frequency corresponding to the time of full rotation of
the shaft by the motor. This sequence can start from different values, which depends only on the initial
position of the motor shaft at the moment of starting the drive system.
To simplify the procedure for determining the motor shaft position, only the error distributions
corresponding to the same combination of the Hall sensor states can be analyzed, which in the case of
the motor under test reduces the number of errors analysed from 24 to only 4.
Figure 10 shows possible error distributions for the selected sensor states (H1 = 0, H2 = 0, H3 = 1).
For a motor with four pairs of poles, four different sequences of assigning the error distribution to the
absolute position of the motor rotor are possible (depending on the initial position of the rotor after
power-up).
Energies 2020, 13, x FOR PEER REVIEW 7 of 9
algorithm that looks for a specific pattern of error distribution. Determination of the rotor position
163
can be done in relation to the reference value of the 𝑒𝑟𝑒𝑓 error distribution, stored in the electrically
164
erasable programmable read-only - EEPROM memory of the motor control device.
165
The sequence of errors is repeated with the frequency corresponding to the time of full rotation
166
of the shaft by the motor. This sequence can start from different values, which depends only on the
167
initial position of the motor shaft at the moment of starting the drive system.
168
To simplify the procedure for determining the motor shaft position, only the error distributions
169
corresponding to the same combination of the Hall sensor states can be analyzed, which in the case
170
of the motor under test reduces the number of errors analysed from 24 to only 4.
171
Figure 10 shows possible error distributions for the selected sensor states (H1 = 0, H2 = 0,
172
H3 = 1). For a motor with four pairs of poles, four different sequences of assigning the error
173
distribution to the absolute position of the motor rotor are possible (depending on the initial position
174
of the rotor after power-up).
175
176
Figure 10. Possible error distributions for determining the rotor speed for the Hall sensor state for
177
different initial rotor positions : (a); (b); (c); (d) for H1 = 0; H2 = 0; H3 = 1; (d) reference distribution
178
stored in the memory of the motor controller.
179
5. Example Implementation
180
The procedure for determining the absolute position of the rotor is, in the simplest possible
181
implementation, to search for the largest 𝑒 error for the same selected sequence of their state, and
182
to assign its sector number to the sector number of the maximum error of the reference distribution.
183
For example, sectors 1, 7, 13 and 19 (Figure10) correspond to the sequence (H1 = 0; H2 = 0; H3 = 1).
184
After determining the speed calculation errors, the algorithm compares the results obtained with the
185
reference values stored in the controller's non-volatile EEPROM memory. If the reference error
186
values stored for specific sensor signals reached the maximum value, e.g. for Sector 7, then after
187
searching the newly obtained error distribution, and finding the maximum error value, e.g., for
188
sector 13, one should change the sector number in which the greatest error of speed measurement
189
was found. In this particular case, sector number 13 will be changed by the algorithm determining
190
the position of the shaft to number 7, and the remaining sector numbers will be automatically
191
assigned to the following sectors of Figure11 by means of software incrementation.
192
The operation of this algorithm allowed to unambiguously assign the BLDC motor sector
193
number to the angle of the mechanical rotor position by analyzing the reference pattern of error
194
distribution, stored e.g., in the EEPROM memory.
195
Figure 10. Possible error distributions for determining the rotor speed for the Hall sensor state for
different initial rotor positions: (a–c); (d) for H1 = 0; H2 = 0; H3 = 1; (d) reference distribution stored in
the memory of the motor controller.
5. Example Implementation
The procedure for determining the absolute position of the rotor is, in the simplest possible
implementation, to search for the largest esec error for the same selected sequence of their state, and to
assign its sector number to the sector number of the maximum error of the reference distribution.
For example, sectors 1, 7, 13 and 19 (Figure 10) correspond to the sequence (H1 = 0; H2 = 0; H3 = 1).
After determining the speed calculation errors, the algorithm compares the results obtained with
the reference values stored in the controller’s non-volatile EEPROM memory. If the reference error
values stored for specific sensor signals reached the maximum value, e.g. for Sector 7, then after
searching the newly obtained error distribution, and finding the maximum error value, e.g., for sector
13, one should change the sector number in which the greatest error of speed measurement was found.
In this particular case, sector number 13 will be changed by the algorithm determining the position
of the shaft to number 7, and the remaining sector numbers will be automatically assigned to the
following sectors of Figure 11 by means of software incrementation.
The operation of this algorithm allowed to unambiguously assign the BLDC motor sector number
to the angle of the mechanical rotor position by analyzing the reference pattern of error distribution,
stored e.g., in the EEPROM memory.
When the differences between the errors of individual sectors are small, i.e., when the shaft
position detection system was made precisely at the factory, it is possible to expand the algorithm of

8. Energies 2020, 13, 1607 8 of 9
searching for similarity between measured values and reference values stored in the EEPROM memory.
The minimum error value and even all errors in all sectors can also be analysed.
Energies 2020, 13, x FOR PEER REVIEW 8 of 9
13
14
15
19
24
7
1
H1 = 0; H2 = 0; H3 = 1
Compare sector errors
find „maximum”
1 7 13 19
max
Compare EEPROM saved sector
errors find „maximum”
1 7 13 19
max
7
+1
+1
+1
+1
8
9
13
18
esec
eref
H1 = 0; H2 = 0; H3 = 1
H1 = 0; H2 = 0; H3 = 1
H1 = 0; H2 = 0; H3 = 1
196
Figure 11. Algorithm for determining the position of the BLDC motor rotor on the basis of the
197
comparison of 𝑒 errors with the reference value of the error distribution 𝑒 , for a given state of
198
the Hall sensors (H1 = 0; H2 = 0; H3 = 1).
199
When the differences between the errors of individual sectors are small, i.e., when the shaft
200
position detection system was made precisely at the factory, it is possible to expand the algorithm of
201
searching for similarity between measured values and reference values stored in the EEPROM
202
memory. The minimum error value and even all errors in all sectors can also be analysed.
203
6. Conclusions
204
During the research, four BLDC motors with four pairs of poles each were analysed. Thanks to
205
the proposed method, it was possible to find the mechanical position of a multi‐pole BLDC motor,
206
despite the fact that the sequence of the Hall sensor states changes in the same sequence p‐times per
207
mechanical revolution. What is very important is that theoretically, if the shaft position monitoring
208
system was made very accurately at the factory, it would not be possible to achieve motor
209
diagnostics results characteristic enough to assign a specific position of the motor rotor to them.
210
Hence, the conclusion that applying the method described in the article can correct a factory’s
211
inaccurate fixing of motor components, and result in the possibility of using such machines in drive
212
systems, where the absolute position of the shaft must be determined for the correct operation of the
213
whole system. Another interesting issue is the possibility of identifying a specific electrical machine
214
by analyzing the distribution of errors in speed measurement, using its shaft position sensors. This
215
gives the possibility of the quick, simple and cost‐free detection of the replacement of the drive
216
motor, even if it is characterized by the same nominal data. Determination of the angular position of
217
the BLDC motor shaft on the basis of the analysis of the error spectrum allows the application of
218
advanced algorithms for the correction of commutation errors, which result from errors in the
219
position of sensors in multi‐pole BLDC motors [3]. This contributes to the reduction of the noise
220
emitted by the drive as well as the use of semiconductor connectors with a lower rated current for
221
the controller.
222
Funding: This research received no external funding.
223
Conflicts of Interest: The authors declare no conflict of interest.
224
225
Figure 11. Algorithm for determining the position of the BLDC motor rotor on the basis of the
comparison of esec errors with the reference value of the error distribution eref , for a given state of the
Hall sensors (H1 = 0; H2 = 0; H3 = 1).
6. Conclusions
During the research, four BLDC motors with four pairs of poles each were analysed. Thanks
to the proposed method, it was possible to find the mechanical position of a multi-pole BLDC
motor, despite the fact that the sequence of the Hall sensor states changes in the same sequence
p-times per mechanical revolution. What is very important is that theoretically, if the shaft position
monitoring system was made very accurately at the factory, it would not be possible to achieve
motor diagnostics results characteristic enough to assign a specific position of the motor rotor to
them. Hence, the conclusion that applying the method described in the article can correct a factory’s
inaccurate fixing of motor components, and result in the possibility of using such machines in drive
systems, where the absolute position of the shaft must be determined for the correct operation of the
whole system. Another interesting issue is the possibility of identifying a specific electrical machine
by analyzing the distribution of errors in speed measurement, using its shaft position sensors. This
gives the possibility of the quick, simple and cost-free detection of the replacement of the drive motor,
even if it is characterized by the same nominal data. Determination of the angular position of the
BLDC motor shaft on the basis of the analysis of the error spectrum allows the application of advanced
algorithms for the correction of commutation errors, which result from errors in the position of sensors
in multi-pole BLDC motors [3]. This contributes to the reduction of the noise emitted by the drive as
well as the use of semiconductor connectors with a lower rated current for the controller.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.

9. Energies 2020, 13, 1607 9 of 9
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