parametric analytical-experimental modeling in bldc engines … · 2019. 6. 5. · parametric...
TRANSCRIPT
Parametric Analytical-experimental Modeling in
BLDC Engines Applied to Propulsion Systems on
Chopper Motorcycles
Héctor C. Terán, Guido R. Torres, Oscar Arteaga, Jonathan J. Morales, Byron P. León, Daniel A. Morales Universidad de las Fuerzas Armadas ESPE/ Energy and Mechanics, Sangolquí, Ecuador
Email: [email protected], {grtorres, obarteaga, jjmorales7, bpleon, damorales12}@espe.edu.ec
Abstract— The present experimental research proposes to
analyze the behavior and performance of a three-phase
Brushless BLDC electric motor when used as an alternative
propulsion system in Chopper-type motorcycles. To carry
out the study of the BLDC motor, an electric mathematical
model is created and a mathematical model of the
mechanical transmission system , with software the models
are merged creating a universal equation applied to electric
Chopper motorcycles, analytically with the variation of the
electric angle (θe), the operating parameters were calculated:
electromagnetic torque, power and efficiency, which are
used as reference standards and compared with those
obtained with field tests in the laboratory with the
MotorLab Kit MLK-B, performing rpm tests under the
standard IEC 60034-30, efficiency with the standard IEC
60034-31, Rated current with IEC 60038 and power under
IEC 60034-30, with the experimental results is proposed
optimized parameters for idle and load operation and
implement them in the programming of a motor inverter
and achieve the highest performance while meeting the
quality standard for motorcycles set in EN 15194.
Index Terms—BLDC electric motor, mathematical model of
the propulsion system, Chopper motorcycle
I. INTRODUCTION
The most commonly used vehicles in urban mobility to move after cars are two-wheeled vehicles; both motorcycles and bicycles, establishing a model for the implementation of an electric propulsion system [1]. The operation of this type of vehicle is based on a three-phase electric motor coupled to a wheel driven to generate motorcycle movement [2]. A characteristic of electric motors is that approximately 80% of the electrical energy is used by the industrial sector [3].
For the correct operation of a three-phase electric
motor in vehicles and motorcycles requires an electric
regulator, which is a system composed of several
electrical and electronic subsystems called three-phase
inverters [4], which will physically code the voltages,
converting them into optimum push buttons, so that the
motor rotation is efficient [5].
A three-phase electric motor is a rotating electric
machine [6], responsible for converting the supplied
three-phase electric energy into mechanical energy [7]. A
Manuscript received July 26, 2018; revised May 29, 2019.
BLDC brushless motor is ideal for electric motorcycles
because of its high power, good speed torque
characteristics, high efficiency, wide speed ranges and
low maintenance [8], unlike other types of motors, the
power loss at activation is minimal, it also has reduced
space with low noise levels [9].
Most BLDC motors have three-phase stator windings,
while their rotors can have several pairs of magnetic
poles [10]. BLDC motors have a fast dynamic response
due to their low inertia rotor (permanent magnet)
compared to ind3uction motors [11].
BLDC motors use permanent magnets instead of coils
in the armature therefore, do not require any brushless.
This experimental research analyzes the waveforms of the
BLDC motor, with its phase current and electromotive
force [12] by offsetting the rotation each π rad to
determine the mathematical model through differential
equations and its electrical components and fuse them
with the propulsion system of the motorcycle Chopper
[13].
In Fig. 1, the connection configuration of the inverter
with respect to the BLDC motor is shown, points a, b and
c that are the outputs of the inverter installed in the motor
to generate a potential three-phase excitation in the
electric motor in the terminals.
Figure 1. BLDC motor drive system configuration and reference current generation.
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International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 4, July 2019
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II. SYSTEM MODELING
For the mathematical modeling of the Chopper electric
motorcycle, it was divided into an electrical subsystem
(batteries, three-phase inverter and electric motor), as
well in a mechanical subsystem (transmission and
pneumatic), as shown in Fig. 2.
Figure 2. Electric motorcycle components.
It was considered a BLDC motor that is part of two
systems in both the mechanical and electrical systems, in
charge of transforming the electrical energy supplied into
mechanical energy necessary for the movement of the
motorcycle Chopper according to the capacity of the
batteries
A. Engine Modeling
For the analysis of the BLDC engine the following
equations are established:
[
𝑣𝑎𝑣𝑏𝑣𝑐] = [
𝑟𝑠 0 00 𝑟𝑠 00 0 𝑟𝑠
] [
𝑖𝑎𝑖𝑏𝑖𝑐
] + ⋯
…+ [𝐿 − 𝑀 0 00 𝐿 −𝑀 00 0 𝐿 − 𝑀
]𝑝 [
𝑖𝑎𝑖𝑏𝑖𝑐
] + [
𝑒𝑎𝑒𝑏𝑒𝑐]
𝜏𝑒𝑚 − 𝜏𝑙𝑜𝑎𝑑 − 𝐵𝜔𝑚 = 𝐽𝑑𝜔𝑚
𝑑𝑡, (1)
The electromagnetic torque is defined:
𝜏𝑒𝑚 = (𝑒𝑎𝑖𝑎 + 𝑒𝑏𝑖𝑏 + 𝑒𝑐𝑖𝑐)/𝜔𝑚 , (2)
For the electrical subsystem:
𝑝 is the operator 𝑑
𝑑𝑡
𝑣𝑎, 𝑣𝑏 , 𝑣𝑐: three-phase voltages feeding the BLDC
motor respectively in Volts (𝑉) 𝑟𝑠: electrical resistance at stator windings in ohms (Ω). 𝑖𝑎, 𝑖𝑏 , 𝑖𝑐: three-phase currents flowing through the
stator windings (𝐴). L: inductance of the stator windings in Henrios (𝐻). M: mutual inductance in (𝐻). [𝑒𝑎 𝑒𝑏 𝑒𝑐]𝑇 : vector representing the voltage
generated in the stator windings due to the relative
movement of the rotor with respect to the mechanical
system.
𝜏𝑒𝑚: electromagnetic torque (𝑁.𝑚) 𝜏𝑙𝑜𝑎𝑑 : load torque (𝑁.𝑚)
B: coefficient of viscous friction (𝑁𝑚𝑠
𝑟𝑎𝑑)
J: rotor inertia (𝑘𝑔.𝑚2)
𝜔𝑚: mechanical speed (𝑟𝑎𝑑
𝑠)
The BLDC motor has the magnetic flux winding that
circulates through the iron to the iron, the waveform is
trapezoidal, thus generating a voltage e_(a,b,c) of the
same waveform as the stator windings, these voltages are
a function of the electric angle θe of the motor and is
represented by:
𝑒𝑎 = 𝜆𝜔𝑟𝑓(𝜃𝑒), (3)
𝑒𝑏 = 𝜆𝜔𝑟𝑓 (𝜃𝑒 −2𝜋
3),
𝑒𝑐 = 𝜆𝜔𝑟𝑓 (𝜃𝑒 −4𝜋
3),
Where:
𝜆: flow bond established by the permanent magnet of
the rotor.
𝜔𝑟: rotor speed.
𝑓(𝜃𝑒), 𝑓 (𝜃𝑒 −2𝜋
3) , 𝑓 (𝜃𝑒 −
4𝜋
3) represent the shape of
a trapezoid bounded in the closed range 1,1.
These were generated analytically in MATLAB and
their model is described below:
𝑓(𝜃𝑒) =
{
𝜃𝑒
6
𝜋0 ≤ 𝜃𝑒 <
𝜋
6
1𝜋
6≤ 𝜃𝑒 <
5𝜋
6
(𝜋 − 𝜃𝑒)6
𝜋−1
(𝜃𝑒 − 2𝜋)6
𝜋
5𝜋
6≤ 𝜃𝑒 <
7𝜋
67𝜋
6≤ 𝜃𝑒 <
11𝜋
611𝜋
6≤ 𝜃𝑒 < 2𝜋}
,
𝑓 (𝜃𝑒 −2𝜋
3) =
{
−1 0 ≤ 𝜃𝑒 <
𝜋
2
(𝜃𝑒 −2𝜋
3)6
𝜋
𝜋
2≤ 𝜃𝑒 <
5𝜋
6
1
(5𝜋
3− 𝜃𝑒)
6
𝜋−1
5𝜋
6≤ 𝜃𝑒 <
3𝜋
23𝜋
2≤ 𝜃𝑒 <
11𝜋
611𝜋
6≤ 𝜃𝑒 < 2𝜋}
,
𝑓 (𝜃𝑒 −4𝜋
3) =
{
1 0 ≤ 𝜃𝑒 <
𝜋
6
(𝜋
3− 𝜃𝑒)
6
𝜋
𝜋
6≤ 𝜃𝑒 <
𝜋
2
−1
(𝜃𝑒 −4𝜋
3)6
𝜋−1
𝜋
2≤ 𝜃𝑒 <
7𝜋
67𝜋
6≤ 𝜃𝑒 <
3𝜋
23𝜋
2≤ 𝜃𝑒 < 2𝜋}
,
At all times for the functions to operate correctly it is
necessary to know the angle 𝜃𝑒.
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For this mechanical subsystem it is necessary to
determine the electromagnetic torque, so when replacing
(3) in (2) the electromagnetic torque is:
𝜏𝑒𝑚 =𝜆𝜔𝑟𝜔𝑚
(𝑖𝑎𝑓(𝜃𝑒) + 𝑖𝑏𝑓 (𝜃𝑒 −2𝜋
3) + ⋯
…+ 𝑖𝑐𝑓 (𝜃𝑒 −4𝜋
3)) (4)
Equation (4) requires knowledge of the initial velocity
other than zero. However, it is considered that 𝜃𝑒 = 𝑛𝑝𝜃𝑚
and 𝜔𝑟 = 𝑛𝑝𝜔𝑚, to determine the electromagnetic torque
which does not depend directly on the motor speed but on
the number of poles (5).
𝜏𝑒𝑚 = 𝑛𝑝𝜆(𝑖𝑎𝑓(𝜃𝑒) + 𝑖𝑏𝑓 (𝜃𝑒 −2𝜋
3) + ⋯
…+ 𝑖𝑐𝑓 (𝜃𝑒 −4𝜋
3)) (5)
Where:
𝑛𝑝: Number of pole pairs.
Equation (5) allows the theoretical electromagnetic
torque to be calculated from a mathematical model to
compare the laboratory tests to be carried out on the
BLDC vacuum motor.
B. Transmission modeling
An electric motorcycle for propelled with ease must
contain a drive system consisting of an electric motor,
drive pulley, driven pulley, rear pulley, rear tire and a
synchronized belt as shown in Fig. 3:
Figure 3. Schematic of electric motorcycle transmission system
The torque required to drive the motorcycle is set
according to the parameters described (6)
𝜏 =𝑟
𝑛𝑔𝐺𝐹𝑡𝑒, (6)
Where:
r: rim spoke (m).
𝑛𝑔: transmission efficiency.
G: transmission angular velocity reduction ratio.
𝜏: torque generated in the motor shaft (N.m).
𝐹𝑡𝑒: tractive force that drives the electric motorcycle
(N).
The relationship between the angular speed of the
motor 𝜔𝑚 and the linear speed v of the motorcycle is:
𝜔𝑚 = 𝐺𝑣
𝑟 (7)
The forces acting on the electric motorcycle are
determined from the free body diagram, Fig. 4:
Figure 4. Electric motorcycle Free-Body Diagram
The tractive force 𝐹𝑡𝑒 that drives the bike and is
provided by the BLDC engine through the transmission.
The friction force between the tyre and the surface is
given by:
𝐹𝑟𝑟 = 𝜇𝑟𝑟𝑚𝑔 cos𝜑, (8)
Where:
𝜇𝑟𝑟: friction coefficient
m: total mass of the motorcycle (kg).
𝑔 = 9,8 (𝑚 𝑠2⁄ ) gravitational acceleration constant.
𝜑 angle of inclination of the surface on which the
motorcycle is moving.
The frictional force of the wind is given by:
𝐹𝑎𝑑 =1
2𝜌𝐴𝐶𝑑𝑣
2, (9)
Where:
𝜌 = 1,67𝑘𝑔
𝑚3 air density.
A: front area of the bike (𝑚2).
𝐶𝑑: aerodynamic coefficient.
𝑣: linear speed of the motorcycle (𝑚/𝑠),
The component 𝐹ℎ𝑐 generated by the weight of the
bike with an angle of inclination 𝜑, is given by:
𝐹ℎ𝑐 = 𝑚𝑔 sin𝜑, (10)
Applying Newton's second law:
𝐹𝑡𝑒 − 𝐹𝑟𝑟 − 𝐹𝑎𝑑 − 𝐹ℎ𝑐 = 𝑚. 𝑎, (11)
Where:
𝑎: motorcycle acceleration (𝑚
𝑠2).
By replacing the corresponding expressions of the
forces acting on the motorcycle (11), the tractive force is
obtained by 𝐹𝑡𝑒:
𝐹𝑡𝑒 = 𝑚𝑎 + 𝜇𝑟𝑟𝑚𝑔cos𝜑 +1
2𝜌𝐴𝐶𝑑𝑣
2 +𝑚𝑔 sin 𝜑, (12)
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The equation does not include parameters related to the
engine and transmission, therefore, based on the
mechanical system expressed in (1), the electromagnetic
torque is obtained 𝜏𝑚 engine:
𝜏𝑒𝑚 = 𝐽𝑑𝜔𝑚
𝑑𝑡+ 𝜏𝑙𝑜𝑎𝑑 + 𝐵𝜔𝑚, (13)
The total inertia in the engine's mechanical system is
the sum of the BLDC motor's rotor inertia and the
motorcycle's inertia:
𝐽 = 𝐽𝑀𝐵 + 𝐽𝑀𝐸, (14)
The inertia of the electric motorcycle EM is 𝐽𝑀𝐸 =1
2𝑚
𝑟2
𝐺2. Substituting (6) and (14) in (13) gives:
𝜏𝑒𝑚 = 𝐽𝑑𝜔𝑚
𝑑𝑡+ 𝐵𝜔𝑚 +
𝑟𝐹𝑡𝑒
𝑛𝑔𝐺, (15)
With equation (12) in (15),
𝜏𝑒𝑚 = 𝐽𝑑𝜔𝑚
𝑑𝑡+ 𝐵𝜔𝑚 +
𝑟
𝑛𝑔𝐺(𝑚𝑎 + 𝜇𝑟𝑟𝑚𝑔 cos𝜑 +⋯
…+1
2𝜌𝐴𝐶𝑑𝑣
2 +𝑚𝑔 sin𝜑),
(16)
The resulting electromagnetic torque contains both the
angular velocity of the rotor 𝜔𝑚 as the linear speed v of
the EM electric motorcycle.
The model of the mechanical system is determined
from terms of 𝜔𝑚. This is achieved with the equation (7).
Therefore, the expression of the mechanical system is:
𝑑𝜔𝑚
𝑑𝑡=
1
𝐽[𝜏𝑒𝑚 − 𝐵𝜔𝑚 −
𝑟
𝑛𝑔𝐺(
𝜇𝑟𝑟𝑚𝑔 cos𝜑 + ⋯
…+1
2𝜌𝐴𝐶𝑑 (
𝑟
𝐺)2
𝜔𝑚2 +𝑚𝑔 sin𝜑
)]. (17)
Equation (17) represents the mechanical system of the
electric motorcycle EM shown in Fig. 2, this including
the BLDC engine with the mechanical transmission
system, the tests are performed from this mathematical
modeling with load to perform its experimental analysis.
III. EXPERIMENTAL TESTING
To test a BLDC motor, the MLK-B MotorLab
equipment is used, see Fig. 5, which provides data on
motor performance with no-load speed as a function of
time, where results were obtained: torque, speed, current,
voltage, input power, output power, power factor and
efficiency with methods and standards established in the
Table I.
Figure 5. BLDC engine test equipment.
With regard to the dynamic tests, they will be carried
out once the electrical and mechanical systems have been
assembled and installed on the motorcycle Chopper.
TABLE I. STANDARD BLDC ENGINE TESTING
Tests Evaluation Standard and
Method
Equipment
and Materials
N rpm Engine working speed IEC 60034-30
MotorLab
Kit
Efficienc
y
Relationship between
the Pout (mechanical) and the Pin (electrical)
IEC 60034-31
Current Electrical charge flow through the motor
IEC 60038
Power
output
Speed of mechanical
work IEC 60034-30
In Table II, measured values of the BLDC motor are
established, which were carried out under no load
conditions.
TABLE II. BLDC ENGINE TEST PARAMETERS
U [V] I [A] Pin [W] PF [%] N [rpm] Pout [W]
100 200 20000 1 5000 20000
99,53 180,65 18062,32 0,9 4887 18009,28
99,05 161,30 16124,64 0,8 4774 16018,57
98,58 141,95 14186,95 0,7 4660 14027,85
98,11 122,60 12249,27 0,6 4547 12037,14
97,63 103,25 10311,59 0,5 4434 10046,42
97,16 83,90 8373,91 0,4 4321 8055,70
96,69 64,55 6436,23 0,3 4208 6064,99
96,22 45,20 4498,54 0,2 4094 4074,27
95,74 25,85 2560,86 0,1 3981 2083,56
95,27 6,49 623,18 0 3868 92,84
To perform the BLDC motor tests from the
mathematical models represented by equations (5) and
(17), the electromagnetic torque is determined as a
function of the electrical angle. 𝜃𝑒 considering the
parameters of factory, see Table III.
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TABLE III. DATASHEET OF BLDC ENGINE AND TRANSMISSION
SYSTEM
Parameter Magnitude Parameter Magnitude
𝜆 0,262 𝑉𝑠/𝑟𝑎𝑑 𝜌 1,67 𝑘𝑔/𝑚3
𝐵 1𝑥10−5 𝑁𝑚𝑠/𝑟𝑎𝑑 𝐴 0,6 𝑚2
𝑛𝑝 4 𝐶𝑑 0,5
𝐿 −𝑀 0.0012 H 𝐺 2,4
𝐽 0,022 𝑘𝑔𝑚2 𝑢𝑟𝑟 0,015
𝑟𝑠 0,121 Ohms 𝑛𝑔 0,95
𝐵𝑢𝑠 𝑑𝑒 𝑐𝑑 100V 𝑚 175 𝑘𝑔
𝑟 0,3778 𝑚 𝑔 9,8 𝑚/𝑠2
In Table IV, values calculated using mathematical
vacuum models and implemented the transmission
system on the electric motorcycle.
TABLE IV. RESULT OF THE MATHEMATICAL MODEL UNDER LOAD
AND UNDER LOAD
𝜃𝑒 (rad) No-load torque (N.m) Loaded torque (N.m)
0 0 -15251,823
0,9076 190,227 -6605,144
1,815 380,454 2041,534
2,723 570,681 10688,212
3,630 760,908 19334,891
4,538 951,135 27981,569
4,573 958,451 28314,134
4,608 965,767 28646,698
4,643 973,084 28979,263
4,678 980,4 29311,828
4,712 0 0
IV. ANALYSIS OF RESULTS
The parameters obtained are used to determine the
characteristic curves of efficiency, power, speed current
as a function of torque with intervals of: efficiency 80%,
power 19907.16 W, current 193.51 A, with a speed
variation between 3868 and 5000 rpm, see Fig. 6.
Figure 6. BLDC motor no-load work curves.
Figure 7. Theoretical electromagnetic torque of BLDC motor.
With the results of the vacuum test calculated from the
mathematical models Table IV, the electrical angle θe
varies proportionally with respect to torque reaching
maximum values of 4.67 rad at 980.4 Nm, see Fig. 7.
From the results of Table IV using mathematical
models of the propulsion system of the Chopper
motorcycle, the minimum electric angle θe to overcome
the load is 1,815 rad, see Fig. 8.
Figure 8. Electromagnetic torque with transmission system.
The values obtained in the no-load tests when
implementing the BLDC motor in the MotorLab Kit
MLK-B equipment, are compared with the mathematical
models, analyzing the amount of mechanical energy
delivered by the BLDC motor with respect to the
electrical energy consumed to establish the appropriate
operating performance. The analysis of the amount of
energy lost is also carried out to obtain the efficiency of
an electric motor Chopper, see Table V.
TABLE V. BLDC ENGINE TEST VALUES ON AN ELECTRIC CHOPPER
MOTORCYCLE
Description Charging
point
Increased efficiency
Maxi
mum power
Maximum torque
Maximum point
Nominal rotation
Voltage (V)
95,97
95,53
95,41
95,41
95,41
95,41
Current (A)
6,494
77,446
161,14
2
161,142
161,173
161,142
Pin (W)
623,18
7391,74
15376,
06
15376,06
15377,01
15376,06
Torque
(mN.m)
707,5
15347,3
32641,
1
32641,1
32646,7
32641,1
Revolutions
(RPM)
4762
4245
3868
3868
3868
3868
Pout (W)
352,79
6754,33
13220,
94
13220,94
13222,75
13220,94
Eff (%)
56,6
91,4
86
86
86
86
With the operating curves of the parameters set out in
Table V: voltage, current, input power, efficiency and
load point as a function of torque, the load point curve is
determined which sets minimum parameters for operation,
see Fig. 9.
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Figure 9. Input power, output and torque curves
The Current and Voltage curves are based on Table 5,
the nominal current reached is 77.44 A with a maximum
efficiency of 91.4%; keeping the constant voltage of
95.7V in function of efficiency, s
ee Fig. 10.
Figure 10. Voltage and Current curves
The speed curve as a function of the efficiency
generated in Table V determines that the maximum
efficiency is 91.4% at 4245 rpm, see Fig. 11.
Figure 11. Revolutions and efficiency curve
V. CONCLUSIONS
Modeling of the drive system of the BLDC engine
including transmission and satisfies the operating
parameters required by EN 15194 for electrically assisted
vehicles.
The minimum electric angle θe of 1,815 rad is the one
necessary to overcome the inertia resistance of the
chopper motorcycle to start the start.
Considering the parameters determined for the load
point, the critical point related to the efficiency is
determined as 91.4 %, stabilizing the input and output
power at 86% efficiency.
The maximum torques reached by a mathematical
model with load are 29311.828 Nm and the real torque
needed to move the motorcycle is 32646.7 Nm, this
difference in torques is due to the friction between the
parts of the transmission system.
The optimum parameters found were entered into the
programming of the inverter controller, obtaining the
highest real efficiency for electric motorcycle
applications of the Chopper type.
In future research, it is planned to analyze the
mathematical model of the batteries to establish their
autonomy according to the circuit and driving modes.
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Conference on Electrical and Computer Engineering (CCECE), pp. 419-424, 2011.
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Hector C. Terán, born in New York, USA in
1982, obtained his MSc. in Energy Management from the Technical University of
Cotopaxi in 2014, and his engineering degree
in Electromechanics from the Polytechnic School of the Army in 2008.
He works as a tenured research professor at the
University of the Armed Forces ESPE, in the Department of Energy and Mechanical
Sciences in the area of Manufacturing and
Energy, has written some books being the latest works "The Metallurgy for welding" and the paper "Mobile robotic table with artificial
intelligence applied to the separate and classified positioning of objects
for computer-integrated manufacturing" His research areas are manufacturing, materials, automation and mechatronics.
Msc. Terán is a member of the Research Group on Robotic Automation
and Intelligent Systems ARSI of the National Research and Education
Network of Ecuador.
Guido R. Torres, born in Alausi Ecuador
1960, obtained MSc. in Energy Management
from the Technical University of Cotopaxi in 2014, MSc. in Higher Education from the
Indo-American Technological University in
2002 and his engineering degree in Mechanics from the Polytechnic Superior School of
Chimborazo in 1990.
He is working as a research professor at the University of the Armed Forces ESPE, in the
Department of Energy and Mechanical Sciences, for the area of Design
and Computational Mechanics, has written the book of the “Fundamentals of fluid mechanics” and paper “Redesign of the rear
suspension of the prototype vehicle for competition in the SAE formula”
and others, his research area is based on the design of conventional and alternative vehicles.
Msc. Torres Muñoz is a member of the Mechanical Engineers
Association of Ecuador.
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