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State space modeling and simulation of sensorless permanent magnet BLDC motor

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Brushless DC (BLDC) motor simulation can be simply implemented with the required control scheme using specialized simulink built-in tools and block sets such as simpower systems toolbox. But it requires powerful processor requirements, large random access memory and long simulation time. To overcome these drawbacks this paper presents a state space modeling, simulation and control of permanent magnet brushless DC motor. By reading the instantaneous position of the rotor as an output, different variables of the motor can be controlled without the need of any external sensors or position detection techniques. Simulink is utilized with the assistance of MATLAB to give a very flexible and reliable simulation. With state space model representation, the motor performance can be analyzed for variation of motor parameters.
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N. Muruganantham et. al. / International Journal of Engineering Science and Technology
Vol. 2(10), 2010, 5099-5106
STATE SPACE MODELING AND
SIMULATION OF SENSORLESS
PERMANENT MAGNET BLDC MOTOR
N. MURUGANANTHAM *, Member, IEEE
Asst. Professor of Electrical & Electronics Engineering,
Periyar Maniammai University, Periyar Nagar,
Vallam – 613 403, Thanjavur (d.t),
Tamil Nadu, India.
DR. S. PALANI
Dean of Electronics Engineering,
Sudharsan Engineering College,
Sathiyamangalam - 622 501, Pudukkottai (d.t),
Tamil Nadu, India.
Abstract:
Brushless DC (BLDC) motor simulation can be simply implemented with the required control scheme using
specialized simulink built-in tools and block sets such as simpower systems toolbox. But it requires powerful
processor requirements, large random access memory and long simulation time. To overcome these drawbacks this
paper presents a state space modeling, simulation and control of permanent magnet brushless DC motor. By reading
the instantaneous position of the rotor as an output, different variables of the motor can be controlled without the
need of any external sensors or position detection techniques. Simulink is utilized with the assistance of MATLAB
to give a very flexible and reliable simulation. With state space model representation, the motor performance can be
analyzed for variation of motor parameters.
Keywords: BLDC Motor; State Space Model; Sensorless.
.1. Introduction
With rapid developments in power electronics, power semiconductor technologies, modern control theory for motors
and manufacturing technology for high performance magnetic materials, the brushless DC motors (BLDCM) have
been widely used in many fields. Due to the advancement of small size, good performance, simple structure, high
reliability and large output torque, BLDC motors have attracted increasing attention. However, the application of
position sensor makes the motor body heavy, as well as lots of wires are needed, which in turn brings complication
and interference in the design. Thus the position sensorless control technology attracts increasing research interest
and currently becomes one of the most promising trends of BLDCM control system. The modeling and simulation
analysis for BLDCM depends on computer engineering and can effectively shorten development cycle of position
sensorless BLDCM control system and evaluate rationality of the control algorithm imposed on the system. This
provides a good foundation for system design and verify novel control strategy. MATLAB [1] possessing powerful
scientific computing and graphics processing function is an interactive software system developed by Mathworks
company for system simulation.
In paper [2], BLDC motor has been designed based on transfer function model. Though the transfer function
model provides us with simple and powerful analysis and design techniques, it suffers from certain drawbacks such
as transfer function is only defined under zero initial conditions. Further it has certain limitations due to the fact that
the transfer function model is only applicable to linear time-invariant systems and there too it is generally restricted
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to single input single output systems. Another limitation of the transfer technique is that it reveals only the system
output for a given input and provides no information regarding the internal state of the system.
In this paper, the motor is designed based on state space model to get information about the state of the system
variables at some predetermined points along the flow of signals. By adopting this model, powerful processor
requirement, large random access memory can be avoided with more design flexibility and faster results can be
obtained.
2. State Space Modeling
2.1. Assumptions
1) The motor’s stator is a star wound type
2) The motor’s three phase are symmetric, including their resistance, inductance and mutual inductances [3].
3) There is no change in rotor reluctance with angle due to non-salient rotor.
4) There is no misalignment between each magnet and the corresponding stator winding.
2.2. Modeling Brushless DC Motor
The coupled circuit equation [4] of the stator winding in terms of motor electrical constants are
 
 
 
00
0
0
00

  
  
  

(1)
where Rs is the stator resistance per phase, Ia Ib Ic are the stator phase currents, p is the time derivative operator, Ea
Eb Ec are the back emfs in the respective phases in (1), Vn is the neutral point node voltage given by
  
 (2)
where  means summing up the individual phase emfs on an instant to instant basis.
Based on equation (1), the equivalent circuit of motors can be obtained as shown in Fig. 1.
Fig. 1. Equivalent circuit for stator windings
The induced emfs are all assumed to be trapezoidal, whose peak value is given by
Ep = (BLv)N = N(Blrω) = NФω = λω (3)
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where B is the flux density of the field in webers, L is the rotor length, N is the number of turns per phase, ω is the
electrical angular speed in rad/sec, Ф represents flux linkage = BLr, λ represents the total flux linkage given as the
product number of conductors and flux linkage/conductor.
If there is no change in rotor reluctance with angle because of non-salient rotor and assuming three symmetric
phases, inductances and mutual inductances are assumed to be symmetric for all phases as in [5]. Hence (1) becomes
100
010
001






(4)
Simplifying (4) further we get the following
100
010
001

 0 0
00
00


(5)
The generated electromagnetic torque is given by
TEIEIEI
(in Nm) (6)
The induced emfs can be written as



 (7)


where fa(θ), fb(θ), fc(θ) are functions having same shapes as back emfs. The values from (6) can be substituted in (5)
to obtain the value of torque. Also,

  (8)
where Tl is the load torque, J is the moment of inertia, B is the friction coefficient. Electrical rotor speed and
position are related by

 
 (9)
where P is the number of poles in the motor. From the above equations, the system state equations are written in the
following form
 (10)
where the states are chosen as x(t) = [ Ia Ib Ic ω θ]T (11)
Thus the system matrices as given below,
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Vol. 2(10), 2010, 5099-5106

/0 0 
 / 0
0
/ 0 
 / 0
0

/
0
0

/
0
/ 
 / 0

/ / 0
0 /2 0
(12)

1/00 0
01/
0 0
0
0
00
0
01/0
0 1/
0 0
(13)
The input vector is defined as u(t) = [ Va Vb Vc Tl]T (14)
where Ll = L – M, L is the self inductance of the winding per phase, M is the mutual inductance per phase and Va,
Vb, Vc are the per phase impressed voltage on the motor windings.
3. Simulation Blocks And Operation
The simulation has five main blocks. They are BLDC motor, controller block, inverter block, estimate block and
changer block shown in Fig. 2. Each main block has several sub-blocks. Some blocks are logical and some are made
using S-Function. The BLDC motor block contains state space sub-block where matrices A, B, C, D are located with
the provision that the initial condition can be varied. In the S-Function, coding file is linked and is shown in Fig. 3.
The sequence of operation of the above blocks are described by the flowchart shown Fig. 4. The simulation starts
with a starter block (No. 1 in chart) that generates 3Φ input voltage to the system’s core block (No.2 in chart) for
one cycle. A changer block is used to close the control loop after the random ramping of the motor. Once the loop is
closed, the starter block will be disconnected from the system and the motor will start receiving the phase voltages
from the connected controller through inverter.
Fig. 2. Simulink model of BLDC motor
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Fig. 3. Inside the core block in BLDC motor block
The PID controller is tuned by Ziegler Nichols method. By this method, the values of Kp=16.61, Ki=0.0134 and
Kd=0 are chosen. An S-Function block is connected to the state space block to choose the motor specifications such
as, the number of conductor turns per phase, resistance per phase, rotor dimensions etc as defined by the user. The
S-Function will read the instantaneous position among twelve position which are separated by 30º. Depending on
the position [6], the back e.m.f and torque in each phase will be defined. The estimate block contains the PID
controller. The block again is an M-file S-Function. This block calculates the reference phase current from the speed
and required torque. Required torque is calculated by actual speed and the speed error value. The above value will
be read and used in a PID controller [7]. The required torque is calculated as follows,
 0.5
 0.5
 (15)
where E is the angular speed error, E-1 is the previous time step error in angular speed, ts is the sampling time, Kp,
Ki, Kd are proportional, integral and derivative constants.
The required current is calculated from the instantaneous required torque. Then it is converted by means of an
approximated Park’s Transformation to three phase currents. The approximated park’s transformation gives the
corresponding phase current to every stator phase according to the rotor’s position. A hold block (No.3 in chart) is
used to hold on both the required and instantaneous current values in the open loop. Once the changer block closes
the control loop, the hold block will give an access to the current values to pass to the present controller scheme. In
this simulation, hysteresis controller function is chosen. Usually, the controller is used to fire the gates of six step
inverter switches, as in [8].
Fig. 4. Detailed flow chart for the whole control process
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However, each firing scheme determines certain voltage in each phase in the stator. According to the change in
the firing angle, stator voltage received by the inverter block changes and the motor speed is varied.
4. Simulation Results And Discussion
The motor specifications used in this simulation are shown in Table. 1. The simulation was run for 0.13seconds
(simulation time). When the reference speed equals 4000 rpm, the simulation curves of 3Φ back emfs, 3Φ currents,
3Φ torques and rotor position are shown in Fig. 5, 6, 7, 8, and Fig. 9. Load torque is applied at 0.01 seconds. The
motor speed stabilizes in 0.058 seconds with 0% overshoot. From Fig. 5 and Fig. 6, the back emf is almost
trapezoidal with 120º phase difference. Since the three phase torques are calculated from 3Φ currents, it gives 120º
phase difference between each phases as shown in Fig. 7. From Fig. 8, the rotor position can be analyzed under
various aligned and unaligned conditions.
Fig. 5. Three phase back EMF
Fig. 6. Three phase currents
Fig. 7. Three phase torque
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Fig. 8. Rotor position
Fig. 9. Speed
TABLE 1. BLDC motor specifications
Current 34 A Rotor length 30 cm
Torque 0.9 N.m Rotor radius 20 cm
Self inductance per
winding 2.72 mH No. of turns per
phase 100
Mutual inductance
between windings -1.5 mH Flux density 0.8167 wb
Motor inertia 0.0002 Coulomb friction 0.0178 N
Rated speed 4500 RPM Static friction 0.089 N
Number of poles 4 Viscous friction 0.002 N
Number of phases 3 Input dc voltage 160 V
Winding resistance per
phase 0.7 No. of slots per pole
per phase 100
5. Conclusions
BLDC motor analysis based on state space model can be easily carried out using MATLAB 7.3 version. This model
has many advantages over transfer function model. The simulation study using state space model has been validated
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with the results obtained using transfer function model. Further using state space model, the performance
characteristics of the BLDC motor can be evaluated for different machine parameters, which can be easily varied in
the simulation study and useful information can be obtained. The simulation results demonstrate that the simulated
waveforms fit theoretical analysis well. However, the simulation involves solving many simultaneous differential
equations and the results obtained are highly dependent upon the choice of the system solver, where some solver
gives highly accurate results, but need longer time to terminate. Through the modularization design, a lot of time
spent on design can be saved and the design efficiency can be promoted rapidly. The method proposed in this paper
provides a novel and effective tool for analyzing and designing the control system of brushless DC motor.
References
[1] MATLAB 7.3 (2006), The Mathworks Inc..
[2] Navidi, N.; Bavafa, M,: Hesami, S. (2009): A new approach for designing of PID controller for a linear brushless DC motor with
using ant colony search algorithm. IEEE Power & Energy Engineering c onference, pp. 1-5.
[3] Figueroa, J.; Brocart, C.; Cros, J.; Viarouge, P. (2003): Simplified methods for ployphase brushless DC motors. Mathematics and
Computer in Simulation, 63 (3-5), pp. 209-224.
[4] Duane, C. Hanselman. (1994): Brushless permanent-magnet motor design. New York, McGraw-Hill.
[5] Krishnan, R. (2007): Electric motor drives-modeling, analysis and control. Pearson prentice Hall, India.
[6] Dixon, J. W.; Rodriguez, M.; Huerta, R. (2002): Position estimator and simplified current control strategy for brushless DC motor using
DSP technology. IEEE Industrial Electronics Conference, IECON’02, pp. 5-8.
[7] Palani, S. (2010): Contol system engineering. Second edition, McGraw Hill, India.
[8] Somanatham, R.; Prasad, P. V. N.; Rajkumar, A. D. (2006): Modeling and simulation of sensorless control of PMBLDC motor using
zero-crossing back EMF detection. IEEE International Symposium on Power Electronics, Electric Drives, Automation and Motion,
SPEEDAM 2006, pp. 984-989.
N.Muruganantham received B.E. degree in Electrical and Electronics Engineering from
Periyar University, Salem, and M.E. degree in Power Electronics and Drives from Anna
Unversity, Chennai. He is working as an Assistant Professor in the Department of Electrical
& Electronics Engineering at Periyar Maniammai University, Vallam, Thanjavur (d.t), Tamil
Nadu, India. Currently he is pursuing Ph.D. in Electrical & Electronics Engineering at Anna
University Coimbatore. He is a member in IEEE (Power Electronics Society) and ISTE. He
has presented papers in 4 National amd International conferences and published 2 research
papers in reputed International journals. His research interest are soft switching converters,
solid state drives, artificial intelligence and motion control.
Dr.S.Palani is the Dean and Professor, Dept. of ECE, Sudharsan Engg. College, Pudukkottai
(d.t), Tamil Nadu, India. He has wide teaching experience of over four decades. He started his
teaching career at the erstwhile Regional Engineering College (now N.I.T), Tiruchirapalli,
where as a Professor and Head, he took the responsibility of establishing Instrumentation and
Control Engg. Dept. He has also served as Principal, Sudharsan Engg. College, Pudukkottai,
Director, ECE Dept in K.S.R. College of Engg. Tiruchengode and Dean, Sona College of
Technology, Salem. He has published more than 54 research papers in reputed national and
international journals. He is the author of the books titled Control Systems Engineering,
Signals and Systems, and Digital Signal Processing. His area of interest in research includes
Artificial Intelligence Tecniques in the design of controllers for dynamic systems besides Digital Signal Processing.
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... where A act and B act are state dependent and they are defined as follows [145]: ...
... where, x act (t) = [I a,act , I b,act , I c,act , ω act , θ act ] T and u act (t) = [V as , V bs , V cs , T l ] T are the motor state and input respectively. R s , λ, J, B f and P are the stator resistance per phase, [145] and are related by: ...
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This paper describes a different way to sense the phase currents and to estimate DC rotor position of a brushless DC (BLDC) motor. For current control, the current is sensed taking the absolute value of two of the three phases, transforming this information in a DC current I<sub>MAX</sub>, which is finally compared with a reference value. With this method of control, all the transistors of the inverter are commutated with the same PWM signal. Based on the this characteristic, the paper proposes a method to estimate the instantaneous position of the rotor. The method is based on the determination of the current slopes of the PWM during the conduction periods. The main characteristic of this type of motor, fed with quasi-square-wave currents, is that it only needs a six-position sensor, and one current controller for its full torque control. The system was implemented using a fast digital signal processor (TMS320F241) which is programmed with a closed loop PI control for the phase currents. The processor also makes all the calculations required for position estimation. Because of a lack of BLDC in laboratory, the motor tested was a 12 kW (16 HP) permanent magnet synchronous motor (PMSM), with sinusoidal back EMF. This machine has been tested with quasi-square current waveforms using the method proposed, and fed with an IGBT inverter working at 15 kHz commutation frequency. Experimental results of the currents and the way the slopes of current locus are obtained are shown.
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Medical system engineering is a theoretical discipline which unifies the methods of analysis and synthesis of complex medical systems, medical technology, and public health services in general. The work of medical institutions typically involves an abundance of medical and economic information, basically in statistical form. Mathematical models of medical institutions establish regularities in the mass of empirical information and remove distortions in the statistical data, thereby aiding the work of planning organizations. Considering medical institutions as complex queueing systems involving cost estimates of the institutions themselves and implicit losses connected with delay times, gives a theoretical basis for choosing the parameters of the medical institutions to be economically optimal. To determine these parameters we must solve problems of nonlinear programming. Some particular problems of public health services are studied: i.e., determining the optimal number of beds in a hospital and the optimal number of ambulances, and determining periodicities in conducting prophylactic examinations. The study of the systems of several medical institutions and their influence on the flow of patients leads to the use of the method of statistical simulation. A simulation using the Markov process, changing the random waiting times of individual cases to their mean values, thereby decreasing the variance of the statistical estimates, is considered. A numerical example is given, illustrating one heuristic approach to estimating the accuracy of the results of this type of simulation. The contradiction between the level of scientific achievements of medicine and the possibilities of their wide practical realization is discussed in the framework of the present system of public health services where individual forms of labor prevail, hampering the effective use of medical technology. The importance of computer automation in processing medical information is pointed out.
degree in Power Electronics and Drives from Anna Unversity, Chennai. He is working as an Assistant Professor in the Department of Electrical & Electronics Engineering at Periyar Maniammai University
  • N Muruganantham
N.Muruganantham received B.E. degree in Electrical and Electronics Engineering from Periyar University, Salem, and M.E. degree in Power Electronics and Drives from Anna Unversity, Chennai. He is working as an Assistant Professor in the Department of Electrical & Electronics Engineering at Periyar Maniammai University, Vallam, Thanjavur (d.t), Tamil Nadu, India. Currently he is pursuing Ph.D. in Electrical & Electronics Engineering at Anna University Coimbatore. He is a member in IEEE (Power Electronics Society) and ISTE. He has presented papers in 4 National amd International conferences and published 2 research papers in reputed International journals. His research interest are soft switching converters, solid state drives, artificial intelligence and motion control.
Currently he is pursuing Ph.D. in Electrical & Electronics Engineering at Anna University Coimbatore. He is a member in IEEE (Power Electronics Society) and ISTE. He has presented papers in 4 National amd International conferences and published 2 research papers in reputed International journals
  • M E Salem
N.Muruganantham received B.E. degree in Electrical and Electronics Engineering from Periyar University, Salem, and M.E. degree in Power Electronics and Drives from Anna Unversity, Chennai. He is working as an Assistant Professor in the Department of Electrical & Electronics Engineering at Periyar Maniammai University, Vallam, Thanjavur (d.t), Tamil Nadu, India. Currently he is pursuing Ph.D. in Electrical & Electronics Engineering at Anna University Coimbatore. He is a member in IEEE (Power Electronics Society) and ISTE. He has presented papers in 4 National amd International conferences and published 2 research papers in reputed International journals. His research interest are soft switching converters, solid state drives, artificial intelligence and motion control.