PREDICTION OF PULL-IN AND PULL-OUT
TORQUE CHARACTERISTICS OF THE PERMANENT
MAGNET STEP MOTOR
MIROSLAV BJEKIĆ1, ALENKA MILOVANOVIĆ2, BRANKO KOPRIVICA3
Key words: Permanent magnet step motor, Electromechanical characteristics,
Measurements.
The aim of this paper is to present a novel procedure for the prediction of the stepping
motor characteristics, in terms of determining its electromechanical properties. For this
purpose, a magnetic circuit of this motor was solved using the finite element method,
and its self- and mutual inductances were calculated. The static torque characteristic of
the motor was solved analytically and the algorithm for determination of the dynamic
characteristics was developed. On the basis of these results, the simulation of the pull-in
and pull-out characteristics of the permanent magnet step motor was done. The obtained
characteristics were compared with those obtained by measurement and a very good
agreement of these characteristics was obtained
1. INTRODUCTION
The Permanent Magnet Step Motor has many applications, owing to their
advantages over other types of electric servomotors, mainly, and the possibility of
simple digital control. Thus, analysis of this motor has been matter of interest for
many authors [1–8]. While its construction is relatively simple, analysis of its
dynamic characteristics is very complicated. The papers devoted to this issue generally
apply only to certain parts of the problem, such as geometry, non-linearity, higher
harmonics of current, and other. In these analyses, authors used various simplifications
in order to reach solutions, and to obtain a motor model [7–9]. The following
approximations have been used: magnetic circuit is linear; distribution of the
magnetic flux is simple; periodic hysteresis; and eddy currents losses are negligible. In
some models, such simplifications have led to solutions that are significantly
different from those obtained by measuring. So, main characteristics of this type of
motor have been obtained by measurements after its production.
1
University of Kragujevac, Faculty of Technical Sciences, Cacak, Svetog Save 65, 32000
Cacak, Serbia, mbjekic@gmail.com
2
alenka.milovanovic@ftn.kg.ac.rs
3
branko.koprivica@ftn.kg.ac.rs
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 60, 1, p. 29–38, Bucarest, 2015
30
Miroslav Bjekić, Alenka Milovanović, Branko Koprivica
2
In this paper, the authors have presented a new procedure for predicting the
stepping motor electromechanical characteristics without any simplification or idealization. The application of this procedure facilitates the easier design of the step motor,
and provides determination of the dynamic characteristics during the projecting
process. Characteristics of the analyzed permanent-magnet stepper motor are:
− No. of Phase: q = 3
− No. of Pole: p' = 8
− Step Angle: θ = 150
− Steps Per Revolution: N = 24
− Rated voltage: U = 24V
− Rated Current [A/phase]: I = 0.8A
− DC Coil Resistance [Ω/phase]: R = 30Ω
− Inductance [mH/phase]: L = 74.7mH
− Pull-out torque: Mgr = 2.2 Nm.
2. PROBLEM DESCRIPTIONS AND PROCEDURE FOR SOLVING
Equation of motion of the step motor can be expressed as:
J
d2θ
dθ
+F
= M mot (θ) − M load ,
2
dt
dt
(1)
where J is the moment of inertia, F is the coefficient of viscous friction, Mmot(θ) is
the electromagnetic torque and Mload is the load torque. Solving of this equation is
very complicated in case of the motor with complex construction, when all
nonlinearities have been taken into consideration. The step motor is movement
affected by successive control pulses which excite each phase by predicted order
and frequency. The characteristic of static torque is stepwise in moments of time,
when the pulse stops in one phase and starts in another. Therefore, Mmot(θ) cannot
be expressed analytically, but it can be presented as the sum of torques created by
each phase in the time intervals in which these phases have been excited. Also, for
motors with variable reluctance the reluctance torque must be considered:
M mot = M f 1 + M f 2 + M f 3 + M rel .
(2)
The electromagnetic torque of each phase depends on time t, stator currents
i1, i2, i3 and rotor position relative to the stator θ, Mfi = f(t,i1, i2, i3, θ). Currents i1
depend on the control unit voltage u1 the winding resistance Ri, the self-inductance
Lii, the mutual inductance Lij and the rate of change of magnetic flux, created by
3
Characteristics of pull-in and pull-out torque for step motor
31
permanent magnets, Ψi in i-th winding, ii = f (Ri, Lii, Lij, Ψi). Self-inductances Lii
and mutual inductances Lij are constant only in linear operating area, when the step
motor is not saturated. In general, the inductances depend on the stator currents,
and the rotor position relative to the stator (because of the stator grooves and the
existence of the reluctance torque). Flux ψ i is a function of the rotor position
relative to the stator. If the magnetic circuit is not linear this flux depends on stator
currents. Therefore, Lii, Lij, Ψi = f(i1, i2, i3, θ). From these dependencies all the
complexity of the problem that should be solved can be seen.
The concept of a new numerical simulation for predicting the stepping motor
electromechanical characteristics consists of several steps. At the start, based on
the desirable construction of the stepping motor, and taking into account the actual
geometry and materials of the motor, the analysis of the magnetic circuit was
performed by using FEM (Fig. 1).
Fig. 1 – Triangular mesh with 22,871 nodes and 45,651 elements.
Its self- and mutual inductances have been calculated taking into account
initial magnetization and magnetic nonlinearity [10–13]. Also, values of the static
torque for each rotor position and all stator currents have been calculated. Fig. 2
shows the variation of the static torque and its components with angle, obtained by
using FEMM software, when two phases have been excited simultaneously.
32
Miroslav Bjekić, Alenka Milovanović, Branko Koprivica
4
Fig. 2 – Static torque and its components.
After analysis of the graphs presented in Fig. 2, it can be concluded that the
step motor works in a linear working regime and that total electromagnetic torque
can be calculated as a sum of the torques created by each phase and the reluctance
torque.
In the second step, these values have been used to obtain the general analytical
expression of the electromagnetic torque of the motor. For that purpose the
trigonometric collocation polynomial has been used [13]
M mot (θ) =
3
∑C
i
(θ)ii + M rel (θ),
(3)
i =1
Ci (θ) = 130
180
(ψ f i (θ' ') − ψ f i (θ')),
π
(4)
M rel ( θ) = 0.13614 sin (18θ),
(5)
ψ f i (θ ) = Bi1 sin( θ) + Bi 3 sin(3θ) + Bi 5 sin(5θ) + Bi15 sin(15θ) + Bi17 sin(17θ) +
+ Ai1 cos( θ) + Ai 3 cos(3θ) + Ai 5 cos(5θ) + Ai15 cos(15θ) + Ai17 cos(17θ).
(6)
Table 1
Calculated coefficients Aν and Bν
ψf1
ψf2
ψf3
Bν
Aν
Bν
Aν
Bν
Aν
ν=1
0.01597
0.45914
–0.40557
–0.21574
0.38968
–0.24337
ν=3
0.00309
0.02954
0.00308
0.02952
0.00309
0.02954
ν=5
0.00046
0.00265
0.00205
–0.00172
0.00252
–0.00091
ν = 15
–0.00013
–0.00024
–0.00013
–0.00023
–0.00014
–0.00023
ν = 17
–0.00020
–0.00030
–0.00016
–0.00032
–0.00036
–0.00002
5
Characteristics of pull-in and pull-out torque for step motor
33
At the end, using results obtained from first and second steps, the dynamic
equation of motion of the rotor was solved [13, 14]. So, the transient state simulations
were conducted.
By multiple consecutive repetition of the simulation for different motor
torque and rotor positions the pull-in and pull-out characteristics of the motor has
been obtained. These characteristics are the most significant results of a new
procedure proposed in this paper.
3. SIMULATION RESULTS
Motor Data: windings resistance R = 30 Ω, self and mutual inductances
Lfe = = 0.074706 H, Mfe = –0.049165 H, moment of inertia Jfe = 0.00086 kgm2,
reduced load torque Jload = 0.01 Nm2/(rad/s), friction coefficient
Ffe = 0.05 Nm2/(rad/s).
Working Regime Data: plug-in voltage U = 18.6 V, impulse duration T = 5 s,
load torque and its character (active or reactive) Mload= 0.77254 Nm − initial angle
and initial rotation speed θ0= 0 rad, ω0= 0 rad/s, sequence of exciting the phases
N = 500, number of positions per step t = T/500 = 6 ms.
Phase Excitation Sequence: F1: 10-1; F2: 011; F3: -1-10.
This part of the programme solves the dynamic movement equation in which
its coefficients were changed for each new position. Simulation results have been
presented in Fig. 3.
4. DETERMINATION OF PULL-IN AND PULL-OUT TORQUE
Pull-in and pull-out characteristic of the step motor have been derived, using
the presented procedure by multiple repetition and dynamic simulation by changing
the load torque. If dynamic simulation shows that the motor stays synchronized, the
load torque should be increased by half of its previous value. If the motor lost
synchronicity the load torque should be decreased by half of its previous value.
After 5 iterations the accuracy achieved is 3.125%. For 7 iterations the accuracy
increases to 0.78%.
Figure 4 show defined motion of the motor used in the determination of the
pull-out characteristic, in the case of simultaneous excitation of only one stator
phase. The simulated pull-in and pull-out characteristic of the step motor, for active
and reactive load and two stator currents values are presented in Fig. 5.
34
Miroslav Bjekić, Alenka Milovanović, Branko Koprivica
M [Nm]
ω [rad/s]
Fig. 3 – Stepping movement with the simultaneous excitation of two phases.
Fig. 4 – Motion of the motor for pull-out characteristic determination.
6
7
Characteristics of pull-in and pull-out torque for step motor
35
Fig. 5 – The simulated pull-in and pull-out characteristics.
5. MEASUREMENTS RESULTS
In order to confirm the obtained simulated results, several measurements
were carried out.
Figure 6 presents graphs of simulated and measured voltage and currents in
all three phases, for load torques when only one phase has been excited.
The results have been found to be in very good agreement. In particular, it is
very important to notice that the good agreement between these results, with
minimal deviation, has been obtained when the frequency of the control pulses was
high (in this case the influence of the induced electromotive forces from permanent
magnets is significant). This confirms the validity of the voltage equation used for
calculation of all three currents. Since this currents cause the electromagnetic
torque, in the transient change in the position of the rotor, in moments of turning on
or turning off of certain phases, it was necessary to calculate them as accurately as
possible.
Comparison of the simulated and the measured pull-out characteristic is
presented in Fig. 7.
36
Miroslav Bjekić, Alenka Milovanović, Branko Koprivica
Fig. 6 – Simulated and measure values of voltages and current.
Fig. 7 – Two phases simultaneously excited In = 0.8 A, pull-out characteristic.
8
9
Characteristics of pull-in and pull-out torque for step motor
37
6. CONCLUSION
A novel general procedure for projecting step motors was proposed in this
paper. Using this procedure, it is possible to predict pull-in and pull-out characteristics
of the step motor in the projecting phase. Also, the geometrical, electrical and
mechanical parameters of the motor, which should satisfy the specified requirements,
can be obtained by using this procedure. Therefore, it is possible to find the optimal
construction of the motor without manufacturing of the prototype. The accuracy of
the results obtained by using the proposed procedure was confirmed by the
measurements.
ACKNOWLEDGMENTS
This paper has been supported by the Scientific Project TR 33016 financed
by the Ministry of Education, Science and Technological Development of the
Republic of Serbia.
Received on September 17, 2014
REFERENCES
1. K. C Kim, D. S. Ryu, Torque Characteristic With Respect to the Load Angle of a Permanent
Magnet Motor, IEEE Transactions on Magnetics, 48, 11, pp. 4200–4203, 2012.
2. J. J. Cathey, Electric Machines analysis and design applying MATLAB, McGraw Hill, International
edition, 2001.
3. W. Fei, P. C. K. Luk, J. X. Shen, Torque Analysis of Permanent-Magnet Flux Switching Machines
With Rotor Step Skewing, IEEE Transactions on Magnetics, 48, 10, pp. 2664–2673, 2012.
4. S. Bifaretti, V. Iacovone, A. Rocchi, P. Tomei, C. M. Verrelli, Nonlinear speed, tracking, control,
for sensorless PMSM swith unknown load torque: From theory to practice, Control Engineering
Practice, 20, pp.714–724, 2012.
5. Z. Li, Y. Luo, X. Liu, Z. Jian Pull out Torque Computation for Line Start Permanent Magnet
Motor, ICEMS 2008, Proceedings of the 11th International Conference on Electrical Machines
and Systems, Wuhan, China, 2008, pp. 3270–3273.
6. K. Hameyer, C. Schlensok, Gracia, M. Herranz, Combined numerical and analytical method for
geometry optimization of a PM motor, The 15th Conference on the Computation of Electromagnetic Fields, Chenuang, China, 2005.
7. H. Rowlands, An Automatic test system for stepper motor selection, Proceedings 15th Symposium
Incremental Motion Control System & Devices, Illinois, 1986.
8. X. Wang, R.Eichenwald, Torque vs speed analysis and simulation for PM step motor, Proceedings
15th Symposium Incremental Motion Control System & Devices, Illinois, 1986, pp. 241–250.
9. K. Kasetty, Math Modeling and computer simulation of stepper motor and carriage system,
Proceedings 14th Symposium Incremental Motion Control System & Devices, Illinois, 1985.
38
Miroslav Bjekić, Alenka Milovanović, Branko Koprivica
10
10. *** http://www.femm.info
11. Bjekić M., Numerical methods of solving magnetic circuit of permanent magnet step motor,
International PhD Seminar Computation of Electromagnetic Field, Budva, Monte Negro,
2004, pp. 11–16.
12. M. Bjekić, Numerical computation dynamic characteristics of permanent-magnet step motor,
International PhD Seminar Numerical Field Computation and Optimization in Electrical
Engineering, Ohrid, Macedonia, 2005, pp. 11–16.
13. M. Bjekić, Determining static torque characteristics of permanent magnet step motor, International
PhD Seminar Computational Electromagnetics and Technical Applications, Banja Luka, Bosnia
and Hercegovina, 2006, pp. 5–10.
14. M. Bjekić, Numerical simulation, measuring and foreseeing dynamic characteristics of the
energetic permanent magnet step motor, 51st Internationales Wissenschaftliches Kolloquium,
Technische Universitat Ilmenau, 2006.