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Modeling of hybrid stepper motors for closed loop operation
6th IFAC Symposium on Mechatronic Systems The International Federation of Automatic Control April 10-12, 2013. Hangzhou, China
Modeling of hybrid stepper motors for closed loop operation B. Henke ∗ O. Sawodny ∗ S. Schmidt ∗∗ R. Neumann ∗∗ ∗
Institute for System Dynamics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany ∗∗ Festo AG & Co. KG, Ruiter Straße 82, 73734 Esslingen, Germany Abstract: In automation and handling, many motion tasks are accomplished using servo motors. For simple motion tasks, hybrid stepper motors can be used in closed loop operation as a cheaper replacement of expensive servo motors. Therefore a model of the electromechanical behaviour is necessary for controller design. A detailed model of the hybrid stepper motor is derived, taking the saliencies of the rotor and the stator into account. Least-squares optimal parameter values are identified for important model parameters. Validation experiments show the high accuracy of the model. Keywords: automation, motor control, ac motors, step motors, modelling, parameter identification 1. INTRODUCTION In automation and handling, many motion tasks are accomplished by means of electric drives. Most of these drives are servo motors (also known as EC-motors or permanent magnet synchronous motors) in a cascade structured position control loop as depicted in Fig. 1. Here, the motor current i is controlled in an inner control loop, whereas the velocity ω and the position θ are controlled in the outer loops. The setpoint values for the inner loops are successively calculated by the outer control loops. θd
Kp
ωd
id
Kv
ω
Ki
diff
u
Tload θ
Table 1. Comparison of motor types Motor type
Operation
Price
Servo motor
+ Closed loop current control - Open loop step operation
- High
Hybrid stepper motor Hybrid stepper motor in closed loop operation
i
Fig. 1. Block diagram of a position control loop in cascade structure However, not every motion task requires the high performance of a modern servo motor. Therefore, the question arises whether it is possible to replace the servo motor in the control loop (Fig. 1) for simple motion tasks by a cheaper motor. Here, a hybrid stepper motor is considered as a replacement for the servo motor. This is particularly reasonable, as a short comparison of these two motor types shows. First of all, both servo motor and hybrid stepper motor share the same principle of torque generation: The supply current flows through the stator coils, which operate as an electromagnet and attract the magnetic rotor. Whereas the servo motor is operated in closed loop current control, the hybrid stepper motor is usually operated in open loop step operation, where a rectangular pattern of terminal voltages results in a stepwise movement of the rotor. Due to this open loop operation it is less robust to load disturbances. Being produced in large quantities makes the hybrid stepper motor available at much lower cost than a 978-3-902823-31-1/13/$20.00 © 2013 IFAC
servo motor. A hybrid stepper motor operated in closed loop current control combines increased performance and robustness with a low price due to high quantities. Table 1 summarizes this comparison.
177
+ Closed loop current control
+ Low, due to high quantities + Low, due to high quantities
Stepper motors are usually divided in three types: permanent magnet, variable reluctance and hybrid stepper motors. As the name implies, permanent magnet stepper motors use a permanent magnet rotor that is attracted by the magnetic field of the stator coils. Variable reluctance stepper motors use the effect that an iron rotor, subject to a magnetic field, adjusts itself such that the magnetic resistance (reluctance) is minimal. Hybrid stepper motors combine both effects. The rotor of the hybrid stepper motor is made up of two iron endcaps with salient poles, as shown in Fig. 2, that are magnetized by an axially mounted permanent magnet. This results in one north pole endcap and one south pole endcap that is displaced by a half pole pitch. The two endcaps superpose to alternating north and south poles. A typical hybrid stepper motor as used in industry applications has 50 pole pairs and eight stator windings, combined into two electrical stator phases as shown in Fig. 3. Note that the stator teeth are also salient, with a similar shape as the rotor poles. The magnetic attraction between the salient stator teeth and the permanent magnet in the rotor 10.3182/20130410-3-CN-2034.00042
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
The aim of this paper is to derive a model for the two-phase hybrid stepper motor, taking into account rotor and stator saliencies and a detailed model of the detent torque. The rest of this paper is organized as follows: First of all, the model stucture is introduced in Sec. 2. Models for the current dynamics and the torque generation are then derived in Sec. 3 and Sec. 4 and combined into an overall mechatronic model in Sec. 5. The results are summarized in Sec. 6. 2. MODEL STRUCTURE The physical design of the hybrid stepper motor with its 50 pole pairs and different tooth pitches at the rotor and the stator, that can be seen in Fig. 3, can be simplified for mathematical treatment. One can introduce a transformation ϕ = p · θ, (1) where θ is the actual mechanical rotor angle, p is the number of pole pairs (here p = 50) and ϕ is the transformed quantity, the so-called electrical rotor angle. This transformation can be interpreted as considering only one pole pair that has to rotate p times the mechanical angle. Fig. 4 illustrates this interpretation. Note that in this simplified model the two stator phases are displaced by an electrical angle of 90 degrees. This reflects the electrical behaviour resulting from the different tooth pitches of rotor and stator.
Fig. 2. Rotor of a hybrid stepper motor
1 ϕ
Fig. 3. Cross-section of stator and rotor of a hybrid stepper motor
A comprehensive overview of stepper motors, models and control is provided by Kenjo and Sugawara (1994). Several authors have pointed out the possibility of closed loop control of stepper motors (e.g. Bodson et al. (1993); Zribi and Chiasson (1991)), yet without taking rotor saliencies into account. Rotor saliencies have been modeled by Yan et al. (2006) for a permanent magnet synchronous motor. Mizutani et al. (1993) use a model of the magnetic circuit to model rotor saliencies for a 3-phase hybrid stepper motor, before simplifying the model for control design. Many results have been reported on compensating the detent torque (also cogging torque in the case of permanent magnet synchronous motors) as for example by Holtz and Springob (1996) for a permanent magnet synchronous motor and more recently by Hwang and Seok (2007), who develop an observer based torque ripple compensation for a linear hybrid stepper motor. Tsui et al. (2009) use a detent torque model to compensate for the detent torque in a three-phase stepper motor.
178
N
The saliency of rotor and stator and the detent torque are effects that are useful for step operation but are disturbing continuous motion and need to be compensated for by closed loop control. Thus, a detailed model of the hybrid stepper motor that includes effects caused by the saliency of stator and rotor is needed for control design.
2'
S
is present even with deenergized stator windings. It results in the so-called detent torque that is typical for motors containing permanent magnets.
2
1' Fig. 4. Simplified model structure with one pole pair Mathematical modelling of electrical motors results in the following basic model structure: di = f (u, i, ϕ, ˙ ϕ) (2) dt Tel = Tel (i, ϕ) (3) ϕ (4) θ= p dω 1 = (Tel − Tdet (ϕ) − Tload ) (5) dt J dθ =ω (6) dt Here, (2) is the equation for the current dynamics in the stator windings that is dependent on the terminal voltages u ∈ R2 , the currents i ∈ R2 , the (electrical) rotor angle ϕ and its time derivative ϕ. ˙ The currents flowing through the stator windings result in an electrical torque Tel . Equation (4) is just the transformation (1), solved for the mechanical rotor angle θ. Finally, the equations of motion (5) and (6) describe the movement of the rotor, i.e. it’s angular
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
The electric circuit in the stator is modeled by the voltage equation dΨ u1 i (7) =R 1 + i2 u2 dt that contains the terminal voltages u1 , u2 and currents i1 , i2 of the two electrical phases, the resistance R of the stator windings and the motor cable and the induced voltage dΨ dt . The magnetic flux linkage Ψ herein can be calculated as i cos(ϕ) Ψ = L(ϕ) 1 + ΨM (8) sin(ϕ). i2
The first part, containing the inductance matrix L(ϕ) ∈ R2×2 , is the flux linkage resulting from the stator windings, whereas the second part, containing the flux constant ΨM , results from the permanent magnetic flux caused by the permanent magnet in the rotor. Combining these equations results in the differential equation for the current dynamics di 1 i1 u1 −1 dt − R = L(ϕ) di2 i2 u 2 dt ∂L i1 − sin(ϕ) − ϕ˙ (9) − ΨM ϕ˙ cos(ϕ) ∂ϕ i2 Similar equations have been used previously, for example by Bodson et al. (1993) or Zribi and Chiasson (1991), but without taking the dependency of the inductance matrix L(ϕ) on the rotor angle ϕ into account. This dependency will be examined in detail in the following. 3.1 Inductance model The dependency of the inductance on the rotor angle is caused by the saliency of the rotor and stator teeth. As depicted in Fig. 5, the part of the rotor, that is included in the magnetic cirucit between the stator windings, changes depending on the rotor angle ϕ.
3.2 Inductance measurements To examine these effects, step responses of the currents subject to unity steps of the terminal voltages were recorded for different angular positions of the rotor. During the measurement, the rotation of the rotor was suppressed by a second motor connected to the hybrid stepper motor. By these means one step responses could be measured for one constant rotor angle at a time. Two examples of these step responses are shown in Fig. 6. They clearly show different settling times due to the different inductances at different angular rotor positions.
current i [A]
3. CURRENT DYNAMICS
magnetic resistance (reluctance) can be included in the model by an angle-dependent induction cos(2ϕ) sin(2ϕ) 10 . (10) + L1 L(ϕ) = L0 sin(2ϕ) − cos(2ϕ) 01 It consists of a constant self-inductance L0 and an angledependent part with amplitude L1 , that is caused by the rotation of the rotor and contributes self- and mutual inductance terms.
1.5 1 0.5 0 0
0.01 0.02 time t [s]
0.03
(a) step response of i1 for ϕ = 2.8rad
current i [A]
velocity ω and rotor angle θ, subject to the electrical torque Tel , the detent torque Tdet and the load torque Tload , that also incorporates friction. J is the rotor inertia. From a system theoretical point of view, the system hybrid stepper motor has the input u and the measurable outputs i and θ.
1.5 1 0.5 0 0
0.01 0.02 time t [s]
0.03
(b) step response of i1 for ϕ = 3.6rad
Fig. 6. Step responses of the current i1 subject to a unity step of the terminal voltage u1 at different angular positions ϕ. Actual measurements (black) and leastsquares optimal fitting (blue). The two step responses clearly show different settling times caused by the angle-dependent inductance.
Fig. 5. Sketch of the rotor in different angular positions. The field lines of the magnetic field are sketched between stator windings 1 and 1’ to illustrate the change of the airgap between stator and rotor teeth. The bigger the airgap between the stator and rotor teeth is, the bigger is the magnetic resistance. This varying
179
The inductance can be identified from step responses using the voltage equation (7). As the angular rotor position ϕ is kept constant during the measurement, i.e. ϕ˙ = 0, this equation simplifies to di 1 u1 i1 dt . + L(ϕ) di =R (11) 2 i2 u2 dt Assuming discrete measurements uk1 , uk2 , ik1 , ik2 with the superscript k = 1, ..., N denoting the individual time
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
samples and N being the total number of samples, this equation can be rewritten point-wise as R k k △ik1 △ik2 L 0 i1 △t △t 0 u1 11 L12 . (12) k k k = △i △i k u2 0 △t1 △t2 L21 i2 0 L 22 bk Ak
characterize the variation of all four components of the inductance matrix. 4. TORQUE GENERATION The hybrid stepper motor uses two components for torque generation: On the one hand the Lorentz force, that is the ×10−3 4
However, this equation is only satisfied if the model perfectly fits the measurements. Consequently, one can define a model error e = bk − Ak ξ. (13) The parameter vector ξ containis the resistance R and the components of the inductance matrix L L L = 11 12 . (14) L21 L22 Note that the rotor angle ϕ is constant during each step response. The parameter vector ξ can now be computed by minimizing the overall quadratic error, N k T k etotal = e e , (15) k=1
A
b
2
0
0
180
5 el. angle ϕ [rad]
10
(b) mutual inductance L12
×10−3 2
0
−2 0
5 el. angle ϕ [rad]
10
(c) mutual inductance L21
×10−3 4
inductance L22 [H]
The measured self-inductances show systematic deviations from the model, that might indicate that there exist unmodelled higher frequencies in the variation of the inductance. Due to the low resolution of the encoder used for position measurement, this can not be decided from the available measurements. The encoder has a resolution of 500 lines per revolution which results with quadrature evaluation in only 40 position increments per pole pitch. However, Fig. 7 shows that the proposed model reflects the dominant frequency in the variation of the inductance. Note that only two parameters, L0 and L1 , are needed to
10
−2
Simulation with these parameters results in the step responses shown with blue lines in Fig. 6.
Evaluation of step responses for 80 distinct angular positions distributed over two full pole pitches (i.e. 4π rad electrical angle) reveals the variation of the inductance depending on the rotor angle. Fig. 7 shows the values resulting from the measurements as well as the values predicted by the proposed inductance model (10). This variation of the inductance repeats over the full rotation of the rotor with a period of π rad electrical angle.
5 el. angle ϕ [rad]
0
(16)
The exact solution to this problem is given for example in Golub and van Loan (1983) and the optimal parameter vector ξ ∗ minimizing the overall quadratic error is −1 AT b. (17) ξ ∗ = AT A
0
×10−3 2
inductance L21 [H]
that is by solving the least-squares problem 2 1 b 1 A . . ! . ξ − .. = min. . A N b N
2
(a) self inductance L11
inductance L12 [H]
A difference quotient is used to discretize the equation.
inductance L11 [H]
ξ
2
0 0
5 el. angle ϕ [rad]
10
(d) self inductance L22
Fig. 7. Variation of inductance over an angular range of 2 pole pitches (4π rad). Measurements (black) and proposed inductance model (blue). The inductance model reflects the dominant frequency in the variation of the inductance.
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
T
T
Here, the vectors i12 = [i1 i2 ] and u12 = [u1 u2 ] are introduced for the sake of brevity. Using the voltage equation (7), the transformation (4) and the inductance model (10), this energy balance equation can be solved for the electromagnetic torque T T ∂L i1 1 i − sin(ϕ) i Tel = pΨm 1 + p 1 . (19) cos(ϕ) i2 2 i2 ∂ϕ i2 permanent magnetic torque
torque Tdet [Nm]
The torque generated by these two effects can be calculated from an energy balance equation, balancing the change of magnetic energy stored in the magnetic field of the stator windings, the supplied electrical power, the ohmic power losses and finally the mechanical power generated by the motor: d 1 T i12 L(ϕ)i12 = iT12 u12 − Ri212 − Tel ω. (18) dt 2
to be constant and can be removed from the torque measurement as a constant offset. The frequency spectrum of this measurement, also shown in Fig. 8, is obtained by a discrete Fourier transformation. 0.5 0 −0.5 10 20 15 el. angle ϕ [rad]
5
25
30
(a) Measured detent Torque
amplitude [Nm]
attractive or repulsive force between the permanent magnet in the rotor and the electromagnetic stator windings. On the other hand the reluctance force, that is the attraction force between the electromagnetic stator windings and the salient iron end caps of the rotor.
0.05
0 0
5
10 20 15 frequency f /fel [-]
25
(b) frequency spectrum
reluctance torque
As mentioned above, the torque divides into a permanent magnetic torque resulting from the Lorentz force and a reluctance torque resulting from the reluctance force. The reluctance torque is only present if the inductance is not constant, that is ∂L ∂ϕ = 0. It is often neglected, resulting in the simplified torque equation for permanent magnet stepper motors in Bodson et al. (1993) and Zribi and Chiasson (1991). 4.1 Detent torque
A close look reveals that a third component is contributing to the motor torque: The permanent magnetic rotor attracts the salient iron teeth of the stator. Note that this force is not electromagnetic but purely permanent magnetic and therefore always present, even if the stator windings are deenergized. As this force determines the steady states of the rotor when the motor is deenergized, the resulting torque is known as detent torque. The detent torque is expected to repeat with every stator phase and every rotor pole. It is therefore often modeled as Tdet = a sin(4ϕ), (20) as for example in Holtz and Springob (1996) and in Zribi and Chiasson (1991). However, in a real stepper motor effects such as assymetries alter the shape of the reluctance torque. Therefore a Fourier-approach is used that accounts for multiple spectral components: n Tdet = ak · sin(kϕ) (21) k=1
Here, ak is the amplitude of the k-th component and n is the total number of modeled spectral components. Fig. 8 shows a measurement of the detent torque of a hybrid stepper motor. A second motor was used to rotate the hybrid stepper motor with constant speed during the measurement. At constant speed, friction is assumed
181
Fig. 8. Measured detent torque of a hybrid stepper motor and frequency spectrum of the measurement, showing three dominant spectral components. With respect to the electrical frequency defined as ϕ˙ , (22) fel = 2π rad the dominant spectral components can be observed at the frequencies fel , 2fel and 4fel . Taking only these dominant frequencies into account for the model, the Fourier sum (21) reduces to Tdet = a1 · sin(ϕ) + a2 · sin(2ϕ) + a4 · sin(4ϕ). (23) This equation is similar to the models obtained by Hwang and Seok (2007) and Tsui2009. In a similar fashion as in Sec. 3.2 the parameter-affine structure of this equation can be utilized to determine the least-squares optimal parameter values. Assuming discrete k measurements ϕk and Tdet with the superscript k = 1, ..., N denoting the individual time samples and N being the total number of samples, this equation can be rewritten point-wise as
a1 k k k k Tdet = sin(ϕ ) sin(2ϕ ) sin(4ϕ ) a2 . (24) a3 k d Ck ζ
Then, the optimal amplitudes ζ ∗ , minimizing the overall quadratic error 2 1 d1 C . . ! . ξ − (25) .. = min, . N N d C d C
2
can be calculated as
−1 C T d. ζ∗ = CT C
(26)
0.5
2 0
−2 4.5
0 −0.5 10 15 20 el. angle ϕ [rad]
5
25
5.5 5 time t [s]
6
(a) current i1
30
Fig. 9. Model of the detent torque with least-squares optimal parameters taking three spectral components into account. The model (black) fits the measurement (blue) well.
current i2 [A]
torque Tdet [Nm]
Fig. 9 shows the model of the detent torque according to (23) with least-squares optimal parameters. It shows a good fit of the measured shape of the detent torque.
current i1 [A]
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
2 0
−2 4.5
5 5.5 time t [s]
6
(b) current i2
The overall mechatronic model of the hybrid stepper motor consists of the current dynamics (9), the inductance model (10), the equations for the electromagnetic torque (19) and the detent torque (23), the transformation (4) and the mechanical equations of motion (5) and (6). Inputs to the system are the terminal voltages u1 , u2 , the measurable outputs are the currents i1 , i2 and the rotor angle θ. The load torque Tload acts as a disturbance on the system.
Fig. 11. Currents resulting from the terminal voltages shown in Fig. 10. The model (blue) closely matches the measurements (black).
voltage u1 [V]
The model of the hybrid stepper motor is validated through experiments. The motor is driven in step operation, where rectangular terminal voltages, as shown in Fig. 10, lead to the currents shown in Fig. 11. This results in a stepwise movement of the rotor, shown in Fig. 12. These experiments show the high accuracy of the model for both, the current dynamics and the mechanical rotor dynamics. 2
0.4 0.2 4.5
5 5.5 time t [s]
6
Fig. 12. Step-wise rotor movement caused by the rectangular terminal voltages shown in Fig. 10. The model (blue) closely matches the measurements (black).
2
the stator and rotor teeth is important for the current dynamics of a hybrid stepper motor. The saliency is included in the model as a variable inductance L(ϕ), dependent on the rotor angle ϕ. The inductance model is identified from step responses by minimization of the least-squares error. A detailed model of the detent torque is derived from measurements. The frequency spectrum shows that three dominant spectral components contribute to the detent torque. Here again, least-squares optimal values for the model parameters are calculated.
0
Common models for two-phase stepper motors can be derived as simplifications of this model.
0
−2 4.5
5 5.5 time t [s]
6
(a) terminal voltage u1
voltage u2 [V]
angle θ [rad]
5. OVERALL MECHATRONIC MODEL
(b) terminal voltage u2
The model is highly accurate and is nonetheless simple enough for analytical treatment. It is therefore a proper foundation for model based control design. Further work will aim at model based design of closed loop current and position controllers for the hybrid stepper motor.
Fig. 10. Rectangular terminal voltages for the validation experiment
ACKNOWLEDGEMENTS
−2 4.5
5 5.5 time t [s]
6
6. CONCLUSION The proposed model shows high accuracy, demonstrated in validation experiments. It is shown that the saliency of
182
The authors want to thank Dipl.-Ing. Armin Hartmann and Dr.-Ing. Alexander Hildebrandt at Festo AG & Co. KG for the fruitful discussions, explanations and great support.
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
REFERENCES Bodson, M., Chiasson, J., Novotnak, R., and Rekowski, R. (1993). High-performance nonlinear feedback control of a permanent magnet stepper motor. IEEE Transactions on Control Systems Technology, 1, 5–14. Golub, G.H. and van Loan, C.F. (1983). Matrix Computations. John Hopkins University Press. Holtz, J. and Springob, L. (1996). Identification and compensation of torque ripple in high-precision permanent magnet motor drives. IEEE Transactions on Industrial Electronics, 43, 309–320. Hwang, T.S. and Seok, J.K. (2007). Observer-based ripple force compensation for linear hybrid stepping motor drives. IEEE Transactions on Industrial Electronics, 54, 2417–2424. Kenjo, T. and Sugawara, A. (1994). Stepping motors and their microprocessor controls. Clarendon Press. Mizutani, K., Hayashi, S., and Matsui, N. (1993). Modeling and control of hybrid stepping motors. In Conference Record of the IEEE Industry Applications Society Annual Meeting. Tsui, K.W.H., Cheung, N.C., and Yuen, K.C.W. (2009). Novel modeling and damping technique for hybrid stepper motor. IEEE Transactions on Industrial Electronics, 56, 202–2011. Yan, Y., Zhu, J., Guo, Y., and Lu, H. (2006). Modeling and simulation of direct torque controlled PMSM drive system incorporating structural and saturation saliencies. In Conference Record of the IEEE Industry Applications Conference. Zribi, M. and Chiasson, J. (1991). Position control of a PM stepper motor by exact linearization. IEEE Transactions on Automatic Control, 36, 620–625.
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Modeling of hybrid stepper motors for closed loop operation B. Henke ∗ O. Sawodny ∗ S. Schmidt ∗∗ R. Neumann ∗∗ ∗
Institute for System Dynamics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany ∗∗ Festo AG & Co. KG, Ruiter Straße 82, 73734 Esslingen, Germany Abstract: In automation and handling, many motion tasks are accomplished using servo motors. For simple motion tasks, hybrid stepper motors can be used in closed loop operation as a cheaper replacement of expensive servo motors. Therefore a model of the electromechanical behaviour is necessary for controller design. A detailed model of the hybrid stepper motor is derived, taking the saliencies of the rotor and the stator into account. Least-squares optimal parameter values are identified for important model parameters. Validation experiments show the high accuracy of the model. Keywords: automation, motor control, ac motors, step motors, modelling, parameter identification 1. INTRODUCTION In automation and handling, many motion tasks are accomplished by means of electric drives. Most of these drives are servo motors (also known as EC-motors or permanent magnet synchronous motors) in a cascade structured position control loop as depicted in Fig. 1. Here, the motor current i is controlled in an inner control loop, whereas the velocity ω and the position θ are controlled in the outer loops. The setpoint values for the inner loops are successively calculated by the outer control loops. θd
Kp
ωd
id
Kv
ω
Ki
diff
u
Tload θ
Table 1. Comparison of motor types Motor type
Operation
Price
Servo motor
+ Closed loop current control - Open loop step operation
- High
Hybrid stepper motor Hybrid stepper motor in closed loop operation
i
Fig. 1. Block diagram of a position control loop in cascade structure However, not every motion task requires the high performance of a modern servo motor. Therefore, the question arises whether it is possible to replace the servo motor in the control loop (Fig. 1) for simple motion tasks by a cheaper motor. Here, a hybrid stepper motor is considered as a replacement for the servo motor. This is particularly reasonable, as a short comparison of these two motor types shows. First of all, both servo motor and hybrid stepper motor share the same principle of torque generation: The supply current flows through the stator coils, which operate as an electromagnet and attract the magnetic rotor. Whereas the servo motor is operated in closed loop current control, the hybrid stepper motor is usually operated in open loop step operation, where a rectangular pattern of terminal voltages results in a stepwise movement of the rotor. Due to this open loop operation it is less robust to load disturbances. Being produced in large quantities makes the hybrid stepper motor available at much lower cost than a 978-3-902823-31-1/13/$20.00 © 2013 IFAC
servo motor. A hybrid stepper motor operated in closed loop current control combines increased performance and robustness with a low price due to high quantities. Table 1 summarizes this comparison.
177
+ Closed loop current control
+ Low, due to high quantities + Low, due to high quantities
Stepper motors are usually divided in three types: permanent magnet, variable reluctance and hybrid stepper motors. As the name implies, permanent magnet stepper motors use a permanent magnet rotor that is attracted by the magnetic field of the stator coils. Variable reluctance stepper motors use the effect that an iron rotor, subject to a magnetic field, adjusts itself such that the magnetic resistance (reluctance) is minimal. Hybrid stepper motors combine both effects. The rotor of the hybrid stepper motor is made up of two iron endcaps with salient poles, as shown in Fig. 2, that are magnetized by an axially mounted permanent magnet. This results in one north pole endcap and one south pole endcap that is displaced by a half pole pitch. The two endcaps superpose to alternating north and south poles. A typical hybrid stepper motor as used in industry applications has 50 pole pairs and eight stator windings, combined into two electrical stator phases as shown in Fig. 3. Note that the stator teeth are also salient, with a similar shape as the rotor poles. The magnetic attraction between the salient stator teeth and the permanent magnet in the rotor 10.3182/20130410-3-CN-2034.00042
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
The aim of this paper is to derive a model for the two-phase hybrid stepper motor, taking into account rotor and stator saliencies and a detailed model of the detent torque. The rest of this paper is organized as follows: First of all, the model stucture is introduced in Sec. 2. Models for the current dynamics and the torque generation are then derived in Sec. 3 and Sec. 4 and combined into an overall mechatronic model in Sec. 5. The results are summarized in Sec. 6. 2. MODEL STRUCTURE The physical design of the hybrid stepper motor with its 50 pole pairs and different tooth pitches at the rotor and the stator, that can be seen in Fig. 3, can be simplified for mathematical treatment. One can introduce a transformation ϕ = p · θ, (1) where θ is the actual mechanical rotor angle, p is the number of pole pairs (here p = 50) and ϕ is the transformed quantity, the so-called electrical rotor angle. This transformation can be interpreted as considering only one pole pair that has to rotate p times the mechanical angle. Fig. 4 illustrates this interpretation. Note that in this simplified model the two stator phases are displaced by an electrical angle of 90 degrees. This reflects the electrical behaviour resulting from the different tooth pitches of rotor and stator.
Fig. 2. Rotor of a hybrid stepper motor
1 ϕ
Fig. 3. Cross-section of stator and rotor of a hybrid stepper motor
A comprehensive overview of stepper motors, models and control is provided by Kenjo and Sugawara (1994). Several authors have pointed out the possibility of closed loop control of stepper motors (e.g. Bodson et al. (1993); Zribi and Chiasson (1991)), yet without taking rotor saliencies into account. Rotor saliencies have been modeled by Yan et al. (2006) for a permanent magnet synchronous motor. Mizutani et al. (1993) use a model of the magnetic circuit to model rotor saliencies for a 3-phase hybrid stepper motor, before simplifying the model for control design. Many results have been reported on compensating the detent torque (also cogging torque in the case of permanent magnet synchronous motors) as for example by Holtz and Springob (1996) for a permanent magnet synchronous motor and more recently by Hwang and Seok (2007), who develop an observer based torque ripple compensation for a linear hybrid stepper motor. Tsui et al. (2009) use a detent torque model to compensate for the detent torque in a three-phase stepper motor.
178
N
The saliency of rotor and stator and the detent torque are effects that are useful for step operation but are disturbing continuous motion and need to be compensated for by closed loop control. Thus, a detailed model of the hybrid stepper motor that includes effects caused by the saliency of stator and rotor is needed for control design.
2'
S
is present even with deenergized stator windings. It results in the so-called detent torque that is typical for motors containing permanent magnets.
2
1' Fig. 4. Simplified model structure with one pole pair Mathematical modelling of electrical motors results in the following basic model structure: di = f (u, i, ϕ, ˙ ϕ) (2) dt Tel = Tel (i, ϕ) (3) ϕ (4) θ= p dω 1 = (Tel − Tdet (ϕ) − Tload ) (5) dt J dθ =ω (6) dt Here, (2) is the equation for the current dynamics in the stator windings that is dependent on the terminal voltages u ∈ R2 , the currents i ∈ R2 , the (electrical) rotor angle ϕ and its time derivative ϕ. ˙ The currents flowing through the stator windings result in an electrical torque Tel . Equation (4) is just the transformation (1), solved for the mechanical rotor angle θ. Finally, the equations of motion (5) and (6) describe the movement of the rotor, i.e. it’s angular
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
The electric circuit in the stator is modeled by the voltage equation dΨ u1 i (7) =R 1 + i2 u2 dt that contains the terminal voltages u1 , u2 and currents i1 , i2 of the two electrical phases, the resistance R of the stator windings and the motor cable and the induced voltage dΨ dt . The magnetic flux linkage Ψ herein can be calculated as i cos(ϕ) Ψ = L(ϕ) 1 + ΨM (8) sin(ϕ). i2
The first part, containing the inductance matrix L(ϕ) ∈ R2×2 , is the flux linkage resulting from the stator windings, whereas the second part, containing the flux constant ΨM , results from the permanent magnetic flux caused by the permanent magnet in the rotor. Combining these equations results in the differential equation for the current dynamics di 1 i1 u1 −1 dt − R = L(ϕ) di2 i2 u 2 dt ∂L i1 − sin(ϕ) − ϕ˙ (9) − ΨM ϕ˙ cos(ϕ) ∂ϕ i2 Similar equations have been used previously, for example by Bodson et al. (1993) or Zribi and Chiasson (1991), but without taking the dependency of the inductance matrix L(ϕ) on the rotor angle ϕ into account. This dependency will be examined in detail in the following. 3.1 Inductance model The dependency of the inductance on the rotor angle is caused by the saliency of the rotor and stator teeth. As depicted in Fig. 5, the part of the rotor, that is included in the magnetic cirucit between the stator windings, changes depending on the rotor angle ϕ.
3.2 Inductance measurements To examine these effects, step responses of the currents subject to unity steps of the terminal voltages were recorded for different angular positions of the rotor. During the measurement, the rotation of the rotor was suppressed by a second motor connected to the hybrid stepper motor. By these means one step responses could be measured for one constant rotor angle at a time. Two examples of these step responses are shown in Fig. 6. They clearly show different settling times due to the different inductances at different angular rotor positions.
current i [A]
3. CURRENT DYNAMICS
magnetic resistance (reluctance) can be included in the model by an angle-dependent induction cos(2ϕ) sin(2ϕ) 10 . (10) + L1 L(ϕ) = L0 sin(2ϕ) − cos(2ϕ) 01 It consists of a constant self-inductance L0 and an angledependent part with amplitude L1 , that is caused by the rotation of the rotor and contributes self- and mutual inductance terms.
1.5 1 0.5 0 0
0.01 0.02 time t [s]
0.03
(a) step response of i1 for ϕ = 2.8rad
current i [A]
velocity ω and rotor angle θ, subject to the electrical torque Tel , the detent torque Tdet and the load torque Tload , that also incorporates friction. J is the rotor inertia. From a system theoretical point of view, the system hybrid stepper motor has the input u and the measurable outputs i and θ.
1.5 1 0.5 0 0
0.01 0.02 time t [s]
0.03
(b) step response of i1 for ϕ = 3.6rad
Fig. 6. Step responses of the current i1 subject to a unity step of the terminal voltage u1 at different angular positions ϕ. Actual measurements (black) and leastsquares optimal fitting (blue). The two step responses clearly show different settling times caused by the angle-dependent inductance.
Fig. 5. Sketch of the rotor in different angular positions. The field lines of the magnetic field are sketched between stator windings 1 and 1’ to illustrate the change of the airgap between stator and rotor teeth. The bigger the airgap between the stator and rotor teeth is, the bigger is the magnetic resistance. This varying
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The inductance can be identified from step responses using the voltage equation (7). As the angular rotor position ϕ is kept constant during the measurement, i.e. ϕ˙ = 0, this equation simplifies to di 1 u1 i1 dt . + L(ϕ) di =R (11) 2 i2 u2 dt Assuming discrete measurements uk1 , uk2 , ik1 , ik2 with the superscript k = 1, ..., N denoting the individual time
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
samples and N being the total number of samples, this equation can be rewritten point-wise as R k k △ik1 △ik2 L 0 i1 △t △t 0 u1 11 L12 . (12) k k k = △i △i k u2 0 △t1 △t2 L21 i2 0 L 22 bk Ak
characterize the variation of all four components of the inductance matrix. 4. TORQUE GENERATION The hybrid stepper motor uses two components for torque generation: On the one hand the Lorentz force, that is the ×10−3 4
However, this equation is only satisfied if the model perfectly fits the measurements. Consequently, one can define a model error e = bk − Ak ξ. (13) The parameter vector ξ containis the resistance R and the components of the inductance matrix L L L = 11 12 . (14) L21 L22 Note that the rotor angle ϕ is constant during each step response. The parameter vector ξ can now be computed by minimizing the overall quadratic error, N k T k etotal = e e , (15) k=1
A
b
2
0
0
180
5 el. angle ϕ [rad]
10
(b) mutual inductance L12
×10−3 2
0
−2 0
5 el. angle ϕ [rad]
10
(c) mutual inductance L21
×10−3 4
inductance L22 [H]
The measured self-inductances show systematic deviations from the model, that might indicate that there exist unmodelled higher frequencies in the variation of the inductance. Due to the low resolution of the encoder used for position measurement, this can not be decided from the available measurements. The encoder has a resolution of 500 lines per revolution which results with quadrature evaluation in only 40 position increments per pole pitch. However, Fig. 7 shows that the proposed model reflects the dominant frequency in the variation of the inductance. Note that only two parameters, L0 and L1 , are needed to
10
−2
Simulation with these parameters results in the step responses shown with blue lines in Fig. 6.
Evaluation of step responses for 80 distinct angular positions distributed over two full pole pitches (i.e. 4π rad electrical angle) reveals the variation of the inductance depending on the rotor angle. Fig. 7 shows the values resulting from the measurements as well as the values predicted by the proposed inductance model (10). This variation of the inductance repeats over the full rotation of the rotor with a period of π rad electrical angle.
5 el. angle ϕ [rad]
0
(16)
The exact solution to this problem is given for example in Golub and van Loan (1983) and the optimal parameter vector ξ ∗ minimizing the overall quadratic error is −1 AT b. (17) ξ ∗ = AT A
0
×10−3 2
inductance L21 [H]
that is by solving the least-squares problem 2 1 b 1 A . . ! . ξ − .. = min. . A N b N
2
(a) self inductance L11
inductance L12 [H]
A difference quotient is used to discretize the equation.
inductance L11 [H]
ξ
2
0 0
5 el. angle ϕ [rad]
10
(d) self inductance L22
Fig. 7. Variation of inductance over an angular range of 2 pole pitches (4π rad). Measurements (black) and proposed inductance model (blue). The inductance model reflects the dominant frequency in the variation of the inductance.
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
T
T
Here, the vectors i12 = [i1 i2 ] and u12 = [u1 u2 ] are introduced for the sake of brevity. Using the voltage equation (7), the transformation (4) and the inductance model (10), this energy balance equation can be solved for the electromagnetic torque T T ∂L i1 1 i − sin(ϕ) i Tel = pΨm 1 + p 1 . (19) cos(ϕ) i2 2 i2 ∂ϕ i2 permanent magnetic torque
torque Tdet [Nm]
The torque generated by these two effects can be calculated from an energy balance equation, balancing the change of magnetic energy stored in the magnetic field of the stator windings, the supplied electrical power, the ohmic power losses and finally the mechanical power generated by the motor: d 1 T i12 L(ϕ)i12 = iT12 u12 − Ri212 − Tel ω. (18) dt 2
to be constant and can be removed from the torque measurement as a constant offset. The frequency spectrum of this measurement, also shown in Fig. 8, is obtained by a discrete Fourier transformation. 0.5 0 −0.5 10 20 15 el. angle ϕ [rad]
5
25
30
(a) Measured detent Torque
amplitude [Nm]
attractive or repulsive force between the permanent magnet in the rotor and the electromagnetic stator windings. On the other hand the reluctance force, that is the attraction force between the electromagnetic stator windings and the salient iron end caps of the rotor.
0.05
0 0
5
10 20 15 frequency f /fel [-]
25
(b) frequency spectrum
reluctance torque
As mentioned above, the torque divides into a permanent magnetic torque resulting from the Lorentz force and a reluctance torque resulting from the reluctance force. The reluctance torque is only present if the inductance is not constant, that is ∂L ∂ϕ = 0. It is often neglected, resulting in the simplified torque equation for permanent magnet stepper motors in Bodson et al. (1993) and Zribi and Chiasson (1991). 4.1 Detent torque
A close look reveals that a third component is contributing to the motor torque: The permanent magnetic rotor attracts the salient iron teeth of the stator. Note that this force is not electromagnetic but purely permanent magnetic and therefore always present, even if the stator windings are deenergized. As this force determines the steady states of the rotor when the motor is deenergized, the resulting torque is known as detent torque. The detent torque is expected to repeat with every stator phase and every rotor pole. It is therefore often modeled as Tdet = a sin(4ϕ), (20) as for example in Holtz and Springob (1996) and in Zribi and Chiasson (1991). However, in a real stepper motor effects such as assymetries alter the shape of the reluctance torque. Therefore a Fourier-approach is used that accounts for multiple spectral components: n Tdet = ak · sin(kϕ) (21) k=1
Here, ak is the amplitude of the k-th component and n is the total number of modeled spectral components. Fig. 8 shows a measurement of the detent torque of a hybrid stepper motor. A second motor was used to rotate the hybrid stepper motor with constant speed during the measurement. At constant speed, friction is assumed
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Fig. 8. Measured detent torque of a hybrid stepper motor and frequency spectrum of the measurement, showing three dominant spectral components. With respect to the electrical frequency defined as ϕ˙ , (22) fel = 2π rad the dominant spectral components can be observed at the frequencies fel , 2fel and 4fel . Taking only these dominant frequencies into account for the model, the Fourier sum (21) reduces to Tdet = a1 · sin(ϕ) + a2 · sin(2ϕ) + a4 · sin(4ϕ). (23) This equation is similar to the models obtained by Hwang and Seok (2007) and Tsui2009. In a similar fashion as in Sec. 3.2 the parameter-affine structure of this equation can be utilized to determine the least-squares optimal parameter values. Assuming discrete k measurements ϕk and Tdet with the superscript k = 1, ..., N denoting the individual time samples and N being the total number of samples, this equation can be rewritten point-wise as
a1 k k k k Tdet = sin(ϕ ) sin(2ϕ ) sin(4ϕ ) a2 . (24) a3 k d Ck ζ
Then, the optimal amplitudes ζ ∗ , minimizing the overall quadratic error 2 1 d1 C . . ! . ξ − (25) .. = min, . N N d C d C
2
can be calculated as
−1 C T d. ζ∗ = CT C
(26)
0.5
2 0
−2 4.5
0 −0.5 10 15 20 el. angle ϕ [rad]
5
25
5.5 5 time t [s]
6
(a) current i1
30
Fig. 9. Model of the detent torque with least-squares optimal parameters taking three spectral components into account. The model (black) fits the measurement (blue) well.
current i2 [A]
torque Tdet [Nm]
Fig. 9 shows the model of the detent torque according to (23) with least-squares optimal parameters. It shows a good fit of the measured shape of the detent torque.
current i1 [A]
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
2 0
−2 4.5
5 5.5 time t [s]
6
(b) current i2
The overall mechatronic model of the hybrid stepper motor consists of the current dynamics (9), the inductance model (10), the equations for the electromagnetic torque (19) and the detent torque (23), the transformation (4) and the mechanical equations of motion (5) and (6). Inputs to the system are the terminal voltages u1 , u2 , the measurable outputs are the currents i1 , i2 and the rotor angle θ. The load torque Tload acts as a disturbance on the system.
Fig. 11. Currents resulting from the terminal voltages shown in Fig. 10. The model (blue) closely matches the measurements (black).
voltage u1 [V]
The model of the hybrid stepper motor is validated through experiments. The motor is driven in step operation, where rectangular terminal voltages, as shown in Fig. 10, lead to the currents shown in Fig. 11. This results in a stepwise movement of the rotor, shown in Fig. 12. These experiments show the high accuracy of the model for both, the current dynamics and the mechanical rotor dynamics. 2
0.4 0.2 4.5
5 5.5 time t [s]
6
Fig. 12. Step-wise rotor movement caused by the rectangular terminal voltages shown in Fig. 10. The model (blue) closely matches the measurements (black).
2
the stator and rotor teeth is important for the current dynamics of a hybrid stepper motor. The saliency is included in the model as a variable inductance L(ϕ), dependent on the rotor angle ϕ. The inductance model is identified from step responses by minimization of the least-squares error. A detailed model of the detent torque is derived from measurements. The frequency spectrum shows that three dominant spectral components contribute to the detent torque. Here again, least-squares optimal values for the model parameters are calculated.
0
Common models for two-phase stepper motors can be derived as simplifications of this model.
0
−2 4.5
5 5.5 time t [s]
6
(a) terminal voltage u1
voltage u2 [V]
angle θ [rad]
5. OVERALL MECHATRONIC MODEL
(b) terminal voltage u2
The model is highly accurate and is nonetheless simple enough for analytical treatment. It is therefore a proper foundation for model based control design. Further work will aim at model based design of closed loop current and position controllers for the hybrid stepper motor.
Fig. 10. Rectangular terminal voltages for the validation experiment
ACKNOWLEDGEMENTS
−2 4.5
5 5.5 time t [s]
6
6. CONCLUSION The proposed model shows high accuracy, demonstrated in validation experiments. It is shown that the saliency of
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The authors want to thank Dipl.-Ing. Armin Hartmann and Dr.-Ing. Alexander Hildebrandt at Festo AG & Co. KG for the fruitful discussions, explanations and great support.
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
REFERENCES Bodson, M., Chiasson, J., Novotnak, R., and Rekowski, R. (1993). High-performance nonlinear feedback control of a permanent magnet stepper motor. IEEE Transactions on Control Systems Technology, 1, 5–14. Golub, G.H. and van Loan, C.F. (1983). Matrix Computations. John Hopkins University Press. Holtz, J. and Springob, L. (1996). Identification and compensation of torque ripple in high-precision permanent magnet motor drives. IEEE Transactions on Industrial Electronics, 43, 309–320. Hwang, T.S. and Seok, J.K. (2007). Observer-based ripple force compensation for linear hybrid stepping motor drives. IEEE Transactions on Industrial Electronics, 54, 2417–2424. Kenjo, T. and Sugawara, A. (1994). Stepping motors and their microprocessor controls. Clarendon Press. Mizutani, K., Hayashi, S., and Matsui, N. (1993). Modeling and control of hybrid stepping motors. In Conference Record of the IEEE Industry Applications Society Annual Meeting. Tsui, K.W.H., Cheung, N.C., and Yuen, K.C.W. (2009). Novel modeling and damping technique for hybrid stepper motor. IEEE Transactions on Industrial Electronics, 56, 202–2011. Yan, Y., Zhu, J., Guo, Y., and Lu, H. (2006). Modeling and simulation of direct torque controlled PMSM drive system incorporating structural and saturation saliencies. In Conference Record of the IEEE Industry Applications Conference. Zribi, M. and Chiasson, J. (1991). Position control of a PM stepper motor by exact linearization. IEEE Transactions on Automatic Control, 36, 620–625.
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