Investigation of torque and sensitivity analysis of a two-phase hybrid stepper motor

Stepper motors are recognized as synchronous motors ideally suited for low-speed, high-torque applications that demand precise position control. This is attributed to their characteristic step-by-step rotation, enabling them to lock into each step and maintain their position without interruption [1].

Based on their structure, stepper motors come in three main types: variable reluctance, permanent magnet, and hybrid. Variable reluctance stepper motors feature a lightweight, non-magnetized toothed rotor, facilitating rapid responses and accelerations, making them ideal for applications requiring swift reactions [1]. Permanent magnet stepper motors share similarities with variable reluctance types but possess a magnetized rotor with fewer teeth. Hybrid stepper motors (HSMs), a combination of the two, offer the capability to deliver high torques at low speeds, owing to their rotor structure comprising both permanent magnets and toothed metal. These characteristics have rendered HSMs increasingly prevalent, emerging as the preferred choice across various applications.

Stepper motors are very common and find extensive use across a myriad of applications, including positioning [2, 3], robotics [4,5,6,7,8,9], and even visual instruments such as cameras [10]. Based on this, there are many articles about stepper motors, but most of them are about presenting different methods for speed and position controlling in these actuators. Jenkins et al. [11] have introduced a method for optimizing the balance between the magnets in the rotor and the coils in the stator to obtain the desired static torque. They could calculate the motor‘s parameters for different situations and the rotor’s different positions from the stator by preparing the motor‘s equivalent magnetic circuit. Based on these parameters they made their desired changes to the motor‘s structure. Jung et al. [12] also have introduced a method for optimizing static torque in permanent magnet stepper motors. They analyzed the HSM with a few selected parameters such as teeth form and the number of wire-turns in each stator pole using the 3D finite element method (FEM) along with the design of experiments (DOE) method. Kosaka and Matsui [13] also performed a nonlinear magnet analysis in a HSM. Their aim in this paper was to introduce a method for increasing 3D FEM analysis speed while FEM itself is significantly time-consuming.

Matsui et al. [14] have presented a method to analyze continuous torque in HSMs using its magnetic circuit. They divided the air gap between rotor and stator teeth into five separate spaces and calculated permeability for these sections and the motor‘s continuous torque using these calculations. Rao and Prasad [15] have analyzed HSM’s torque using its equivalent magnetic circuit along with 2D FEM. They used the sectioning method to divide the space between the teeth of the rotor and the stator into five sections to analyze it easier.

Stuebig and Ponick [16] have achieved a faster method to calculate the stepper motor parameters by combining FEM and the analysis method. They first prepared the motor‘s equivalent magnetic circuit, then with the use of 2D FEM, they calculated the magnetic permeability for each element. And after the calculation of the required parameters, they could achieve the motor‘s torque. Kang and Lieu [17] have proposed a parametric magnetic model for a 5-phase HSM and analyzed that along with 2D FEM to calculate its torque. Lim et al. [18] also have used FEM, equivalent magnetic circuit, and neural network to estimate and calculate unknown parameters for a 5-phase stepper motor.

A lot of research has been done on different methods to control stepper motors. Besides, several new geometries for stepper motors have been represented by researchers. Bendjedia et al. [19] could estimate speed and rotor position in a HSM without using any mechanical sensor but by the use of digital signal processing and Kalman filter. Lin and Zheng [20] also could control a HSM’s torque practically and in simulation without the usage of any sensor and just by using digital signal processing. Butcher et al. [21] have proposed a new electrical model usable in stepper motor control applications.

Le-Huy et al. [22] have proposed a comprehensive model that can be used in MATLAB-SIMULINK to calculate desired parameters using this environment. Yazdani et al. [23] also have presented a new PID controller that can improve motor speed tracking. Lu et al. [24] have accomplished the designing and fabrication of a novel stepper motor with permanent magnets in its stator structure. Kavitha and Umamaheswari [25] also have designed and analyzed a novel disk-type stepper motor. Last but not least, Hojati et al. [26, 27] have designed three new structures that have increased HSMs torque density dramatically.

Despite this wealth of research, limited attention has been directed toward investigating the behavior of HSMs under varying conditions and their sensitivity to different parameters. Hence, this study aims to address this gap comprehensively. Section 2 outlines the modeling of a real HSM, followed by an analytical study in Section 3. Finally, Section 4 delves into the motor’s sensitivity to different parameters.

To investigate different parameters of a HSM, a two-phase HSM was chosen and modeled (Figure 1). The specifications and dimensions of this motor are shown in Table 1.

Figure 1:

This stepper motor has a stator with eight poles and a rotor consisting of two half-rotors which are mounted together with a 3.6° mechanical phase difference. Since some of the motor specifications such as the magnet, rotor, and stator materials were unknown, a trial-and-error method had to be used. So, at first, a few different coil currents were applied to one of the stator phases, and the motor holding torque was calculated. The testbed to measure this torque is shown in Figure 2.

Figure 2:

Since only one phase of the motor is excited, the stepper motor remains locked in its position and the holding torque is measurable. To calculate the produced holding torque in this state, different weights were placed on a 60 mm arm coupled with the motor shaft. The weights for each test mode were added until the motor was released from its locked position. In each case, a digital balance was used to measure the weights. Finally, the motor torque was assessed with this data. Table 2 shows the results of this experiment.

Using the obtained data, a computer model of the motor in Ansys Maxwell was designed and analyzed with 1.5 A coil excitation. The results of this simulation were similar to practical tests with a good approximation that can be seen in Figure 3, and the simulated motor is illustrated in Figure 4.

Figure 3:

Figure 4:

After preparing the computer model, the actual motor and the modeled motor performances were measured and compared at several points to ensure the precision of all motor parameters and consequently the simulation accuracy. The results confirmed the accuracy of the simulated model. Figure 3 shows the output holding torque of the simulated model in Ansys Maxwell software for the states corresponding to Table 2.

By modeling the motor to study it analytically, the motor’s specifications and output torque can be calculated quickly and accurately. This method is very useful for studying motor sensitivity to its various parameters. According to the papers [14, 15], the magnetic permeability of the air gap between the rotor and stator teeth is divided into five zones (Figure 5). Based on the parameters of these five regions and the applied current, the output voltage, flux linkage, and motor torque can be calculated.

Figure 5:

Based on Eqs (1)–(5), the permeances P1 to P5 can be calculated [14]: (1) P1=hμ0t−xg {P_1} = h{\mu _0}{{t - x} \over g} (2) P2=hμ02πln1+πx2g {P_2} = h{\mu _0}{2 \over \pi }ln \left( {1 + {{\pi x} \over {2g}}} \right) (3) P3=hμ01πlng+2d−πx2g+πx2 {P_3} = h{\mu _0}{1 \over \pi }ln \left( {{{g + 2d - {{\pi x} \over 2}} \over {g + {{\pi x} \over 2}}}} \right) (4) P4=hμ02πlng+2dg+2d+πx2 {P_4} = h{\mu _0}{2 \over \pi }ln \left( {{{g + 2d} \over {g + 2d + {{\pi x} \over 2}}}} \right) (5) P5=hμ0s−x−4dπg+2d {P_5} = h{\mu _0}{{s - x - {{4d} \over \pi }} \over {g + 2d}}

In the above equations, h is the rotor and stator height, μ₀ is the vacuum magnetic permeability coefficient, and t is the slot thickness; x is the amount of nonalignment between rotor and stator teeth. The parameter g is the air gap length and d is the depth of the rotor and stator teeth. These dimensions are listed in Table 3. The total permeance of each pole can be calculated from Eq. (6). In this equation, Z_s is the number of teeth per stator pole, and it is five for this motor: (6) Pt=ZsP1+2P2+P3+P4+P5 {P_t} = {Z_s}\left[ {{P_1} + 2\left( {{P_2} + {P_3} + {P_4}} \right) + {P_5}} \right]

The torque waveform created by each tooth is sinusoidal over the electrical angle (θ). The Eq. (7) is used to calculate this angle. In this equation, Z_r is the number of rotor teeth which is 50 for this motor and θ_m is the mechanical angle: (7) θ=Zrθm \theta = {Z_r}{\theta _m} Eqs (8) and (9) according to Figure 6, are used to calculate the flux linkage in α and β phases. In these equations, N_s is the number of wire-turns per stator winding, i_α and i_β are the excitation currents of α and β phases, respectively, and F_m is the equivalent mmf of the permanent magnet: (8) ψα=2Pα+Pα′Ns2iα+Pα−Pα′NsFm {\psi _\alpha } = 2\left( {{P_\alpha } + P_\alpha ^\prime} \right)N_s^2{i_\alpha } + \left( {{P_\alpha } - P_\alpha ^\prime} \right){N_s}{F_m} (9) ψβ=2Pβ+Pβ′Ns2iβ+Pβ−Pβ′NsFm {\psi _\beta } = 2\left( {{P_\beta } + P_\beta ^\prime} \right)N_s^2{i_\beta } + \left( {{P_\beta } - P_\beta ^\prime} \right){N_s}{F_m}

Figure 6:

Based on the values calculated in the above equations, Eq. (10) determines the torque generated by the motor. In Eq. (6), the values of inductances L_α and L_β are calculated according to Eqs (11) and (12).

To calculate the static torque waveform for an electric angle period (θ = 0 − 2π), the above equations and MATLAB were used. In Table 3, the parameters used in this analysis are outlined. (10) T=Zr1∂Lα2∂θiα2+1∂Lβ2∂θiβ2+iα∂ψα∂θ+iβ∂ψβ∂θ T = {Z_r}\left( {{{1\partial {L_\alpha }} \over {2\partial \theta }}i_\alpha ^2 + {{1\partial {L_\beta }} \over {2\partial \theta }}i_\beta ^2 + {i_\alpha }{{\partial {\psi _\alpha }} \over {\partial \theta }} + {i_\beta }{{\partial {\psi _\beta }} \over {\partial \theta }}} \right) (11) Lα=2Pα+Pα′Ns2 {L_\alpha } = 2\left( {{P_\alpha } + P_\alpha ^\prime} \right)N_s^2 (12) Lβ=2Pβ+Pβ′Ns2 {L_\beta } = 2\left( {{P_\beta } + P_\beta ^\prime} \right)N_s^2

Figure 7 also shows the comparison of the torque curve of the numerical simulation and analytical model. The difference in the graphs is due to the simplifications used in analytical equations.

Figure 7:

Many parameters influence the design of a stepper motor, with varying degrees of impact on its final performance. While some parameters have minimal effects, others significantly alter motor behavior with slight adjustments. This section examines key parameters in HSMs. The methodology involves keeping all parameters in the motor model constant except for the one under investigation. This specific parameter is varied from a minimum to a maximum value, and its effects on the motor’s output torque are measured. Through this approach, the motor’s sensitivity to each parameter can be comprehensively analyzed.

Meshing is very important in finite element analysis; therefore, different meshes were used in this step. The meshing method and the size of the meshes for each zone were based on their influence on the magnetic flux path and the purpose of the analysis. Figure 8 illustrates a sample meshing used for this analysis. In this picture, the parts of the stator that are attracted by the rotor have the finest meshes, and the parts of it that do not have any significant influence on the magnetic flux, have larger meshes.

Figure 8:

a.

Magnet height

The strength of a magnet depends on its material and its dimensions. Besides, the magnitude of the magnetic field produced by the magnets depends only on its material; and the magnets’ dimensions do not have any influence on its magnitude. The magnet’s dimensions only influence the magnetic flux density around it. In this motor, a cylindrical magnet is used; as the magnet height increases, it appears that the power of the motor and thus, the produced torque should increase. However, as can be seen in Figure 9, the results obtained by the software show that this conclusion is incorrect and the magnet height has no specific and important effect on motor performance, therefore, the motor has no sensitivity to this parameter. In Figures 9 and 10, all the parameters are the same as in Table 1 except for the magnet height. As shown in Figure 10, the cause of this insensitivity is how magnetic flux is distributed in the motor. As the magnet height increases or decreases, the magnetic flux distribution and its density on the teeth and the air gap do not change; therefore, the motor torque remains constant. In these figures, the motor was analyzed magneto-statically for magnet heights from 9 mm to 21 mm. As illustrated, the magnetic flux density in the teeth is relatively uniform at about 1 Tesla.

Figure 9:

Figure 10:

As can be seen in Figures 10B and 10C, the only effect that the magnet height has on the motor is the increase in flux density at the upper or lower end of the rotor, but this effect is negligible at the air gap because the magnetic flux must still pass the same distance to reach this area.

This trend is also the same when the motor is not excited. As it is represented in Figure 11, the motor produces very small torque levels when the rotor is rotated which is called the motor cogging torque. This torque for different magnet heights is almost the same which again proves that the magnet height does not affect the motor performance.

Figure 11:

b.

Magnet diameter

It has been shown in the previous section that the motor is not sensitive to the magnet height, but it seems that the performance of the motor is affected by the magnet diameter as the magnetic flux in the teeth varies with the variations in the magnet edge distance from the rotor teeth. This effect is illustrated in Figure 12. In Figures 12 and 13, all the parameters are the same as in Table 1 except for the magnet diameter.

Figure 12:

Figure 13:

As can be seen in Figure 12, despite the excitation of the coils with a current of 1.5 A, the motor torque has changed significantly with a slight change in the magnet diameter, indicating a high sensitivity of the motor to the magnet diameter. In Figure 13, the variations of the magnetic flux density on the rotor teeth caused by two magnets of 20.8 mm and 10.8 mm in diameter can be seen clearly.

By comparing Figures 13A and 13B, it can be seen clearly that with increasing magnet diameter, the flux density increases dramatically which is the result of the magnet near the rotor teeth, and consequently, the motor torque will increase dramatically.

For different magnet diameters, this trend is also the same when the motor is not excited.

As represented in Figure 14, although the motor-produced torque is very small, it alters for different magnet diameters; and as the magnet diameter shrinks, the produced cogging torque leads to a straight line.

Figure 14:

c.

Magnet flux density

The magnet type and the magnetic flux density created by it play an important role in the amount of torque produced by the stepper motor. This is due to the flux density increase or decrease in the air gap between the rotor and stator teeth. Studies have shown that while the motor core is not saturated, its torque changes linearly with flux density variations. Therefore, the sensitivity of the motor is proportional to the changes in this parameter. In Figure 15, the performance of the motor based on the magnetic flux density changes can be seen vividly. In this graph, the motor is excited with 1.5 A. To change the magnetic flux density in the rotor, the rotor magnet has been replaced with an electric magnet, and with different excitations, the magnetic flux densities shown in this figure are produced. In this simulation, all the other parameters are the same as in Table 1.

Figure 15:

d.

Air gap

One of the most effective parameters on motor performance is the air gap. This parameter is such influential that by decreasing it from 0.1 mm to 0.05 mm the motor torque increases from about 180 N/mm to about 300 mN/m. This is due to its direct impact on the amount of flux density that passes through the motor components. The shorter the air gap, the less the resistance to magnetic flux. However, by decreasing the air gap, the motor fabrication problems increase. Figure 16 shows the effect of this parameter on flux distribution and flux density on the teeth and the air gap. In Figure 17, the torque curve can be seen for several air gaps with 1.5 A excitation in stator coils. In these simulations, all the parameters are the same as in Table 1 except for the air gap distance.

Figure 16:

Figure 17:

e.

Number of wire-turns per coil

One of the factors affecting the performance and power of the motor is the number of wire-turns per coil. As can be seen in Figure 18, with a reduction of two turns of wire per coil, the motor torque declines 20 N/mm which means that by reducing the number of turns to 96%, the motor torque also reduces to 96%. In this diagram, because the motor is excited with 1.5 A, it is still very far from the saturation situation even with 33 turns of wire. In this simulation, all the parameters are the same as in Table 1 except for the wire-turns number.

Figure 18:

f.

Teeth depth in the rotor and stator

The teeth in the rotor and stator are very effective on motor performance. The proper depth of the teeth improves the sinusoidal waveform of the torque and improves symmetry for the positive and negative peak points of the torque curve. Also, it can improve the motor’s torque output. In Figures 19–21, the influence of change in the rotor, stator, and both teeth depth on the torque curve is illustrated respectively. In these simulations, all the parameters are the same as in Table 1 except for the rotor and the stator depth.

Figure 19:

Figure 20:

Figure 21:

g.

Teeth width

The teeth width, as well as their depth, has a great influence not only on the overall behavior of the motor and its output torque but also on the torque waveform shape in terms of symmetry. Therefore, it can be said that the motor is highly sensitive to this parameter. In the motor understudy, the teeth width is 3.6°, but as shown in Figure 22, this angle is not quite optimal and the angles from 3.1° to 2.6° have the best waveforms. In this simulation, all the parameters are the same as in Table 1 except for the teeth width of the rotor and the stator.

Figure 22:

h.

Teeth fillet

The fillet radius on the edge of the rotor and stator teeth (as shown in Figure 23) has a great influence on motor behavior. This effect is similar to the effect of the depth of the rotor and stator teeth, except that it is much more intense and the motor is more sensitive to it. This is due to its effect on the distribution of magnetic flux passing the air gap. When the edges of the teeth are sharp and without fillets, in those sharp points flux concentration is created resulting in saturation on those areas. Thus, this phenomenon affects the flux density on the air gap significantly. Moreover, the fillet radius is effective and important as well.

Figure 23:

In Figure 24, the discontinuity of the flux in the air gap for the sharp teeth situation, and its consistency and uniformity for the filleted teeth can be seen clearly. In addition, in Figure 25, the effect of the fillet with different radiuses is illustrated. In these simulations, all the parameters are the same as in Table 1 except for the teeth fillet radius.

Figure 24:

Figure 25:

i.

Stator pole thickness

Stator poles are used to place the teeth and are essential for the transmission of magnetic flux (Figure 26). The thickness of this part seemed to play a critical role in the motor performance; however, studies showed that the motor was not sensitive to this parameter and therefore, the thickness of this section on the torque curve is negligible. Figure 27 illustrates it clearly. In this simulation, all the parameters are the same as in Table 1 except for the stator pole thickness.

Figure 26:

Figure 27:

This study focused on analyzing the torque of HSMs and investigating their sensitivity to different parameters. A real HSM was first modeled and simulated. After obtaining the motor parameters, the simulated model was compared with the actual HSM to ensure its accuracy. The simulation results were then compared with the analytical model to evaluate these two methods and investigate the influence of various parameters on the output torque waveform.

The study found that the simplifications made in the analytical model caused minor errors, rendering the waveform not entirely accurate. However, these errors did not affect the desired result, which is the maximum produced holding torque. Finally, the effects of different motor parameters were investigated using the 3D FEM method.

The results showed that HSMs are most sensitive to parameters such as magnet diameter, magnetic flux density, air gap, number of wire-turns in the coils, teeth width, and teeth fillet. Small changes in these parameters significantly affect motor performance. In contrast, parameters such as magnet height and stator pole thickness have minimal impact on motor performance, allowing flexibility in designing new HSM motors with higher torque and power density.