1. Introduction

A hysteresis motor is a synchronous motor that is able to self-start and is easy to use. It is usually applied in small applications such as dynamically tuned gyroscopes (DTGs) and centrifuges. Because of the non-linearity of the hysteresis, this motor is very difficult to analyze, unlike a simple structure. Analytical studies using static hysteresis models have assumed that there is no eddy current ⁽¹⁾. Since a hysteresis motor is a synchronous motor, the synchronous entry time is an important point in the analysis.

Recently, researchers have tried to analyze the dynamic characteristics of hysteresis motors with some equivalent circuit models. One model expands the induction motor model ⁽²⁾ by DQ transformation ^(3-4), and another model added an eddy current resistance that is parallel to the resistance component in the rotor impedance component ⁽⁵⁾. Since the equivalent models and approaches are different, the results of calculations are also different. The results are also different from the actual hysteresis motor characteristics.

This paper presents a model in which the eddy current resistance is parallel to the rotor hysteresis impedance ⁽⁶⁾. The impedance, current, torque, and other characteristics are calculat- ed using an equivalent circuit. The motor torque is calculated by separating the hysteresis torque and eddy current torque. We also propose a method that can accurately measure the synchronous entry time by using the phase difference between the line-to-line or equivalent phase voltage and the phase current of the motor. This method can measure the synchronous entry time by measuring the phase difference between the applied voltage and phase current using an oscilloscope or zero-crossing detector circuit. This method does not require a special speed measuring sensor for a very small motor such as a DTG, for which it is difficult to use equipment such as a dynamometer. The proposed method was applied to the developed simulation program, and the results were confirmed by comparing them with experiments.

2. Previous Dynamic Equivalent Circuit Models

Niasar et al. ⁽³⁾ analyzed a hysteresis motor based on a dynamic equivalent circuit the induction machine, as shown in Fig. 1. The output of the hysteresis motor is proportional to the area of the hysteresis loop of the rotor ring. Induction torque is also generated during the start-up due to the eddy current in the rotor ring. Therefore, it is difficult to obtain the hysteresis motor characteristics by directly applying the dynamic analysis method for an induction motor to a hysteresis motor ⁽³⁾. The results of the dynamic analysis ⁽³⁾ are close to the results of a dynamic analysis of an induction motor.

Fig. 1. The equivalent circuit of DQ transformed dynamic model for the hysteresis motor by Niasar.

Fig. 2 shows a one-phase equivalent circuit model of a hysteresis motor by Nitao ⁽¹⁾, who theoretically described a dynamic analysis of a hysteresis motor in detail. In Fig. 2, $V_{s}$ is the phase voltage, $I_{s}$ is the phase current, $I_{m}$ is the magnetization current, $I_{r}$ is the rotor current, $R_{e}$ is the eddy current resistance of the rotor ring, and s is the rotor’s slip. From this, the impedance of the rotor is calculated as follows:

The hysteresis motor is analyzed by considering the eddy current flowing through $R_{e}/s$ in Fig. 2. However, it does not distinguish the torque caused by the eddy current of the hysteresis motor and instead uses the hysteresis loop area of the rotor ring, including the eddy current. In addition, the reactance component of the hysteresis rotor ring is treated as a leakage reactance. As mentioned, the two methods described have very different results because of the different approaches.

Fig. 2. Equivalent circuit of a hysteresis motor from Nitao’s model

3. Proposed Dynamic Analysis Method of Hysteresis Motor

3.1 Equivalent Circuit and Torque Calculation

The loop area of the hysteresis ring is the motor output in the ideal case. The induction torque is generated during start-up in the real case. Fig. 3 shows the addition of an eddy current resistance to the well-known equivalent circuit of a hysteresis motor ⁽¹⁾.

Fig. 3. Equivalent circuit of the hysteresis motor.

$X_{m}$ is the magnetization reactance, $R_{h}$ is the rotor resistance, and $X_{h}$ is the rotor reactance. $R_{h}$ and $X_{h}$ are determined by the permeability $\mu$ and the B-H hysteresis angle $\delta$ or load angle ^(1-5). Equations (3), (4), and (5) show the equations in Fig. 3 ⁽¹⁾.

(3)

$X_{m}=\omega_{b}\dfrac{2m K_{w}^{2}N_{w}^{2}\mu_{0}r_{g}l}{\pi p^{2}l_{g}}$

(4)

$R_{h}=\omega_{b}\dfrac{m K_{w}^{2}N_{w}^{2}V_{r}\mu}{\pi^{2}r_{r}^{2}}\sin\delta$

(5)

$X_{h}=\omega_{b}\dfrac{m K_{w}^{2}N_{w}^{2}V_{r}\mu}{\pi^{2}r_{r}^{2}}\cos\delta$

In this equation, $m$ is the number of phases, $K_{w}$ is the winding constant, $N_{w}$ is the number of turns per phase, is the average radius of the air gap, $l$ is the axial length of the rotor ring, $l_{g}$ is the air-gap length, $V_{r}$ is the rotor ring volume, $\mu$ is the permeability of the rotor ring, and $r_{r}$ is the average radius of the rotor ring. With these equations, the following motor impedances are calculated.

(6)

$\dot Z_{h}=R_{h}+j\omega L_{h}$

(7)

$\dot Z_{r}=\dfrac{1}{\dfrac{1}{\dot Z_{h}}+\dfrac{s}{R_{e}}}$

(8)

$\dot Z_{t}=\dot Z_{s}+\dfrac{1}{\dfrac{1}{j X_{m}}+\dfrac{1}{\dot Z_{r}}}$

Once the motor impedance is determined, $I_{s}$ is calculated. After that, the internal emf $E_{g}$ is determined, and the other currents are calculated. After DQ transformation, the equilibrium three-phase voltage applied to the motor is expressed as a complex value as shown in (9).

(9)

$\dot V = v_{ds}+ j v_{qs}$

$E_{g}$ is calculated using (10), and then the rotor current $\dot I_{r}$ is calculated using (11).

(10)

$\dot{g}_{2}=\dot{V}_{s}-\dot{Z}_{s} \dot{I}_{s}$

(11)

$\dot{I}_{r}=\frac{\dot{E}_{g}}{\dot{Z}_{r}}$

The mechanical output of a hysteresis motor is calculated differently from typical motors. The output of the hysteresis motor is proportional to the area of the hysteresis loop of the hysteresis ring and the rotor volume as shown in (12) ⁽¹⁾:

(12)

$T_{e}=\dfrac{p V_{r}B_{m}^{2}}{2\mu}\sin\delta =\dfrac{p V_{r}S_{H}}{2\pi}$

where $S_{H}$ is the hysteresis loop area of the rotor ring. The hysteresis torque of the motor can be rewritten using electrical terms, as shown in (13) ⁽⁵⁾.

(13)

$T_{h}=\left(\dfrac{m}{2}\right)\left(\dfrac{P}{2}\right)L_{m}\left |\dot I_{r}\right |\left |\dot I_{m}\right |\sin\delta$

This is the torque due to the hysteresis loop. The torque due to the eddy current can be calculated with (14), which is the same equation used for induction motors:

(14)

$T_{e}=m I_{r}^{2}\dfrac{R_{e}}{s}\dfrac{1}{\omega_{bm}}$

where $\omega_{bm}$ is the synchronous motor angular velocity. The sum of the two torques is the torque generated by the motor from start-up to steady state, which is calculated as follows.

(15)

$T_{d}= T_{h}+ T_{e}$

3.2 Ellipse Approximation Model and Load Angle

Ellipse approximation of the hysteresis loop is used to calculate the hysteresis easily because the loop area is same when the fundamental component of the flux density is used ⁽¹⁾. A previous dynamic model ⁽⁵⁾ uses a fixed permeability, and only the load angle or hysteresis angle $\delta$ varies when the motor arrives at the synchronous speed, which is illustrated in Fig. 4(a). $\delta$ can be calculated using torque (15), which is balanced with the load including inertial, friction, and windage losses.

Fig. 4(b) shows the equivalent elliptic loop used to simulate the dynamic behaviors. After entering synchronous speed, the rotor speed begins to oscillate until the rotor reaches steady state according to the load conditions ⁽⁷⁾, which is called hunting. However, Miyairi explained about the loop variation after entering synchronous speed ⁽¹⁾, as shown like Fig. 5(a), where both the hysteresis angle and the permeability $\mu$ vary according to the load conditions.

Fig. 4. Hysteresis loop and elliptical loop variation after synchronization ⁽¹⁾

In this paper, the permeability is also varied after the motor enters the synchronous speed. The variation is continued until the motor enters steady state. Fig. 5(b) shows the proposed elliptical approximation concept. When the motor speed arrives at synchronous speed, the motor adjusts the motor torque to balance the load inertia. At this time, the area of the hysteresis loop varies to balance this condition, which results in motor hunting. The loop area is proportional to the hysteresis torque, so the loop variation determines the motor behaviors.

Fig. 5. Proposed hysteresis loop and elliptical loop variation after synchronization.

4. Synchronization Time Calculation

In the case of a DTG, the synchronous entry time and stabilization time are important characteristics, and the required time to arrive at the steady state should have almost no hunting. However, a proper dynamometer to measure these characteristics is very rare because DTGs are usually very small. Devices such as stroboscopes are also difficult to use for reliable measurements because the measurements are usually rough, and the hunting effect of the rotor can not be measured. Of course, an encoder can not be installed. However, since a hysteresis motor is an electric motor in which the loop area is the output, the load angle changes from the synchronous entry time, and then the hysteresis reactance changes.

As the reactance changes, the current waveform and current phase both change. Unfortunately, the only values that can be measured for this motor are the external line-line voltage and phase current. However, since there is no need to accelerate from the moment of synchronization, the output power decreases along with the hunting phenomenon. This results in a decrease of the load angle after entering the synchronous speed.

Equations (4) and (5) include $\delta$, and $X_{h}$ and $R_{h}$ vary according to. This causes a phase difference $\theta_{d}$ between the line-ling voltage $V_{ab}$ and the phase current $I_{a}$. Therefore, it is possible to measure the synchronous entry time accurately by measuring the phase difference between the line-line voltage and the phase current of the motor. The phase difference can be measured by a memory oscilloscope, which can store the waveform data. The method is simple, but a large amount of memory may be required. Using a zero-crossing detector could be another solution to measure the data. Both methods were tried to confirm the proposed method to detect the synchronous entry time and steady-state entry time.

Table 1 shows the motor parameters, and the synchronous speed is 19,200rpm. Fig. 6 shows the measuring system, including a hysteresis motor and a 3-phase sinusoidal voltage driver. Fig. 7 shows the block diagram of the motor driver.

Fig. 6. Motor and driver system to measure line-line voltage and phase current.

Fig. 7. Block diagram of the motor driver.

5. Simulation and Experiments

A phase shifter and power op amps were used to apply 3-phase voltage of 15V at 640Hz to the hysteresis motor. A memory scope measures the applied voltage and phase current data for 2 seconds, as shown in the figure. From the data, the phase difference $\theta_{d}$ between $V_{ab}$ and $I_{a}$ can be detected. Fig. 8(a) shows the measured $\theta_{d}$ variation from the start. At about 1.5 seconds, the variation increases abruptly, which means the motor begins to enter synchronous speed. The oscillation is caused by motor hunting. The initial permeability is calculated using the inverse Preisach model ⁽⁸⁾ in the simulation. From the equivalent circuit in Fig. 3, $\theta_{d}$ can be calculated using DQ inverse transformation.

Fig. 8(b) shows the variation calculated by a previous method ⁽⁵⁾, which is somewhat different from Fig. 8(a). Fig. 8(c) shows the simulation results ⁽⁶⁾ obtained when the permeability $\mu$ is constant with the model in Fig. 3. Fig. 8(d) shows the $\theta_{d}$ variation obtained by the proposed method, which best matches the measured data in Fig. 8(a).

Fig. 8. Comparison of measured and simulated delay $\theta_{d}$.

Other characteristics were simulated by the proposed model and Nitao’s method. The comparisons of the motor speeds showing expanded hunting are shown in Fig. 9. The motor torques are shown in Fig. 10, and the load or hysteresis angle $\delta$ is shown in Fig. 11. The input and output powers are shown in Fig. 12.

Fig. 9. Comparison of speed.

Fig. 10. Comparison of torques.

Fig. 11. Comparison of $\delta$.

Fig. 12. Comparison of $P_{"\in "}$ and $P_{"out"}$.

6. Conclusion

In this study, a reasonable dynamic model for hysteresis motors was proposed and simulated. In addition, a method was proposed to accurately measure the synchronous entry time of a very-low-torque motor such as a DTG, and the feasibility was confirmed with an experiment. The eddy current resistance was considered and was located parallel to the hysteresis impedance. Two kinds of torques due to hysteresis and eddy current were calculated separately. The rotor impedance was recalculated while considering the variation of the rotor permeability and the load angle after entering synchronous speed. The load angle begins to oscillate at the synchronous entry time, which affects the motor current waveform. The synchronous entry time can be measured by measuring the variation of the phase difference between the line voltage and the phase current. This method is applicable to any synchronous motors when measuring the speed directly is difficult.