#### Abstract

There is an interest in super-high-speed motors in industry applications such as gyroscope, micro gas turbines, centrifuge, machine tool spindle drives, and information storage disk drives. This paper presents the dynamic performance characteristics of hysteresis motors using a Matlab/Simulink software. A nonlinear mathematical model based on a *d-q* axis theory in the rotor reference frame is applied to study the starting and synchronization processes of a hysteresis machine with a circumferential-flux-type rotor. The steady-state and transient responses of the motor to different changes such as the variation in the load torque are provided. The calculation method of the motor parameters in dynamic modeling based on a steady-state model of the motor is presented. The simulation results such as the current,the input power, and power factor are compared with some experimental results in steady-state condition.

#### 1. Introduction

The hysteresis motor is a well-known type of electrical machines. Basically, it is a synchronous motor. However, in spite of synchronous motors, it has a stable performance at asynchronous conditions similar to the induction motors. It is often used for small power applications that needs a very smooth torque at high speeds such as gas centrifuges and gyroscopes. Developing new semihard magnetic materials leads to a higher power range for Hysteresis motors up to 0.5 hp. hysteresis motors have a constant torque from starting to synchronous frequency and their starting current is at most 150% rated current. Nonetheless, the cost of hysteresis motors is high due to the special material of the rotor’s ring. Moreover, hysteresis motors have a low power factor less than 0.5 and efficiency as high as 75% that limits their applications [1].

The structure of hysteresis motors is similar to induction motor. Figure 1 shows a cross-section of a hysteresis motors. Its stator is considered to have a three-phase two poles with sinusoidal distributed winding. The rotor includes two parts: (i) a magnetic ring which is the basic element for torque providing and (ii) the inside part of the rotor that is made of a non-magnetic material that acts as a support for the magnetic ring. The starting of the motor is due to the hysteresis losses induced in the rotor. The starting current is at most 200% of its full load current that can pull into synchronism any load inertia coupled to its shaft.

In this paper, the dynamic performance of hysteresis motor is investigated. The steady-state model of hysteresis motors is explored in Section 2. Then a *d-q* model of the hysteresis motor is developed in a rotor reference frame. Finally, some simulation results under different conditions are presented and compared with some measured results.

#### 2. Steady-State Model of the a Hysteresis Motor

The electric equivalent circuit of a hysteresis motor for the steady-state synchronous operation is shown in Figure 2, including the stator impedances and voltage source . The effect of the rotor magnetic material is to produce a “hysteresis lag angle ” between the stator MMF and the resultant air gap flux density waveforms [1].

The values of air gap inductance , unsaturated incremental inductance and saturated incremental inductance in Figure 2 are calculated from: where, is the stator’s winding turns, is the axial length of stator or hysteresis ring, is the mean radius of air gap, is the mean radius of hysteresis ring on the rotor, is the thickness of hysteresis ring, is the effective width of air gap, is the free space permeability, is the unsaturated relative permeability, and is the relative permeability with as saturated relative permeability [1, 2].

The lag angle is independent of the frequency of rotor magnetization but it depends on the shape of the hysteresis loop. At synchronous speed, the fundamental eddy current torque is zero and the operation of the motor is accomplished exclusively by the hysteresis torque that is developed from hysteresis power of B-H loop. The hysteresis power can be represented as hysteresis resistance as function of lag angle. It can be approximated from the power loss approach as: where, is the number of phases, is the air gap voltage, and are the residual flux density and coercive force of the hysteretic material, is the supply frequency and is the volume of hysteretic material.

At any speed except to synchronous speed, the motor torque is due to both the hysteresis and eddy current effects as shown in Figure 3. The representation of the rotor eddy current is carried out by equivalent resistance as [2, 3]: where, is the specific resistivity, and is the cross sectional area of ring.

#### 3. Dynamic Modeling of a Hysteresis Motor

Figure 4 shows the dynamic model of a hysteresis motor in *qd0* reference frame rotating with rotor speed . The difference between the model of hysteresis and general synchronous motors comes from the modeling of rotor materials that for the hysteresis motor is so different. For this purpose, the steady-state model of the rotor in hysteresis motors from Figure 3 is employed.

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It is convenient to express the voltage and flux linkage equations in term of reactance rather than inductances [4, 5]. Hence, using the hysteresis motor voltage and linkage flux equations in the *qd0* rotating reference frame is represented by
where is the stator coil resistance and is the stator leakage reactance. The quantities and are the magnetizing and rotor leakage reactance that are derived from the following:
The reactance is the apparent reactance due to hysteresis ring and is approximated by
And the rotor apparent resistance is given by
Electromagnetic torque and the rotor speed are obtained from
where is the number of stator poles and is the shaft inertia moment. The definitions and corresponding values of the parameters for the employed hysteresis motor are listed in Table 1.

#### 4. Simulation Results

The developed model has been simulated for a 60000 rpm super-high-speed hysteresis motor. The corresponding parameters are summarized in Table 1. Figure 5 shows the implementation of the dynamic modeling of hysteresis motors in Matlab/Simulink. Simulations are carried out at no-load and under full-load conditions.

Figure 6 shows the simulation results at a rated speed and at no-load condition. RMS value of the stator voltage is 400 volt. And the voltage waveform and its *d-q* components are shown. Figure 7 shows the instance and RMS value of the stator current as well as *d-q* components. The steady-state value of *d-q* components depend on the B-H curve of the rotor material (B_{r} and H_{c}). The RMS value of the stator current in steady state is 0.5 A. The transient time in simulation is about 0.01 sec. Figure 8 shows the variations of the developed torque, rotor speed and input power. There is no load on the rotor and so, the developed torque is zero at the steady state. The input power is equal to the stator copper loss.

Figure 9 shows the variations of the power factor, input active power, input reactive power, and available power of the motor at no-load condition. The power factor is too low, that is common in hysteresis motors. The input power is wasted in the winding of stator.

Figure 10 shows the variations of the motor quantities. Developed torque at the steady state reaches to 1 [pu]. The RMS value of the current is 0.53 [A] that is close to no-load current at the same conditions. Power factor grows up to 0.25 that confirms the power factor of the hysteresis motor is far from the power factor of PM and IM motors. The input active power is 91 [W] that is the summation of output power (60 W) and stator copper loss (31 W).

#### 5. Experimental Results

In this section, some experimental results for a 60000 rpm hysteresis motor, under full-load condition at the steady state are presented. Due to some mechanical limitations and vibration at the starting process of the employed actual hysteresis motor, the starting process is carried out with a smooth ramp that it takes one hour time. So, because of the RAM limitation of the PC, the starting process has been simulated only for one second. Therefore, the simulated and measured results are compared just in steady state under full-load condition.

Figure 11 shows the current and voltage of the hysteresis motor which is fed from a high frequency PWM inverter. Moreover, one period of each waveform as well as corresponding fundamental harmonic are shown. An FFT analysis is carried out to identify the magnitude and frequency of the stator phase current and voltage. The magnitude of the stator current and voltage are close to the simulation results. To show the hunting phenomenon of hysteresis motor, the stator current and voltage are measured for one second in steady state. Figure 12 shows the instant RMS value of the current and voltage as well as the mean value of RMS during one second. The mean RMS value of the stator current and voltage are 0.55 A and 397 V, respectively, that confirms the simulation results. Moreover, variations of power factor and input power during one second are shown. The mean value of the power factor and power are 0.27 and 95 W, respectively, whereas the corresponding simulated results are 0.25 and 91 W. There are a good agreement between simulation and experimental results.

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#### 6. Conclusion

A dynamic model for a circumferential-flux hysteresis motor has been developed and implemented in Simulink. A developed model is based on *d-q* theory and is similar to the model of the synchronous motor. The main point in the proposed model is the modeling of B-H characteristics of the hysteretic material of rotor and including in the dynamic model. There exists a close agreement between simulated and measured results. It is shown that the power factor of the motor is so low rather than PM or IM motors and the stator current does not have much change at no-load and full-load conditions. proposed model can easily be used for fault diagnosis, closed-loop vector control, sensorless control, and parameters identification.