# Dynamic Modeling and Simulation of a Super-High-Speed Circumferential-Flux Hysteresis Motor

Authors: A. Halvaei Niasar; M. Zare; H. Moghbelli
Journal: Journal of Engineering
 (2013)

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## 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 ad-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.


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## Body

## 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. Figure1 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.Figure 1
A cross-section of a circumferential-flux hysteresis motor.In this paper, the dynamic performance of hysteresis motor is investigated. The steady-state model of hysteresis motors is explored in Section2. 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 Figure2, including the stator impedances and voltage source Ep. The effect of the rotor magnetic material is to produce a “hysteresis lag angle β” between the stator MMF and the resultant air gap flux density waveforms [1].Figure 2
A equivalent circuit of a hysteresis motor based on an idealized B-H curve of a hysteretic material  [1].The values of air gap inductanceLg, unsaturated incremental inductance Lo and saturated incremental inductance Lp in Figure 2 are calculated from:
(1)Lg=3π8Ns2μorglge,Lo=3π8Ns2μoμrotrlrh,Lp=3π8Ns2μoμptrlrh,
where, Ns is the stator’s winding turns, l is the axial length of stator or hysteresis ring, rg is the mean radius of air gap, rh is the mean radius of hysteresis ring on the rotor, tr is the thickness of hysteresis ring, ge is the effective width of air gap, μo is the free space permeability, μro is the unsaturated relative permeability, and μp(=μrs∥μro) is the relative permeability with μrsas saturated relative permeability [1, 2].The lag angleβ 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 Rh  as function of lag angle. It can be approximated from the power loss approach as:
(2)Rh=mEg24BrHcfVrsinβ,
where, m is the number of phases, Eg is the air gap voltage, Br and Hc are the residual flux density and coercive force of the hysteretic material, f is the supply frequency and Vr 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 Figure3. The representation of the rotor eddy current is carried out by equivalent resistance Re as [2, 3]:
(3)Re=12ρl104Ah,
where, ρ is the specific resistivity, and Ah is the cross sectional area of ring.Figure 3
A steady-state equivalent circuit of a hysteresis motor.


## 3. Dynamic Modeling of a Hysteresis Motor

Figure4 shows the dynamic model of a hysteresis motor in qd0 reference frame rotating with rotor speed ωr. 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.A hysteresis motor model in rotatingqd0 reference frame.
(a)
(b)It is convenient to express the voltage and flux linkage equations in term of reactance rather than inductances [4, 5]. Hence, using ψ=ωbλ the hysteresis motor voltage and linkage flux equations in the qd0  rotating reference frame is represented by
(4)vqs=rsiqs+ωrωbψds+1ωbdψqsdt,vds=rsids-ωrωbψqs+1ωbdψdsdt,vqr′=0=rr′iqr′+dψqr′dt,vdr′=0=rr′idr′+dψdr′dt,ψqs=(xls+xm)iqs+xmiqr′,ψds=(xls+xm)ids+xmidr′,ψqr′=xmiqs+(xlr′+xm)iqr′,ψdr′=xmids+(xls+xm)idr′,
where rs is the stator coil resistance and xls is the stator leakage reactance. The quantities xm and xlr′ are the magnetizing and rotor leakage reactance that are derived from the following:
(5)xm=(ωLg)∥(ωLo),xlr′=ωLp+ωLh.
The reactance ωLh is the apparent reactance due to hysteresis ring and is approximated by
(6)ωLh=Rhtanβ,
And the rotor apparent resistance is given by
(7)rr′=Rh∥Res
Electromagnetic torque and the rotor speed are obtained from
(8)Tem=32P2ωb(ψdsiqs-ψqsids),Tem-Tmech=2JPdωr(t)dt,
where P is the number of stator poles and J is the shaft inertia moment. The definitions and corresponding values of the parameters for the employed hysteresis motor are listed in Table 1.Table 1
Hysteresis motor parameters.
Number of poles
p
2
Rated output power
P
60 W
Rated voltage
V
s
400 V
Rated frequency
f
s
1000 Hz
Stator leakage reactance
X
l
s
152 Ω/ph
Stator resistance
r
s
36 Ω/ph
Equivalent resistance due to eddy current
R
e
3288 Ω/ph
Equivalent resistance due to hysteresis ring
R
h
127 Ω/ph
Equivalent reactance due to hysteresis ring
X
h
163.7 Ω/ph
Unsaturated incremental reactance
X
o
451 Ω/ph
Saturated incremental reactance
X
p
13 Ω/ph
Air gap equivalent reactance
X
g
1217 Ω/ph


## 4. Simulation Results

The developed model has been simulated for a 60000 rpm super-high-speed hysteresis motor. The corresponding parameters are summarized in Table1. 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 5
A dynamic modelling block diagram of a hysteresis motor in Simulink.Figure6 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 (Br and Hc). 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 6
Stator voltage waveform at 60000 rpm at no-load condition.Figure 7
Stator current waveform at 60000 rpm at no-load condition.Figure 8
Torque/speed/input power at 60000 rpm at no-load condition.Figure9 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 9
Power factor/Pin/Qin/Sin at 60000 rpm at no-load condition.Figure10 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).Figure 10
T
e/Irms/Pf/Pin/Qin/Sin at 60000 rpm under full-load condition.


## 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.Figure11 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.Figure 11
Experimental results and the FFT analysis of the stator current/voltage waveforms at 60000 rpm under full-load condition.Experimental results of the stator current/voltage RMS value, power factor, and input power during 1 second.
(a)
(b)
(c)
(d)


## 6. Conclusion

A dynamic model for a circumferential-flux hysteresis motor has been developed and implemented in Simulink. A developed model is based ond-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.



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