1. Introduction
Energy problems and environmental issues in the 21st century have become one of the world’s major concerns, and switched reluctance motors (SRMs) have received widespread attention from scholars around the world by virtue of their simple structure, high fault tolerance, wide speed range, and reliable operation [
1,
2]. With the development of modern power electronics and vehicle drive technology, SRMs are particularly suitable where the reliability of drive systems is required, such as in electric vehicles, aerospace power systems, and wind power generation [
3,
4,
5]. However, the special structure of SRMs with biconvex poles and the nonlinear current excitation method cause large vibration and noise, so the vibration constraint of SRMs is more demanding, which makes the widespread use of SRMs a great challenge [
6,
7,
8].
The world’s methods for solving SRM radial force analysis and vibration problems are roughly divided into two types: drive control changes and motor mechanical structure optimization [
9,
10,
11]. From the perspective of motor drive control analysis, some researchers have proposed a two-pole commutation active vibration-cancellation control using negative voltage pulse width modulation (PWM).This method can not only effectively reduce the vibration occurring on the stator during commutation but also keep the torque ripple and copper loss of the motor within a reasonable range [
12,
13].The optimization of the voltage fluctuations on the PWM control method does not trigger natural frequency vibrations and has a better damping effect at low- and medium-speed conditions [
14,
15]. In recent years, the literature [
16,
17] has proposed the introduction of direct force control and reference current adapters based on torque control, and this control strategy is effective in reducing vibrations under steady and transient conditions. This method slightly modifies the torque ripple-minimizing reference current, and torque fluctuations remain almost constant, but the accuracy of radial force determination is a major obstacle to the implementation of this technique [
18].
To reduce the vibration and noise of SRMs, prediction of vibration and noise is indispensable in the process of motor design [
19,
20]. Since the SRM works differently from other motors, its electromagnetic force variation pattern with time is closely related to the body structure. Although the direct measurement of SRM vibration characteristics is simple, the disadvantages are the low accuracy of the results and the high cost of construction [
21,
22]. In recent years, the establishment of simple and accurate analytical models of electromagnetic forces has become a hot research topic, and the calculation of electromagnetic forces by analytical methods can provide help for vibration prediction. To improve the accuracy of computational simulation, the literature [
23] proposed a new analytical method of simulation through 3D data in this paper, i.e., considering the excitation of high axial order vibration modes and simulating the vibration law in the 3D domain. Considering that the analysis of the SRM vibration process involves a variety of physical fields, the electromagnetic forces of the stator teeth are calculated based on finite element analysis in the three-dimensional electromagnetic field, and the modal analysis and harmonic response analysis are performed through the laws in the frequency domain to estimate the intrinsic frequency and vibration of the whole motor [
24]. In the literature [
25,
26], the analysis of the SRM multiphysics field and the mutual coupling SRM were carried out for comparison, and a good prediction of the vibration model was calculated by the mode superposition method considering electromagnetic characteristics, mechanical vibrations, and acoustic noise. However, the inhomogeneous electromagnetic forces on the stator teeth and the orthogonal anisotropy of the motor material structure can introduce large errors in the prediction of the vibration and motor modes.
The DDSRG studied in this paper is a generator with a deflectable structure and higher power-generation efficiency [
27,
28]. In the
Section 1, a mechanical model of the DDSRG is developed to illustrate the advantages of the dual-stator structure compared to the common motor structure. In
Section 2, a mathematical model of the electromagnetic field of the generator is developed, and the time and space distribution characteristics of the inhomogeneous electromagnetic force are analytically derived and decomposed for verification. In the
Section 3, the DDSRG electromagnetic model is analyzed based on finite element software, and the electromagnetic and mechanical characteristics of the generator’s internal and external circuits are analyzed in detail. Then the vibration characteristics are predicted by comparing the analysis of the modal characteristics of the motor stator with the vibration process under electromagnetic force loading.
Figure 1 shows the framework of electromagnetic-vibration analysis in this paper. The overall electromagnetic-vibration characteristics of the motor are a combination of the electromagnetic field and modal analysis, and the finite element method is used to analyze the electromagnetic force distribution and the stator–rotor vibration distribution. The overall analysis process includes electromagnetic force finite element analysis, structural transfer function construction, and stator–rotor vibration coupling analysis. The advantage of this analysis method is that it improves the accuracy of the electromagnetic vibration analysis while ensuring the accuracy of the model mesh and parameters, and it is also applicable to the vibration evaluation of other motors. Finally, the experimental platform of DDSRG is built, and the vibration characteristics are tested to verify the validity and accuracy of the proposed simulation results.
2. Design and Principle of DDSRG
As a new type of wind turbine, the switched reluctance generator has the advantages of simple structure, low starting wind speed, high power-generation efficiency, easy rectifier storage, and good fault tolerance. Due to the special double-stator structure of the DDSRG, which combines the compactness and fast system response of conventional generators with alternating inner and outer stator operation, the generator power density and power-generation efficiency can be increased. The DDSRG has the advantage of multi-degree-of-freedom operation at constant-rated generator speed, making full use of wind energy for power generation, increasing efficiency and reducing power-generation costs. The specific mechanical parameters of DDSRG are shown in
Table 1.
Figure 2 is the overall structure distribution map of DDSRG, which is composed of motor shell, stator and rotor, and shaft and base. The advantage of DDSRG is that it has an internal and external double-stator structure and can realize the deflection operation of the motor through the double-stator structure. When the wind direction changes, it can still maintain a good power-generation efficiency. Because the dual stator of DDSRG is composed of internal and external independent loops, magnetic field interference of the internal and external parts of the rotor is prevented by adding an excellent magnetic insulation plate in the middle of the rotor. The bearing connects the rotor through the shaft to realize the motor deflection, reduce the mechanical loss caused by the friction of the motor, and improve the motor efficiency. The shape of internal and external stator teeth is concave spherical, and the shape of rotor tooth pole is convex spherical. It is thanks to the spherical structure design that DDSRG can perform deflection motion.
Table 1.
Mechanical parameters of DDSRG.
Table 1.
Mechanical parameters of DDSRG.
Parameter | Value |
---|
Rated speed | 200 r/min |
Rated power | 2.5 kW |
Rated torque | 70 N*m |
Rated voltage | 380 V |
External diameter of external stator | 300 mm |
Inner diameter of inner stator | 13.5 mm |
Rotor outer diameter | 108 mm |
Rotor inner diameter | 43.3 mm |
Air-gap interval | 2 mm |
Rotor pole | 8 |
Stator pole | 12 |
Rotor deflection angle | 0~17° |
Thickness of magnetic insulation board | 15 mm |
Figure 3 shows the DDSRG deflection diagram. According to the characteristics of the double stators of the motor, the deflection angle α is the angle between the central shaft of the rotor and the central shaft of the whole generator, and the rotor can be deflected in a certain space. The windings of DDSRG internal and external stator and rotor systems are connected to an independent external circuit, respectively, and the motor efficiency is improved by controlling the deflection state of the motor. The power converter is shown in
Figure 4. The rectifier and inverter can be realized by using the dual PWM converter. The generator side is the grid side converter, and the machine side converter is near the generator side. If the converter on one side is disturbed, it will not affect the normal operation of the other side and can better adapt to the fault of the power grid.
3. Analytical Calculation of Radial Electromagnetic Force and Magnetic Field
The doubly salient structure of SRM and the switching characteristics of winding current lead to the obvious nonlinear characteristics of electromagnetic force during operation. The vibration of DDSRG in the deflection state is more complex, and the research on the analytical model of electromagnetic torque and electromagnetic force of motor is the premise to solve the vibration problem of stator and rotor. Since DDSRG has two independent loops for driving and deflection, the magnetic circuit is asymmetric, so the process of magnetic circuit analysis will be more difficult. Different from the vibration process analysis of ordinary single-degree-of-freedom motor, the vibration analysis of DDSRG needs to be extended to the calculation of spatial electromagnetic force distribution so as to accurately obtain the vibration characteristics of the motor under different operating conditions. Therefore, this paper calculates and deduces the vibration of DDSRG in time and space.
Because the power-generation characteristics of internal and external circuits are similar, this paper takes the motor external stator and rotor circuit as an example. Assuming that the permeability of the stator rotor magnet is not saturated,
Figure 5 is the
q-term magnetomotive force distribution map of DDSRG,
Nr is the number of rotor poles of the motor,
Rr is the outer diameter of the outer rotor,
Dr is the width of the single-tooth pole of the outer rotor, and
Nf is the number of turns of the stator single-tooth pole winding.
The
q-phase permeability of DDSRG outer rotor can be expressed as the product of winding magnetomotive force and winding current. The magnetomotive force
Bq of the
q-phase tooth pole can be expressed as [
29]:
where
If is the excitation current amplitude of the stator phase
q winding.
To analyze the spatial order of the magnetokinetic potential, the Fourier series expansion of
Bq can be expressed as:
where
Q is the number of DDSRG phases,
Ns is the number of external stator poles, and the magnetodynamic potential coefficient
Bv is taken as:
Figure 6 shows the time distribution of the current in phase
q. The expression of
Iq can be obtained from the distribution.
where
θc is the DDSRG conduction angle, and
ω is the angular velocity of the motor. The Fourier expansion of the phase
q current
iq with time is:
where the value of
Im is taken as:
Bringing Equations (3) and (5) into Equation (2), the Fourier series expansion of the magnetic potential
Bq(θ,
t) of the phase
q winding can be obtained by deducing:
where
Kvm is the magnetic momentum coefficient associated with
v and
m. Therefore, the value of the magnetic momentum coefficient
Kvm is:
Bringing
n = vNt into Equation (8),
Knm can rewrite Equation (7) as follows [
30]:
Therefore, the DDSRG magnetomotive force coefficient
Knm varies with
n,
m. According to the operating principle of SRM, the different stator winding magnetomotive forces can be expressed as:
Because of the special characteristics of the convex pole structure of the SRM stator and rotor, the saturation characteristics of the motor magnetic circuit need to be considered. At higher currents, the linearity of the magnetic permeability decreases as the rotor rotation angle becomes larger. Due to the phenomenon of magnetic circuit saturation, the magnetic chain appears nonlinear when the current grows, so an accurate solution of the motor permeability is very important. Since the DDSRG stator is not slotted, and the air-gap distance between the stator teeth is large, the permeability between the stator teeth can be neglected. Therefore, the Fourier expansion of the air-gap permeability function Λ
r(θ,
t) is:
where Λ
rz is the coefficient of magnetic permeability as:
where
μ0 is the vacuum permeability, and
go is the outer fixed-rotor radial air-gap distance. Therefore, it can be found that the rotor permeability Λr
(θ,
t) is related to the rotor position angle
θ, and the rotor pole arc coefficient will have an effect on the permeability.
According to the electromagnetic field principle, the radial flux density of phase
q winding is
Brq(θ,
t):
Combining the Maxwell stress equation, it can be seen that the combined force and momentum within a given volume and magnetic mass are equivalent to the sum force of the stator surface tension. By further analysis, it can be obtained that the tangential flux density is much smaller than the radial flux density, so the radial electromagnetic force
Fr can be expressed as:
where
Btq is the tangential flux density; the analytical formula of
Fr(θ,t) can be obtained by combining Equations (11) and (12).
Based on the principle of minimum reluctance, the tangential magnetic density decreases continuously and the radial magnetic density increases continuously during the process of turning from the maximum reluctance position to the minimum reluctance position under the influence of electromagnetic force. Combining with Equation (13), it can be seen that the ERF is increasing with the rotor position during the rotation from the maximum reluctance position to the minimum reluctance position. In this paper, the flux density and the ERF of the inner stator circuit of DDSRG are studied in the same way as that of the outer stator rotor, and the calculation process of the outer stator rotor can be referenced.
5. Experimental Verification
After the above analysis, as shown in
Figure 20, the DDSRG experimental platform is established to verify the generator operating characteristics and vibration law. The experimental platform includes a DDSRG test prototype, a controller, a servo motor, a switching power supply, an oscilloscope, and a vibration meter. A servo motor is used to drive the DDSRG to generate electricity, and a light bulb is used to replace the load and determine the power-generation status by the brightness. A DC power supply is used to provide excitation for the generator, and the power is generated by another excitation method to ensure the stability of the power-generation system. Real-time data on the electromagnetic characteristics of the DDSRG can be measured by an oscilloscope during the power-generation process. The vibration response of the motor at different positions is measured and collected using the probe of the vibrometer TIME7231, and the measurement results are stored by spectrum and curve.
Figure 21a shows the deflection dial of the DDSRG, with the output shaft supported by a fixed guide for deflection motion, and the dial can visually show the deflection angle.
Figure 21b shows the rotor structure of the DDSRG. The rotor material uses a DW465-50 silicon steel sheet with a lamination factor of 0.95, and the stator tooth surface has a special spherical structure to provide support for multi-degree-of-freedom operation.
Figure 22 shows the voltage waveforms of the DDSRG at different rotational speeds. By observation, it is obtained that the voltage fluctuation is larger at lower rotational speeds and gradually decreases as the rotational speed gradually increases and gradually smooths out at rated speed. Because of the convex structure of the generator rotor teeth, the flux change in the generator electromagnetic circuit is amplified at low speed, but when the rated speed is increased, the voltage amplitude and fluctuation are gradually stabilized, which verifies the stability and feasibility of the DDSRG operation. The vibration characteristics of the stator and motor casing are observed in
Figure 23 by measuring the stator and motor casing at rated speed with a vibrometer, and it is found that the casing has a restraining effect on the stator vibration.
6. Conclusions
In this paper, the time and frequency domains of unbalanced electromagnetic forces of a DDSRG with a rated power of 2.5 kW are analytically calculated, and the vibration characteristics of the generator are simulated and analyzed for both self-rotating operation and deflecting operation and finally verified and compared by finite element analysis and experimental results. DDSRG allows for a multi-degree of freedom operation, and the excitation current action produces particularly significant ERF pulsations during generator operation. The calculation of the electromagnetic field of the DDSRG at different deflection angles is carried out using finite element software, and the distribution law of the ERF in time and space is analyzed to verify that the main amplitude order of the generator ERF acting on the stator is 8 f0N(N = 0, 1, 2…). The modal analysis of different stator structures of switched reluctance motors with corresponding materials shows that the vibration order is more intense in the order 2–7. By combining the coupling analysis of electromagnetic-vibration, the vibration response of DDSRG was calculated at the benchmark of rated speed of 200 rpm. The results of harmonic response spectrum analysis corresponded to the modal analysis, and the electromagnetic force wave near the intrinsic frequency had a greater influence on the generator vibration. Finally, by establishing the DDSRG vibration measurement experimental platform, the power-generation characteristics and vibration response were tested, and the experimental results were basically consistent with the finite element analysis results.