Magnetic Vibration in Induction Motor Caused by Supply Voltage Distortion

Abstract

This article discusses magnetic vibrations in squirrel-cage induction motor stators and provides a mathematical description of the process of their excitation. A model of a 30 kW squirrel-cage induction motor was developed in finite element software. This model considers the motor geometry, material properties and stator winding. The electromagnetic and mechanical processes in the motor during the rotation of the rotor were considered. In the course of this study, currents of various harmonic compositions and amplitudes were applied to the motor windings, which caused magnetic noise, vibration and pulsations of the electromagnetic torque. Magnetic noises, vibrations and pulsations of the electromagnetic torque were investigated in the case of imbalance and harmonic distortions of the supply voltage.

1. Introduction

Magnetic vibration is a problem for almost all medium- and high-power electric motors [1,2,3,4,5]. It causes adverse events: a change in the harmonic composition of the supply current [6,7], bearing wear [8], acoustic noise [9,10], an increase in power consumption [11,12] and the destruction of electric motors [13,14]. This problem occurs in the electric drive [15,16] and generators, including wind turbines [17,18].

The most strongly magnetic vibration is manifested in asynchronous squirrel-cage induction motors (SCIM), characterized by a rotating magnetic field relative to the stationary stator of the machine. Waves of radial magnetic force cause periodic deformations of the stators of motors, which manifests primarily in the form of acoustic noise.

Much research from has been devoted to the study of magnetic vibration and the development of methods to suppress it. In the 1970–1980s, research provided a fundamental base for the mathematical description of magnetic vibrations of electrical machines and the structural methods for their reduction [19,20]. In the 1990–2000s, researched methods focused on actively reducing the magnetic vibration occurring in the SCIM stator. These methods are based on adding a compensation harmonic to the supply voltage [4,21,22,23]. Completely different approaches were offered by the methods proposed in the works of the last decade [1,24,25]. In these works, the control of magnetic vibration is achieved by a given change in angles of voltages in the vector control system of an SCIM-based electric drive.

Almost all known methods of reducing magnetic vibrations in SCIM are based on the same theoretical base, within which magnetic fields, radial magnetic forces (Maxwell forces) and stator deformations are waves of a certain frequency, spatial order (mode number) and amplitude. At the same time, the reduction in magnetic vibration is achieved through effective control of the parameters of the motor supply voltage, for which it is necessary to know the relationship of these physical quantities. The task of analytical representation of this relationship is complicated by numerous irregularities and assumptions that cannot be taken into account due to the complex and time-varying geometry of an investigated motor. Therefore, it is advisable to carry out interconnected mechanical, acoustic, and electromagnetic calculations of processes occurring in a motor under operation with finite-element multiphysics design software.

The study is applicable to alternating current motors of medium and high power, from 15 to 150 kW, and with a speed of 1500–3000 rpm. These machines do not have a bevel groove in the rotor, and the air gap reaches 0.5–0.8 mm. Magnetic vibrations are much weaker in low-power motors due to the bevel groove and lower magnetic flux density. Bearing vibration and rotor eccentricity exceed magnetic vibration in such motors. In high-power alternating-current motors, magnetic vibrations are also a serious problem. They are a powerful source of noise. In any alternating current motors, the magnetic vibration is 5–10 times greater than in direct-current machines.

This article is a logical continuation of the work [26,27] devoted to the development of an FEM model of the AIR180M4U3 motor and the study of electromagnetic and mechano-acoustic processes. The method of active reduction of magnetic noise is described in [26]. The method is based on the injection of an additional compensation current of the electric motor to reduce the harmonics of the magnetic field. The developed method reduces the radial magnetic force by 20%, the harmonics of the magnetic force by 70%, and the amplitude of the stator displacement oscillations by 30%. The article [27] describes magnetic forces, mechanical stress fields, sound pressure and stator oscillation modes at sinusoidal voltage.

The purpose of this article is to study the mechano-acoustic processes in this motor when the shape of the supply voltage is distorted due to voltage imbalance and high harmonic distortion.

2. Magnetic Vibration Excitation in SCIM Stator

Waves of radial magnetic forces in defect-free SCIMs, which are the main sources of their vibration [20,26], are caused by the action of the magnetic field B in the air gap on the stator steel [20]:

$$F_r(x,t)=\frac{B^2(x,t)}{2·{\mathsf{\mu }}_0},$$

(1)

where μ₀—vacuum permeability, t—time, and x—angular coordinate of air gap (rad.).

One of the ways to reduce the vibrations of SCIM is to control the spectral composition of the magnetic field in the air gap [28]. The most common method involves active reduction of SCIM vibrations based on stator voltage control. In this case, an additional compensation harmonic or several harmonics generated by PWM are added to the fundamental supply voltage component. The task of the compensation harmonic is to suppress one of the harmonics of the magnetic field generated by the supply voltage, which causes the most intense vibrations of the SCIM stator. A detailed description of the essence and implementation of this method is described in [21,22].

The radial magnetic force generated by the magnetic field in the air gap of the SCIM is powered by the PWM inverter:

$$F_r(x,t)=\frac{1}{2{\mathsf{\mu }}_0}{\left(B_1(x,t)+B^i(x,t)+{\displaystyle \sum _{g=2}^{\infty }{B}_g(x,t)}\right)}^2,$$

(2)

where g—magnetic field harmonic number, B₁(x,t)—first magnetic field harmonic, Bⁱ(x,t)—ith magnetic field harmonic caused by injected current, and B_g(x,t)—higher-order harmonic of the magnetic field due to stator and rotor slots, air gap unevenness, etc.

The injected current should have small amplitude values (no more than 15% of the supply current amplitude for the 3rd harmonic and even less for the others [21]). In this case, the amplitudes of the radial magnetic forces, determined by the values of the magnetic field in the air gap with the values (B_mⁱ)²/4μ₀ and B_mⁱ · ΣB_mg/2·μ₀, will be negligible (the “m” index means the peak value). THD_F in this case will change by no more than 19.3% (in the case of injection of 3, 5, 7, 11 and 13 harmonics with amplitudes of 0.15%, 0.9%, 0.7% 0.4% and 0.3% of the amplitude of the main harmonic of the supply voltage, respectively). Only two harmonics of the radial magnetic force generated by the interaction of the first magnetic field harmonic and the compensatory harmonic have a significant effect. These harmonics are described by the following equation:

$$F_r^i(x,t)=\frac{B_m^i·B_{m1}}{2{\mathsf{\mu }}_0}·(\cos \left[2·Z_p·x-({\mathsf{\omega }}^i+{\mathsf{\omega }}_1)·t-({\mathsf{\phi }}^i+{\mathsf{\phi }}_1)\right]+\cos \left[-({\mathsf{\omega }}^i-{\mathsf{\omega }}_1)·t-({\mathsf{\phi }}^i-{\mathsf{\phi }}_1)\right]),$$

(3)

where B_m1—first magnetic flux density harmonic amplitude, B_mg—gth magnetic flux density harmonic amplitude, $B_m^i$—compensating harmonic amplitude, Z_p—pair of poles number, ω₁—magnetic field angular frequency, φ₁—phase shift of first magnetic field harmonic, ωⁱ—magnetic field angular frequency of injected harmonic, and φⁱ—phase shift of injected harmonic.

These harmonics of radial magnetic force have a mode number of 2·Z_p and 0.

It is necessary to know the basic parameters (angular frequency, amplitude and phase shift) in order to determine the most pronounced harmonics of the magnetic force:

$$F_r(x,t)=F_m·\cos (r·x-\mathsf{\omega }·t-\mathsf{\phi }),$$

(4)

where F_m—radial magnetic force amplitude, r—mode number, ω—angular frequency of magnetic force (ω ≈ 2ω₁), and φ—phase shift of magnetic force.

The harmonic with the mode number r = 0 according to [20] remains and will cause vibrations; however, its amplitude will be much smaller compared to the force Equation (4).

The mode number r = 0 causes uniform radial deformation of the stator along the entire air gap, radial magnetic forces of the second mode number (r = 2) and higher cause bending deformations of the stator, as shown in Figure 1 [20].

3. Calculation of Magnetic Vibration Occurring in SCIM Stator

The radial magnetic forces of one harmonic composition cause vibrations in the SCIM stator of another harmonic composition. This is due to the fact that the SCIM stator is characterized by its own frequency response with one or several eigenfrequencies. Consequently, due to the resonant amplification, the spectral composition of the radial magnetic forces will be strikingly different from the harmonic composition of the stator vibrations.

With a mode number of r = 2 and above, the stator undergoes deformations of a complex spatial shape of an r-gon with an eigen angular frequency equal to [20].

$${\mathsf{\omega }}_r=\frac{r(r^2-1)}{\sqrt{r^2+1}}\sqrt{\frac{E·h^3}{12·m·R_{st}^4}},$$

(5)

where E—the elastic modulus of the stator, h is the height of the stator back, R_st—the stator radius, and m—the reduced mass of the stator yoke determined by

$$m=\frac{m_{st}}{2\mathsf{\pi }{R}_{st}{l}_{st}},$$

(6)

where m_st—the mass of the stator core and l_st—the length of the stator.

Analysis of Equation (5) allows us to conclude that vibrations with the mode number of r = 2 are characterized by the highest intensity, since their frequency is lower than that of other ones [20]. At the same time, vibrations also have a significant effect on the stator magnetic noise in the case of the SCIM power supply from the PWM inverter. The mode number of these vibrations is r = 2·Z_p = 4.

The mechanical impedance z_ω of the SCIM stator at the angular frequency ω of the radial magnetic force is determined as follows:

$$z_{\mathsf{\omega }}=\mathsf{\omega }m-\frac{1}{\mathsf{\omega }{\mathsf{\lambda }}_{st}}=\mathsf{\omega }m-\frac{Eh}{R_{st}^2\mathsf{\omega }},$$

(7)

where λ_st—mechanical flexibility.

Vibrations arise due to the interaction of the first and higher harmonics of the magnetic field in the air gap of the SCIM. Waves of radial magnetic forces can be expressed on condition as follows (on condition ω₀→ω₁):

$$F_r(x,t)={\displaystyle \sum _{g=1}^{\infty }{F}_{rgm}\cos \left((Z_p\pm Z_p)x-2(1\pm g){\mathsf{\omega }}_{1}t-({\mathsf{\phi }}_{12}\pm {\mathsf{\phi }}_{12g})\right),}$$

(8)

where F_rgm—amplitude value of magnetic force harmonic:

$$F_{rgm}=\frac{B_{\mathsf{\delta }}^{2}R{I}_{12g}}{2{\mathsf{\mu }}_0{R}_{st}{I}_{12}},$$

(9)

where B_δ—root mean square amplitude of magnetic flux density in the air gap, R—inner radius of the stator, I_12g and φ_12g—amplitude and phase shift of the gth harmonic of magnetizing current, I₁₂ and φ₁₂—amplitude and phase shift of the first harmonic of magnetizing current.

4. Multiphysics FEM model of SCIM

The theory presented in the previous section allows us to accurately describe the processes of magnetic vibration excitation in the SCIM stators. However, it contains a large number of assumptions and restrictions regarding the shape of the air gap, changes in the geometry of the engine due to rotation of the rotor and deformation of the stator, and the shape and harmonic composition of the supply voltage on the stator windings. Accounting for each of these factors leads to a significant complication of mathematical expressions describing the processes of magnetic vibration excitation and often requires the introduction of additional restrictions, for example, taking into account only two or three harmonics of the supply current when describing magnetic induction waves or taking into account stator and rotor slots [20]. If magnetic induction waves are described using the harmonic conductivities method, then an assumption appears in the equation that the width of the air gap is constant in time when calculating the magnetic conductivity function [24].

In [26], a model of SCIM of an investigated motor with a power of 30 kW and synchronous rotation speed of 1500 rpm, developed in the multiphysics FEM design software, was presented. The motor under investigation was classified as medium power (20–250 kW). Therefore, the waves of radial magnetic force acting on the stator have a mode number of r = 4. Some geometric and technical characteristics of the motor are given in

Table 1. Type 30 kW induction motor parameters.

Parameter	Value	Unit
Rated voltage	380	V
Rated current	57	A
Rated power	30	kW
Frequency	50	Hz
Number of pole pairs	2	–
Stator core outer diameter	0.313	m
Stator core inner diameter	0.211	m
Length of stator and rotor cores	0.185	m
Air gap width	0.0006	m
Stator back height	0.01	m
Stator slot height	0.041	m
Rotor slot height	0.025	m
Stator slot width (inner/outer)	3.2/9	mm
Rotor slot width (inner/outer)	11/8	mm
Number of stator slots	48	–
Number of rotor slots	36	–
Number of conductors in one winding	17	–
Motor efficiency	91.5	%
Power factor	0.87	-
Nomber of phases	3	-
Winding connection	star	-

.

The number of rotor slots was reduced from 39 to 36 to simplify the model and increase the convergence of calculations. The stator winding in each slot is represented by a separate homogeneous multiturn coil. The other end of this coil closes outside the stator. The rotor winding is represented by a solid copper conductor, which is a squirrel cage. The geometry of the SCIM model of the investigated motor and the mesh are shown in Figure 2a,b.

The mesh has the “swept” distribution type at the outer boundary of the model and in the air gap region (Figure 2a). This is necessary for calculations at the boundary between the moving and stationary parts of the model, which describes the processes in the rotor and stator. A special grid was applied manually to the side boundaries of the model sector in order to ensure its symmetry. Thus, continuity conditions act on the side boundaries, which simulate a “full rendering” of the motor geometry. The features of the simulation model of investigated motor are described in more detail in [26].

5. Modeling of Mechano-Acoustic Processes in the Motor Powered by Voltage with Harmonic Distortion

As noted earlier, the main source of SCIM vibration is waves of radial magnetic forces caused by the interaction of the first harmonic of the magnetic field with harmonics of a higher order. Obviously, the vibrations are caused harmonics, which are characterized by a large amplitude and low frequency. The harmonics g = 3, 5, 7, 11, 13 caused vibrations with frequencies f = 400, 600, 800, 1200, 1400 Hz according to Equation (8). These harmonics can be caused by the operation of the inverter and other consumers connected to the network.

It was found in [26] that in the range of 0–2500 Hz, the investigated motor has four modes of natural frequencies: f₁ = 165 Hz; f₂ = 571 Hz; f₃ = 1556 Hz; f₄ = 1764 Hz. According to Equation (8), the first two oscillation modes can be enhanced by the interaction of the first harmonic of the magnetic field with the harmonic g₃ = 3 at r = 0 and with the harmonic g₅ = 5 at r = 4.

Using the simulation model, studies of the variables characterizing the magnetic noise and vibration of the SCIM stator and torque were carried out. The results of simulation of mechanical stresses in the SCIM stator are shown in Figure 3.

An analysis of the mechanical stresses of the stator showed that the amplitude of mechanical stresses in the stator steel decreases when added with high harmonics. Figure 3d shows a graph of the mechanical stresses of the stator when the shape of the supply voltage is distorted by both higher harmonics and amplitude imbalance (the effect of the phase shift of the supply voltage wave was not considered). A similar situation is observed when considering the graphs of acoustic pressure radiated by the motor stator, as shown in Figure 4. The load torque is 10 Nm (5% of the nominal 195 Nm). The coefficients of the negative and positive sequence components k_0u = k_2u = 0% for case I_g3 = I_g5 = 0 A; k_0u = 7%, k_2u = 8% for case I_g3 = 8.8 A, I_g5 = 3.3 A and k_0u = 23%, k_2u = 18% for case I_g3 = 15.4 A, I_g5 = 7.7 A.

According to Figure 4, the acoustic pressure p radiated by the SCIM reaches an amplitude of 2 Pa, which corresponds to a noise level L_p = 100 dB (relative to the reference value of 20 μPa). The expression [29,30] was used in the following calculation:

$$L_p=20\lg \frac{p}{20\times {10}^{-6}}$$

(10)

At the same time, at the stage of frequency analysis, an increase in mechanical deformations caused by the influence of harmonics g₃ and g₅ was revealed. These graphs are shown in Figure 5.

According to Figure 5, the vibration displacement of the stator with the square form of the supply increased two-fold at frequencies f₁ = 165 Hz and f₃ = 1556 Hz and 1.5-fold at f₄ = 1764 Hz. Presumably, the sum of the interacted first harmonic of the supply current with the harmonic g = 3 has a lower peak value than the first harmonic itself, which contributed to the reduction in mechanical stress. According to Equation (1), in this case, significantly lower values of radial magnetic forces arise, which act for longer periods and cause greater stator deformations.

Figure 6 shows the time-dependance diagrams of the electromagnetic torque of the investigated motor.

Figure 6 shows curves of the electromagnetic torque for the investigated motor at no load. At the same time, a slight increase in torque ripples can be observed in the case of harmonic distortions of the supply voltage: the range of peak ripple values at I_g3 = I_g5 = 0 A was M_em = –30 ÷ 9.5 N·m (Figure 6, a), at I_g3 = 8.8 A, I_g5 = 3.3 A: M_em = –34 ÷ 15.5 N·m (Figure 6, b), at I_g3 = 15.4 A, I_g5 = 7.7 A: M_em = –31.5 ÷ 14.5 N·m (Figure 6c). This mode can be explained by the interaction of tooth pulsations of the magnetic field with pulsations caused by harmonic distortions.

6. Modeling of Mechano-Acoustic Processes in the Motor Powered by Voltage with Amplitude Unbalance

The harmonic composition of the magnetic field waves can be changed in the motor air gap. This may be caused by the supply voltage imbalance. Voltage imbalance studied at the three phase voltages differs only in amplitude. The phase relationship is normal–2π/3. The results of the simulation of stator mechanical stresses in the case of voltage imbalance are shown in Figure 7.

Unlike in the case of harmonic distortions of supply voltages, their amplitude imbalance practically does not affect the value of mechanical stresses caused by the action of radial magnetic forces on the steel core of the stator. A similar situation is observed when analyzing the acoustic pressure graphs shown in Figure 8. The sound pressure at a current asymmetry of ±40% (voltage asymmetry of ±5–8%) is reduced by no more than 20%.

Figure 9 shows the frequency responses of the stator vibration displacements caused by the action of radial magnetic forces in the case of amplitude imbalance of the supply voltage.

The graphs (Figure 9) show multiple decreases in the amplitude of vibration displacement harmonics with frequencies f₃ = 1556 Hz and f₄ = 1764 Hz, which correspond to the 15th and 17th harmonics of the magnetic field (and supply voltage). Presumably, these harmonics were caused by the interaction of the first harmonic of the magnetic field with the slot harmonics, which explains their sufficient decrease (by 75% for the harmonics with a frequency f₃ = 1556 Hz; by 50% for the harmonics with a frequency of f₄ = 1764 Hz), with a slight decrease (no more than 15%) of the first harmonic of vibration displacement.

Figure 10 shows the diagrams of the electromagnetic torque of the investigated motor in the case of supply voltage imbalance.

It follows from Figure 10 that at currents I_A = 1.1 I_B = 0.9 I_C the torque ripples increased to M_em = –30.5 ÷ 13.5 N·m; however, at I_A = 1.4 I_B = 0.8 I_C, the torque ripples decreased (M_em = −28.5 ÷ 10.5 N·m), which may have been caused by a decrease in the sum of effective current values in the stator windings from 171.0 A to 162.5 A.

7. Conclusions

In a multiphysics finite element design software environment, a model allowing study of the electromagnetic, mechano-acoustic and energy processes in a squirrel-cage induction motor was developed. The calculations take into account the interconnectedness and mutual influence of processes occurring at different branches of physics.

It was found that the third and fifth harmonics of the supply voltage have a significant effect on these physical variables, even in the absence of a phase mismatch with the first harmonic of the voltage. In the case of a signal close to the meander (I_g1 = 57 A, I_g3 = 15.4 A, I_g5 = 7.7 A), a decrease in the mechanical stresses of the stator and sound pressure is observed. However, the amplitude of the stator vibration displacements more than doubled. This mechano-acoustic mode is explained by a decrease in the peak value of the supply current due to the superposition of the first and third harmonics and an increase in the duration of the half-wave of the radial magnetic force, which reaches its maximum value faster due to the rectangular shape of the supply voltage.

In the case of the supply voltage imbalance, a significant decrease (by 2–4 times) in the stator vibration disturbances in modes with frequencies f₃ = 1556 Hz and f₄ = 1764 Hz was observed, presumably caused by the interaction of the first harmonic of the magnetic field with the slot harmonics. In addition, this decrease is aggravated by a decrease in the sum of the effective values of the currents in the SCIM stator windings from 171.0 A to 162.5 A.

The displacements of the stator of the investigated motor were 0.016 mm at the frequency of the fundamental harmonic of the radial magnetic force of 100 Hz. Displacements increased to 0.022 mm (at I_g3 = 8.8 A, I_g5 = 3.3 A) and up to 0.027 mm (at I_g3 = 15.4 A, I_g5 = 7.7 A) when the supply voltage was distorted by higher harmonics. Voltage imbalance had practically no effect on the motor vibration.

In the future, we plan to use the developed simulation model to further study the effect of the harmonic composition of the supply current and the parameters of its imbalance (phase and amplitude) on the processes of magnetic vibration excitation in the SCIM stators.