Numerical simulation of the magnetic field and electromagnetic vibration analysis of the AC permanent-magnet synchronous motor

3.1. The magnetic field model of the air gap of permanent-magnet synchronous motor

According to the basic principle of the magnetic circuit in the electromagnetic field, the permeability of the motor core is much larger than that of the air. The reluctance of the core is so small and generally negligible. Therefore, magnetic potentials of permanent magnets on rotors and armatures on stators basically land on the air gap. The magnetic potential of the air gap composes of fundamental magnetic potential, harmonic magnetic potential of stators and harmonic magnetic potential of rotors. The magnetic potential of the air gap is:

where ${\omega }_0$ is the current frequency of the fundamental wave of stator; $v$ is the number of the harmonic wave of stator and takes the value of $6k+{\left(-1\right)}^k$; $\mu \text{,}v$ is the number of the harmonic wave of rotor and takes the value of $2k+1$; $k$ takes the value of 1, 2, 3, 4,…; $p$ is the number of pole-pairs; $\theta$ is the rotation angle.

The gap flux density is expressed as [17]:

where $\lambda \left(\theta \right)$ is the magnetic permeability of the air gap. Just considering the average components and first-order tooth harmonic components, the magnetic potential of the air gap can be approximated by:

where ${\mathrm{\Lambda }}_0$ is the average component of the magnetic permeability the air gap; ${\mathrm{\Lambda }}_1$ is the amplitude of first-order tooth harmonic components; $z$ is the number of stator slots.

With simultaneous Eq. (1)-Eq. (3), gap flux density can be deduced:

Because $F_v<F_0\text{,}$$F_{\mu }<F_0$ and ${\mathrm{\Lambda }}_1<{\mathrm{\Lambda }}_0\text{,}$ discarding product terms with small amplitude, Eq. (4) can be simplified to:

This is the air gap flux density model of the permanent-magnet synchronous motor. The air gap field of the permanent-magnet synchronous motor can be obtained by it.

3.2. The theoretical foundation of the numerical simulation of magnetic field

The numerical simulation of magnetic field is based on Maxwell equations, namely [18]:

where $\mathbf{E}$ is the intensity of the electric field; $\mathbf{B}$ is the intensity of magnetization; $\mathbf{H}$ the intensity of the magnetic field; $\mathbf{J}$ is the intensity of the current; $\mathbf{\rho }$ is the density of the charge.

Because mediums meet:

where $\mu$ is the magnetic permeability; $\gamma$ is the conductivity; $\epsilon$ is the dielectric constant.

Taking Eq. (10)-Eq. (12) into Eq. (6)-Eq. (9), the electromagnetic field with sinusoidal characteristics meets:

Because $\nabla \times \mathbf{B}=0$, there is magnetic potential $\mathbf{A}$ to make:

So:

Because:

Combining Eq. (8), Eq. (11) and Eq. (12) show that:

Taking Eq. (6) into Eq. (18), the above equation can be written as:

Combine Eq. (19) with Eq. (17) to get:

Remove $\mathbf{H}$ of Eq. (20) to obtain an equation about $\mathbf{E}$:

Eq. (16), Eq. (20) and Eq. (21) constitute the theoretical foundation of the numerical simulation of magnetic field. In the numerical simulation, in addition to theoretical formulas, boundary conditions are also needed. Boundary conditions of numerical simulation of the magnetic field can be divided into three categories. The first is to directly give the value of an unknown function on the boundary. The second is to give the directional derivative along the normal outside the boundary of an unknown function. The third is to give some linear combinations along the outer normal derivative of an unknown function.

3.3. The model of the numerical simulation of magnetic field

The structural parameters of the motor: the inner diameter and outer diameter of the stator are respectively 16 mm and 23.3 mm; 4 poles and 24 stator slots. According to Fig. 1, the model of the numerical simulation of magnetic field can be drawn as shown in Fig. 2.

Fig. 2The numerical simulation of the three-phase AC permanent-magnet synchronous motor

4.1. The model of electromagnetic force

The main reason for the electromagnetic vibration is the influence of the radial electromagnetic force on stators. Knowing from the Maxwell stress equations, the instantaneous radial electromagnetic force on the inner surface unit area of the stator teeth [19]:

Taking Eq. (5) into Eq. (22). After the prosthaphaeresis of the vibration of stators and ignoring items with high stress wave order, the instantaneous radial electromagnetic force per unit area can be obtained:

The radial force of stators:

where $D_{sf}$ is the inner diameter of stator cores; $L_{sf}$ is the length of stator cores.

4.2. Vibration model

The dynamic model of mechanical vibration can be equivalent to a mass – spring – damper system, so as to meet the force balance equation:

where $\left[\mathbf{M}\right]$, $\left[\mathbf{C}\right]$ and $\left[\mathbf{K}\right]$ are namely mass matrix of the mechanical system, damping matrix and stiffness matrix; $\left\{\mathbf{x}\right\}$ and $\left\{\mathbf{F}\right\}$ are displacement and force of nodes.

To simplify the analysis, the stator is simplified into a cylindrical shell with two end covers. Two ends are free. The equivalent simplified vibration model is shown in Fig. 3.

Fig. 3The equivalent simplified vibration model of the stator

Considering only the radial direction of the stator vibration, when the order of the electromagnetic force equals that of the motor, the amplitude of radial vibration displacement of the stator with order m produced by electromagnetic force is:

where $M$ is the equivalent mass of cylindrical shell; ${\omega }_m$ is m-order natural angular frequency; ${\omega }_r$ is the angular frequency of $r$-order electromagnetic force; $F_{rm}$ is the amplitude of radial electromagnetic force from Eq. (24). ${\xi }_m$ is the modal damping ratio, which is difficult to be analyzed theoretically and got by experimental methods. Its value is influenced by the frequency and winding layers. Empirical formula is:

where $f_m$ is the natural frequency of the cylindrical shell. Formula of its value is:

where ${\nu }_c$ is the Poisson’s ratio of the stator core. The value of ${\mathrm{\Omega }}_m$ is:

where $m\ge 1$. When $m=0$, ${\mathrm{\Omega }}_0=1$. ${\kappa }^2$ is the dimensionless thickness parameter:

where $h_c$ is the thickness of the stator core.

5.1. The analysis of electromagnetic vibration frequency

The form of Eq. (23) shows that the radial electromagnetic force is a kind of wave. Motor vibration amplitude is inversely related to the order of stress wave. Know from the five items in braces of Eq. (23).

The order of stress wave of the first one is 2$p$, and the motor vibration frequency is 2 times as the fundamental wave.

The order of stress wave of the second one is $z-2p$. When the slot number of the stator$\mathrm{z}$is close to the pole number of the motor 2$p$, that is, $z-2p\approx 0$, the motor will produce vibration frequency which is 2 times as the fundamental frequency.

The order of stress wave of the third one is $\left(v-\mu \right)p$. When the harmonic wave number of the stator is equal to that of the rotor, vibration with the frequency of $\left(\mu \pm 1\right){\omega }_0$ will generate. When $\nu =6k-1\text{,}$ take $\left(\mu +1\right){\omega }_0\text{;}$ when $\nu =6k+1\text{,}$ take $\left(\mu -1\right){\omega }_0\text{.}$ For the three-phase synchronous motor, the value formula of harmonic wave numbers of the stator and the rotor shows that when the harmonic wave number is low, if $v$ equals $\mu$ and the value is 5 or 7, the motor will produce vibration frequency which is 6 times as the fundamental frequency.

The order of stress wave of the fourth one is $z-\nu p-p$. When $\nu =6k-1$, the coefficient before the time is 0 and there is no vibration. When $\nu =6k+1$, the coefficient before the time is 2. When the order number of stress wave is close to 0, the motor will produce vibration frequency is 2 times as the fundamental frequency. Therefore, for the three-phase synchronous motor, the vibration when $v$ is 5 or 7 is: when $v=$5, there is no vibration; when $v=\mathrm{}$7, the motor will produce vibration frequency which is 2 times as the fundamental frequency and the number of stator slots $z\approx 8p$.

The order of stress wave of the fifth one is $z-\mu p-p$. When $z-\mu p-p\approx 0$, the motor will produce electromagnetic vibration with the frequency of $\left(\mu +1\right){\omega }_0$. For the three-phase synchronous motor, low harmonic wave will still be analyzed. Take the harmonic wave number of the rotor as 3 or 5. When $z\approx$4$p$ or $z\approx$6$p$, the motor will produce vibration frequency which is 4 or 6 times as the fundamental frequency.

The foregoing analysis shows that the electromagnetic vibration frequency of three-phase permanent-magnet synchronous motor is mainly 2 or 6 times as the fundamental frequency. When the slots number of the stator is close to 4$p$ and 6$p$, the electromagnetic vibration also includes vibration frequency which is 4 or 6 times as the fundamental frequency.

5.2. The simulation analysis of the electromagnetic force

Eq. (24) shows that the electromagnetic force wave is a function influenced by mechanical angle and time. Therefore, the magnetic flux density and electromagnetic force wave were simulated according to spatial location and time.

Parameters of the simulation are shown in Table 3. Take the intermediate mechanical angle of the inner surface of the stator as 30°. Substitute it into Eq. (5). For the three-phase permanent-magnet synchronous motor, when neglecting items with high harmonic wave and the stator is 5 or 7 and rotor is 3, 5 and 7, the curve of magnetic flux density is shown in Fig. 4.

Simulation parameters of Eq. (24) is same with Eq. (5). When the harmonic wave number of the stator takes 5 and 7, the harmonic wave number of the rotor takes 3, 5 and 7 and the mechanical angle is 30°, the curve of stress wave is shown in Fig. 5.

Fig. 4The radial flux density changes with time at the angle of 30°

Fig. 5The radial force changes with time at the angle of 30°

Fig. 6The radial flux density changes with the space angle

Fig. 7The radial force changes with the space angle

Parameters of the simulation are shown in Table 3. When the harmonic wave number of the stator takes 5 and 7, the harmonic wave number of the rotor takes 3, 5 and 7 and the permanent-magnet synchronous motor rotates at a certain moment (set ${\omega }_{0}t=\pi$), the curve of radial flux density changing with the mechanical angle is shown in Fig. 6.

Simulation parameters are as above. When the permanent-magnet synchronous motor rotates at a certain moment (set ${\omega }_{0}t=\pi$), the curve of radial electromagnetic force changing with the mechanical angle is shown in Fig. 7.

From Fig. 4 to Fig. 6, the radial air gap flux density and the radial electromagnetic force wave are periodic from space and time. And the frequency of the radial electromagnetic force wave is 2 times of that of the radial air gap flux density.

5.3. The numerical simulation of the electromagnetic field

According to structure parameters in Table 1 and the electromagnetic field simulation of the three-phase AC permanent-magnet synchronous motor in Fig. 2, simulation results can be shown in Fig. 8 and Fig. 9.

Fig. 8The radial force changes with the space angle

Fig. 9The simulation results of magnetic flux density of the three-phase AC permanent-magnet synchronous motor

Knowing from Fig. 8 and Fig. 9, numerical simulation results of the electrical angle and radial air gap flux density are shown in Fig. 10. Fig. 11 to Fig. 13 are simulation curves of electrical parameters of the three-phase AC permanent-magnet synchronous motor. Fig. 11 shows that the maximum no-load is close to 175 V, the winding current is close to 4 A. In the rated load, electrical angle and phase voltage curve are sinusoidal.

Fig. 10The simulation results of magnetic flux density of the three-phase AC permanent-magnet synchronous motor

Fig. 11The relationship between the electrical angle and winding back EMF

Fig. 12The relationship between the electrical angle and the winding current

5.4. The calculation of electromagnetic vibration

By the motor’s structural parameters of Eq. (26) to Eq. (30) and Table 3, the relationship between the radial electromagnetic force and vibration displacement of the stator of the three-phase AC permanent-magnet synchronous motor in different stress wave and frequency is shown in Table 2.

Fig. 13The relationship between the electrical angle and the phase voltage with the rated load

As can be seen from Table 2, the vibration displacement of the 500 Hz electromagnetic force can be 3.16 μm. As the main component of the electromagnetic vibration, the vibration frequency is 2 times as the fundamental frequency.