**8. Electromagnetic simulation**

User needs to know a lot of information to create a model in RMXprt. One of the items to know is the basic design dimensions. Furthermore, it is necessary to know how to place the windings in the grooves and also the dimensions of the individual wires. All of these param-

**Figure 3.** Models of induction machine generated by RMxprt: a) 2D model and b) 3D model.

200 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

When calculating the vibration of an electrical machine, it is important to note, as in the case of its measurement, the need to take into account the sampling frequency of the search signal. In the case of finite element calculation, this sampling frequency is represented by the time

Results of transient analysis are linear approximate in the Ansys program. This can cause data loss. Example of choosing a time step or sampling frequency is shown in **Figure 4** on the simple signal. The used signal shows the sinus function with frequency 1 Hz. When selecting a large time step (specifically 3 Hz), this function is approximated by a straight line. There is a complete loss of function. When there is use 5 values on signal sampling and there is use linear interleaving function for his reconstruction, then constructed signal has triangle waveform. The calculated signal is the same as the original function only when using a sampling frequency of 36 Hz (and higher). This example shows how the time step can affect the results of the time waveform

eters affect the end result [10] (**Figure 3**).

**7. Determining the time step**

step of the individual calculations.

**Figure 4.** Linear approximation of sin signal.

Electromagnetic simulation is a possible solution in the Maxwell program, which is one of the Ansys software package modules. Maxwell module is used to calculate the magnetic field on 2D and 3D models. The calculation itself is based on Maxwell's equations. These equations can be written in a differential form:

$$\nabla \mathbf{x} E = -\frac{\partial \mathbf{B}}{\partial t} \tag{4}$$

$$
\nabla B = 0\tag{5}
$$

$$
\nabla \mathbf{x} H = \mathbf{J} + \frac{\partial D}{\partial t} \tag{6}
$$

$$
\nabla D = \rho \tag{7}
$$

where *E* is electric field intensity, *B* is magnetic flux density, *H* is magnetic field intensity, *J* is current density on surface, *D* is electric flux density, and *ρ* is volume charge density.

Some of these parameters depend on the properties of the used material:

$$B = \mu\_v \mu\_r H \tag{8}$$

where *μ*<sup>0</sup> is permeability of vacuum and *μr* is relative permeability of material.

$$D = \varepsilon\_0 \varepsilon\_r E \tag{9}$$

where *ε*<sup>0</sup> is permittivity of vacuum and *ε<sup>r</sup>* is permittivity of magnetic material.

$$\mathbf{J} = \sigma \mathbf{E} \tag{10}$$

where *σ* is electric conductivity.

To determine the vibrations in the electric machine model, the force acting between the rotor and the stator must be determined from the magnetic field in the air gap. For the calculation of these forces, it is possible to use the Maxwell stress tensor. Based on this, it is possible to write for two components electromagnetic forces on a 2D model equation:

• For radial direction

$$F\_{ud} = \frac{L\_{sd}}{2\,\mu\_0} \oint (B\_u^2 - B\_l^2) dl\tag{11}$$

• For tangential direction

$$F\_{\rm tan} = \frac{L\_{\rm sub}}{2\,\mu\_0} \oint B\_{\rm n} B\_{\rm i} dl \tag{12}$$

From the point of view of the electromagnetic calculation of electrical machines, the magnetic magnet induction is the most important variable in the air gap. This magnitude depends on the design of the particular electric machine (number of slots, etc.). The figure of this magnitude at half the length of the air gap is shown in **Figure 5**. It is seen that magnetic flux density waveform is not as smooth as the supply voltage waveform. This waveform is displayed on a line representing the half of the air gap. It is clear that the quality of this process is very dependent on the quality of the mesh. The number of elements can affect the resulting signal waveform and also its frequency spectrum. Generally, it is recommended to use at least four elements representing the width of the air gap. Since the width of the air gap of the electric machine can range from tens to millimeters, it is a factor that can greatly influence the solving time and calculation difficulty. The total time course of the absolute value of the forces in the electric machine is shown in **Figure 6**. As shown in **Figure 6**, the influence of linear approximation between the individual

Vibration Simulation of Electric Machines http://dx.doi.org/10.5772/intechopen.72266 203

The mechanical analysis itself in the Ansys program serves to determine the deformations based on the forces applied to the model. The main variable, which in this case describes the vibration, is the displacement of the individual components of the model. It also serves to determine a deformation and displacement of the body at each point in the mesh when finite

[*M*].{*a*} + [*C*].{*v*} + [*K*].{*x*(*t*)} = {*F*(*t*)} (15)

element method is used. The calculation is based on the following equation:

time steps is evident [2, 8, 9].

**Figure 5.** Magnetic field density in the air gap.

**Figure 6.** The time course of forces in the electric machine.

**9. Mechanical simulation**

where *Bn* is normal component of flux density, *Bt* is tangential component of flux density, *l* is length of stator edge, and *Lstk* is stack length of the motor [2, 8, 9].

In the Maxwell environment, the following relationships to determine the individual power components [3] can be used:

$$F\_{val} = F\_x \cos \Theta\_{tip} + F\_y \cos \Theta\_{tip} \tag{13}$$

$$F\_{\rm tan} = -F\_{\rm x} \cos \Theta\_{\rm t\phi} + F\_{\rm y} \cos \Theta\_{\rm t\phi} \tag{14}$$

The radial force component acts perpendicularly to each tip, and the teeth cause radial deformation and vibration. Meanwhile, the tangential force acts on the rotor and produces rotational torque and also causes torsional strains.

When determining the behavior of an electrical machine, the time course of the supply voltage is decisive. This chapter deals with the calculation of vibrations on an asynchronous motor. This type of machine is powered by three-phase alternating voltage. For the simplest case, the time course is harmonious, containing one harmonic. In many real cases, the supply voltage is not smooth. The effect on these waveforms may be the power supply or the function of the inverter connected to the electric machine.

For calculation, the voltage was given in the following phases:


**Figure 5.** Magnetic field density in the air gap.

where *ε*<sup>0</sup>

where *Bn*

is permittivity of vacuum and *ε<sup>r</sup>*

202 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

*Frad* <sup>=</sup> *<sup>L</sup>*\_\_\_\_*stk*

*<sup>F</sup>*tan <sup>=</sup> *<sup>L</sup>*\_\_\_\_*stk*

is normal component of flux density, *Bt*

length of stator edge, and *Lstk* is stack length of the motor [2, 8, 9].

where *σ* is electric conductivity.

• For radial direction

• For tangential direction

components [3] can be used:

*Frad* = *Fx*

*F*tan = −*Fx*

tional torque and also causes torsional strains.

inverter connected to the electric machine.

• Phase B: Umax \* sin(2\*pi\*50\*time-2\*pi/3) • Phase C: Umax \* sin(2\*pi\*50\*time-4\*pi/3)

• Phase A: Umax \* sin(2\*pi\*50\*time)

For calculation, the voltage was given in the following phases:

is permittivity of magnetic material.

)*dl* (11)

.*Bt dl* (12)

is tangential component of flux density, *l* is

.cos Θ*tip* (13)

.cos Θ*tip* (14)

*J* = *σ*.*E* (10)

To determine the vibrations in the electric machine model, the force acting between the rotor and the stator must be determined from the magnetic field in the air gap. For the calculation of these forces, it is possible to use the Maxwell stress tensor. Based on this, it is possible to

> 2.*μ*<sup>0</sup> .∮

*l* (*Bn* <sup>2</sup> − *Bt* 2

2.*μ*<sup>0</sup> .∮

*l Bn*

In the Maxwell environment, the following relationships to determine the individual power

.cos Θ*tip* + *Fy*

.cos Θ*tip* + *Fy*

The radial force component acts perpendicularly to each tip, and the teeth cause radial deformation and vibration. Meanwhile, the tangential force acts on the rotor and produces rota-

When determining the behavior of an electrical machine, the time course of the supply voltage is decisive. This chapter deals with the calculation of vibrations on an asynchronous motor. This type of machine is powered by three-phase alternating voltage. For the simplest case, the time course is harmonious, containing one harmonic. In many real cases, the supply voltage is not smooth. The effect on these waveforms may be the power supply or the function of the

write for two components electromagnetic forces on a 2D model equation:

**Figure 6.** The time course of forces in the electric machine.

From the point of view of the electromagnetic calculation of electrical machines, the magnetic magnet induction is the most important variable in the air gap. This magnitude depends on the design of the particular electric machine (number of slots, etc.). The figure of this magnitude at half the length of the air gap is shown in **Figure 5**. It is seen that magnetic flux density waveform is not as smooth as the supply voltage waveform. This waveform is displayed on a line representing the half of the air gap. It is clear that the quality of this process is very dependent on the quality of the mesh. The number of elements can affect the resulting signal waveform and also its frequency spectrum. Generally, it is recommended to use at least four elements representing the width of the air gap. Since the width of the air gap of the electric machine can range from tens to millimeters, it is a factor that can greatly influence the solving time and calculation difficulty.

The total time course of the absolute value of the forces in the electric machine is shown in **Figure 6**. As shown in **Figure 6**, the influence of linear approximation between the individual time steps is evident [2, 8, 9].
