Magnetic Field Analysis and Structural Optimization of Deflection Double Stator Switched Reluctance Generator

Abstract

Switched reluctance generator has been widely used in industry. With the consumption of energy, the requirement of generator is higher and higher. The energy conversion rate of the traditional generator structure is very low, and it can no longer meet the existing technical indicators. In order to solve the problem of low efficiency of existing generators, this paper proposes a deflectable double stator switched reluctance generator (DDSRG), and performed a finite element analysis on this model. The magnetic density of the new generator is analyzed. A comparison between the proposed mathematical model and the finite element method verifies the accuracy of the model. finally, the core loss was used as the target to optimize the structure of the generator using the response surface method. The experimental results were compared with the simulation. The error was within the allowable range, which verified the correctness of the calculation method and the optimization scheme.

1 Introduction

Switched reluctance generators have very reliable performance and have been widely used in the industrial field. Compared with other types of generators, switched reluctance generators have a smaller size. At the same time, the two stators of the switched reluctance generator with dual stator structure can simultaneously induce output voltage, which is more suitable for industrial production [1]. To further improve the working efficiency of the generator, the concept of multiple degrees of freedom can be combined with the generator structure. The deflectable generator structure can improve the power generation efficiency of the generator. Although the dual-stator structure generator has better power generation characteristics, at the same time its iron loss will also increase to a greater extent. This is not negligible for a switched reluctance motor with a double stator structure [2].

This paper proposes a mathematical model for the proposed new generator structure, and compares the test results with the finite element results to verify the accuracy of the model. Finally, the analysis and optimization results are verified through experiments.

2 Structure and Principle of DDSRG

The structure model of DDSRG is shown in Fig. 1. Both stator and rotor are spherical in shape. The parameters of the generator are shown in Table 1.

Fig. 1

Structure of deflected double stator generator

Table 1 Structural parameters of generator

The tooth pole axis of the inner stator is consistent with that of the outer stator. DDSRG can be regarded as a combination of an inner generator and an outer generator, with a rotor between the inner and outer stators.The rotor part has a magnetic isolation part, which can isolate the magnetic fields of the inner and outer stators from each other.

3 Analytical Calculation of Magnetic Density of DDSR

The magnetic field strength of the main air gap magnetic field, the magnetic density of the main air gap magnetic field, and the magnetic density of the edge air gap magnetic field can be expressed by Eqs. (1)–(3), respectively.

$$ \begin{aligned} H_{m} & = \frac{{B_{s} \left( {2\mu _{r} l_{g} + 2l_{g} + l + 2ll_{g} } \right) + N_{m} i_{m} \mu _{0} \mu _{r} \left( {l + l_{g} } \right)}}{{2\mu _{0} \mu _{r} ll_{g} }} \\ & \quad - \sqrt {\begin{array}{*{20}l} {\left( {\frac{{B_{s} \left( {2\mu _{r} l_{g} + 2l_{g} + l + 2ll_{g} } \right) + N_{m} i_{m} \mu _{0} \mu _{r} \left( {l + l_{g} } \right)}}{{2\mu _{0} \mu _{r} ll_{g} }}} \right)^{2} } \hfill \\ { - \frac{{N_{m} i_{m} \left[ {B_{s} \left( {\mu _{r} l - \mu _{r} l_{g} + l - l_{g} } \right) + \mu _{0} \mu _{r} N_{m} i_{m} } \right]}}{{\mu _{0} \mu _{r} l}}} \hfill \\ \end{array} } \\ \end{aligned} $$

(1)

$$ \begin{aligned} B_{m} & = \frac{{B_{s} \left( {2\mu _{r} l_{g} + 2l_{g} + l + 2ll_{g} } \right) + N_{m} i_{m} \mu _{0} \mu _{r} \left( {l + l_{g} } \right)}}{{2\mu _{r} ll_{g} }} \\ & \quad - \sqrt {\begin{array}{*{20}l} {\left( {\frac{{B_{s} \left( {2\mu _{r} l_{g} + 2l_{g} + l + 2ll_{g} } \right) + N_{m} i_{m} \mu _{0} \mu _{r} \left( {l + l_{g} } \right)}}{{2\mu _{r} ll_{g} }}} \right)^{2} } \hfill \\ { - \frac{{\mu _{0} N_{m} i_{m} \left[ {B_{s} \left( {\mu _{r} l - \mu _{r} l_{g} + l - l_{g} } \right) + \mu _{0} \mu _{r} N_{m} i_{m} } \right]}}{{\mu _{r} l}}} \hfill \\ \end{array} } \\ \end{aligned} $$

(2)

$$ \begin{aligned} B_{\text{f}} & = \frac{{2\mu _{0} \mu _{r} N_{{\text{m}}} i_{{\text{m}}} \left( {\frac{{A_{{\text{s}}} }}{{l - l_{{\text{f}}} }} + A_{{\text{f}}} } \right) + B_{{\text{s}}} \left( {A_{{\text{s}}} l_{{\text{f}}} \mu _{r} + A_{{\text{s}}} l_{{\text{f}}} + A_{{\text{f}}} l - A_{{\text{f}}} l_{{\text{f}}} } \right) - \Phi _{{\text{m}}} \mu _{0} \mu _{r} l_{{\text{f}}} }}{{2\mu _{{\text{r}}} l_{{\text{f}}} N_{{\text{m}}} i_{{\text{m}}} \left( {\frac{{A_{{\text{s}}} }}{{l - l_{{\text{f}}} }} + 2A_{{\text{f}}} } \right)}} \\ & \quad \sqrt {\begin{array}{*{20}l} {\left( {\frac{{2\mu _{0} \mu _{r} N_{{\text{m}}} i_{{\text{m}}} \left( {\frac{{A_{{\text{s}}} }}{{l - l_{{\text{f}}} }} + A_{{\text{f}}} } \right) + B_{{\text{s}}} l_{{\text{f}}} \left( {A_{{\text{s}}} \mu _{r} + A_{{\text{s}}} } \right) - \Phi _{{\text{m}}} \mu _{0} \mu _{r} l_{{\text{f}}} }}{{2\mu _{{\text{r}}} l_{{\text{f}}} N_{{\text{m}}} i_{{\text{m}}} \left( {\frac{{A_{{\text{s}}} }}{{l - l_{{\text{f}}} }} + 2A_{{\text{f}}} } \right)}}} \right)^{2} } \hfill \\ { - \frac{{A_{{\text{s}}} \left( {\frac{{\mu _{0} \mu _{r} A_{{\text{s}}} N_{{\text{m}}} i_{{\text{m}}} }}{{l - l_{{\text{f}}} }} + B_{{\text{s}}} N_{{\text{m}}} i_{{\text{m}}} \mu _{r} } \right) - \Phi _{{\text{m}}} \left( {\mu _{0} \mu _{r} N_{{\text{m}}} i_{{\text{m}}} + B_{{\text{s}}} l - B_{{\text{s}}} l_{{\text{f}}} } \right)}}{{\mu _{r} l_{{\text{f}}} N_{{\text{m}}} i_{{\text{m}}} \left( {\frac{{A_{{\text{s}}} }}{{l - l_{{\text{f}}} }} + 2A_{{\text{f}}} } \right)}}} \hfill \\ \end{array} } \\ \end{aligned} $$

(3)

where A_f1 and A_f2 are the areas where the lines of force pass through the edges on both sides. A_f = A_f1 + A_f2, lf is the average length of the edge magnetic circuit on both sides, l_g is the average air gap length, N_m is the number of turns of the outer stator winding, i_m is the phase current of the winding. B_s is the saturation magnetic density of the material.

4 Finite Element Analysis Results of Generator Magnetic Field

The deflection type double stator switched reluctance generator can be composed of the radial component Br and the tangential component B_θ [3,4,5,6].

In the transient field, the composite flux density and magnetic flux density components of each element are simulated. The analysis radius of the outer rotor yoke is 55–70 mm. The analysis radius of the outer rotor teeth is from 70 to 75 mm. The analysis radius of outer stator teeth is 75.5–80.5 mm. The analysis radius of the outer stator yoke is 80.5–110 mm. Figure 2 shows the changes in the magnetic density of different types of generators.

Fig. 2

Comparison of magnetic flux density by analytical method and FEM

The error between the analytical method and the finite element method is relatively small enough.

5 Structural Optimization of DDSRG

Bertotti et al. believe that the core loss is mainly composed of three types of loss, namely eddy current loss, eddy current loss P_e, hysteresis loss P_h, and residual loss. P_c [7, 8].

$$ P_{{\text{Fe}}} = P_{{\text{e}}} + P_{{\text{h}}} + P_{{\text{c}}} $$

(4)

The change of stator and rotor structure has a great influence on hysteresis loss.

Four design variables are selected: core length L₁, stator outer pole width B₁, stator outer yoke height H1 and outer air gap G₁.

Figure 3 is the schematic diagram of generator optimization parameters. Each optimized parameter has 3 value variables, respectively, the values of the A, B, and C variables determine the changes of the four parameters. The changes of the four parameters are shown in Table 2.

Fig. 3

Generator optimization parameters

Table 2 The optimized parameters and evaluated parameters

Taguchi algorithm helps to greatly reduce the iterative method in the experiment and reduce the cost of the experiment [9]. The core loss of the generator model is analyzed using finite element, and the core loss P_Fe of each combination in the orthogonal table is obtained, as shown in Table 3.

Table 3 Analysis results

Table 3 is the orthogonal experiment table, and the finite element method can be used to calculate the iron loss under different conditions.

$$ YY = 3\sum\limits_{i = 1}^{3} {\left[ {m_{xi} \left( {P_{{{\text{Fe}}i}} } \right) - m\left( {P_{{{\text{Fe}}}} } \right)} \right]}^{2} $$

(5)

where YY is the proportion of each parameter variable affecting the performance, x is the optimized parameter variable, namely the core length L₁, the outer stator pole width B₁, the outer stator yoke height H1, the external air gap G₁, m_xi (P_Fei) as the parameters The average value of the core loss of the variable x under the i-th level variable, m(P_Fe) is the average value of P_Fe in 9 experiments [10].

As shown in Table 4, the iron loss has a great correlation with the proposed parameters. Among them, the iron core length L₁ accounts for 7.04%, which has a small influence on the iron loss, and the influence rate of other structural parameters is very high. The largest part of the outer stator pole width B₁ accounts for 46.93% of the total proportion.

Table 4 The Influence rate of optimization parameters on generator core loss

The outer stator pole width B₁, the outer stator yoke height H1, the outer air gap G₁ as the three parameter variables, the core loss of the generator as the response value, the application of the central composite design (CCD) method to obtain the coding con-version of the parameter variables is shown in Table 5.

Table 5 Comparison of measured and calculated values of generator iron loss

The data is fitted by the least square method, and based on the results of the response surface, the regression equations with the generator core loss as the response value are obtained as

$$ \begin{aligned} Y & = 272.52 + 73.68X_{1} - 22.70X_{2} - 29.08X_{3} \\ & \quad + 12.93X_{1} X_{2} - 17.62X_{1} X_{3} + 0.10X_{2} X_{3} \\ & \quad - 19.53X_{1}^{2} - 11.57X_{2}^{2} - 28.93X_{3}^{2} \\ \end{aligned} $$

(6)

where X₁ is the outer stator pole width B₁, X₂ is the outer stator yoke height H1, and X₃ are the external air gap G₁, where X₁, X₂, X₃, X₁X₂, X₁X₃, X₁₂, X₂₂, and X₃₂ have a significant effect on the core loss of the generator Significant influence, other items are not significant.

It can be seen from Fig. 4 that the outer stator pole width B₁, the outer stator yoke height H1, and the outer air gap G₁ have a significant effect on the response value of the generator core loss.

Fig. 4

The relationship between response surface and parameters

6 Experimental Verification

When the DDSRG is running in a steady state of rotation, The loss can be calculated by formula (7):

$$ P_{{\text{Fe}}} = P_{1} - P_{2} - P_{{\text{Cu}}} - P_{{\text{fw}}} - P_{\text{s}} $$

(7)

where P₁ is input power, P₂ is output power, P_Cu is copper loss, P_fw is mechanical loss, and P_s represents stray loss. As shown in Fig. 5, the experiment plat-form is composed of generator, power converter and oscilloscope.

Fig. 5

Experimental device

7 Conclusion

The deflection type DDSRG has important application value. This paper introduces the basic structure and control principle of the generator, focusing on the magnetic density distribution of the DDSRG under the rotation state. The overall qualitative analysis of the core magnetic density is carried out by using FEM. According to the calculation result of the magnetic density, the Fourier transform is performed to calculate the Iron loss of the DDSRG, and the response surface method is used to optimize the structural parameters of the DDSRG. The indirect measured experimental data verifies the accuracy of the optimization scheme. It provides theoretical support for the subsequent research of the generator.