Force performance investigation of linear induction motor for low-speed Maglev train with an improved equivalent circuit model

Abstract

In order to fast calculate the thrust force and normal force of linear induction motor (LIM) for low-speed maglev train under overall conditions including traction and braking, this paper proposes an improved equivalent circuit (EC) method. Firstly, the improved equivalent circuit model is proposed based on the traditional one by adopting a precision correction factor, which can accurately consider the transverse edge effect of cap-shape secondary conducting sheet. Moreover, the permeability of back-iron and magnetic inductance are amended considering the saturation effect. The corresponding calculation procedure is given out. Then, the precise correction factor is investigated based on 2D finite element (FE) model. By comparison with the traditional Russell and Norsworthy correction factor (RNF), the precision correction factor is proved to be more suitable to LIM with wide-range slip frequency. Finally, the thrust force and normal force of LIM with different slip frequency are predicted under braking and traction conditions by improved EC method. The predicted results and the proposed method are validated by 3D FE method and experiment results.

Keywords

Braking condition equivalent circuit (EC) method linear induction motor (LIM)low-speed Maglev train normal force slip frequency transverse edge effect

1. Introduction

Single-sided linear induction motors (SLIMs) are widely used in low-speed Maglev trains because of the simple structure and easy maintenance [1]. Several papers investigate the performance considering some impact factors of SLIMs, such as saturation effect, transverse edge effect and longitudinal end effect [2–4]. In order to improve the analysis accuracy, the finite element (FE) method is often adopted [5–7]. The 2D FE model of SLIMs with all poles is normally adopted, so that the longitudinal end effect can be included, but without the transverse edge effect. As known, the transverse edge effect is an important phenomenon for the performance of SLIMs, which mainly causes the longitudinal-direction component of eddy current in the secondary. In addition, it causes the magnetic field distorted along the transverse direction. The thrust force is influenced since it is generated by the normal-direction air gap magnetic flux density and the transverse-direction component of secondary eddy current. Therefore, the calculation of eddy current is very important to the thrust force and normal force. That is to say, the transverse edge effect should be taken into consideration by adopting a correction factor on 2D FE models [5–9].

Among the research of LIMs, the influence of normal force on levitation system is rarely investigated, especially under the braking condition with high slip frequency. Due to the specified operation conditions, the normal force performance cannot be conveniently obtained by FE method [2]. Therefore, the equivalent circuit (EC) method is proposed to be adopted in the calculation of the SLIMs with flat-solid secondary, which considers the transverse and longitudinal end effects by four coefficients [10,11]. However, the transverse edge shape of secondary conducting sheet and the force performances are not carefully analyzed under braking condition in this method.

Actually, the transverse edge effect becomes a main influence factor since the dynamic longitudinal end effect is relatively small under low velocity condition. As imaged, without considering the transverse edge effect in EC method, the predicted forces should have big errors. In the traditional T-EC method, the resistances of secondary conducting sheet and back-iron are not individually modified by different transverse edge effect coefficients for covering overall conditions. In addition, the permeability of back-iron and magnetic inductance are required to be amended considering the saturation effect.

In order to accurately calculate the thrust force and normal force under overall conditions, an improved EC model with a precise correction factor is proposed in this paper to well consider the transverse edge effect of cap-shape secondary conducting sheet, which is compared to Russell and Norsworthy correction factor (RNF) based on 2D FE model. By using the improved EC method, both traction and braking characteristics of SLIMs are obtained under different slip frequency. Finally, the 3D FE method and experiments are carried out to validate the proposed method.

2. SLIM topology and parameters

Figure 1 shows the cross section of the SLIM used in low-speed Maglev train. The SLIM has a short primary fixed to the bogie and a long compound-secondary located on the rails. The conducting sheet of compound-secondary is made of aluminum, also called aluminum sheet. It is laid on the F-type track (back-iron). In order to lower the weight of the primary, the material of three phase windings adopt aluminum. Limited by the rail, the overhang length of secondary is only 10 mm. In Fig. 1, x-, y- and z-axis represent the longitudinal direction (the motion direction), the normal direction and the transverse direction of the SLIM, respectively.

Figure 2 shows the primary prototype of SLIM applied in low speed Maglev train, the main geometry parameters are listed in Table 1. Since the width of secondary aluminum plate is the main factor of the transverse edge effect, it is normally enlarged as much as possible to normal LIM. However, for the SLIMs in low speed Maglev train, the overhang length of secondary is only 10 mm limited by the rail, so that a cap-shape conducting sheet is adopted for reducing the transverse edge effect. Unlike the normal LIMs, this SLIMs need to adopt big slip frequency for reducing the attractive normal force, which is normally over than 10 Hz.

Fig. 1.

The cross section of SLIM in low-speed Maglev train.

Fig. 2.

Prototype of the SLIM primary applied in Changsha Line.

Table 1

Structure parameters of the SLIM

Parameters	Values
Maximum line voltage, U (V)	220
Number of slots, Q ₁	80
Number of pole pairs, p	4
Pole pitch, τ (mm)	202.5
Number of phase, m ₁	3
Length of air gap, g ₀ (mm)	13
Primary lamination width, l _δ (mm)	220
Overhang length of the secondary, c (mm)	10
Length of the primary, L (mm)	1807.9
Secondary conducting sheet thickness, d ₁ (mm)	4
Secondary back-iron thickness, d ₂ (mm)	36

3. Improved equivalent circuit

Since the magnetic circuits of SLIMs are open in the motor structure, the longitudinal end effect and transverse edge effect appear. Because the overhang length of secondary aluminum sheet is limited and the number of poles is high in the SLIMs used in the low-speed Maglev train, the transverse edge effect is apparent, while the static longitudinal end effect is relatively small. Therefore, the improved EC is focused on the transverse edge effect.

Figure 3 shows the proposed improved EC, where R ₁ and L ₁ are the primary resistance and leakage inductance. K _r, C _r are the correction factors of resistance for the longitudinal end effect and transverse edge effect, while K _x, C _x are the corresponding correction factors of inductance, respectively. Compared to the traditional EC, the improved EC has two features:

Fig. 3.

Improved T-type equivalent circuit of SLIM.

(1) To the transverse edge effect, the factor 𝜆 is added in order to consider the cap-shape of aluminum plate except the correction factor C _r. That is to say, the transverse edge effect is considered by 𝜆C _r. On this basis, the secondary equivalent resistance $R_{2}^{\prime }$ and inductance $L_{2}^{\prime }$ is corrected considering secondary conducting plate shape, relative permeability of back-iron and magnetic inductance.

(2) By using an iterative algorithm, the magnetic inductance L _m, the permeability of back-iron, depth of penetration d _fe, and magnetic flux density B _g are amended respectively considering the saturation effect.

To the secondary of SLIM, the fluxes penetrate through the aluminum sheet, and then enter the back-iron. Apparently, the secondary equivalent resistance $R_{2}^{\prime }$ is composed of the equivalent resistance of aluminum sheet and back-iron: $\begin{eqnarray}\displaystyle R_{2}^{\prime }=K_{r}C_{r}{\lambda}R_{2Sheet}R_{2Back}/({\lambda}R_{2Sheet}+R_{2Back})s, & & \displaystyle\end{eqnarray}$ (1) where R _2Sheet is the resistance of the secondary conducting aluminum sheet, and R _2Back is the effective resistance of back iron, which considers the depth of penetration d _fe. Since the permeability of back-iron varies with the magnetic flux density in back-iron, d _fe is required to be repeatedly amended.

In the formulas of $R_{2}^{\prime }$ , the amendments account for adding the new factor 𝜆, which corrects the transverse edge effect coefficient by considering the aluminum sheet transverse edge shape: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\lambda}=sG^{\prime }(\text{Re}^{2}[\boldsymbol{T}^{\prime }]+\text{Im}^{2}[\boldsymbol{T}^{\prime }])/C_{r}\text{Re}[\boldsymbol{T}^{\prime }]\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{T}^{\prime }=\boldsymbol{j}[\boldsymbol{{\gamma}}^{2}+\boldsymbol{{\eta}}(1-\boldsymbol{{\gamma}}^{2})\tanh (0.5l_{{\delta}}{\alpha})/0.5l_{{\delta}}{\alpha}]\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{{\eta}}=1/[1+k_{t}\cdot \tanh (0.5l_{{\delta}}{\alpha})\tanh [0.5D(c_{2}-l_{{\delta}})]/\boldsymbol{{\gamma}}].\end{eqnarray}$ (4) Where T ^′ and 𝜂 are intermediate variables which stand for the magnetic flux density distribution in transverse direction, which are calculated based on Maxwell equations. They are complex due to sinusoidal excitation. s is the slip, l _δ is primary lamination width, G’ is modified goodness factor and c ₂ is secondary width. 𝛼 is the function of secondary overhang length, the pitch and 𝛾, and D is the function of slip s and motor structure parameters. The calculation process is complex and corresponding method is listed in [12].

k _t is the amendment factor of transverse edge effect for correcting the transverse edge shape of aluminum sheet, calculated as follows: $\begin{eqnarray}\displaystyle k_{t}=1+1.3(d-d_{1})/d_{1}, & & \displaystyle\end{eqnarray}$ (5) where d is secondary cap-shaped cap height, d ₁ is conducting sheet thickness.

𝛾 is the function of the slip s and modified goodness factor which is modified for considering the eddy current of back-iron. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}^{2}=1/(1+jsG^{\prime })\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle G^{\prime }=2{\mu}_{0}f{\tau}^{2}(d_{1}\cdot {\sigma}_{al}+d_{fe}\cdot {\sigma}_{fe})/{\pi}g.\end{eqnarray}$ (7) Where μ₀ is air permeability, f is primary frequency, τ is pole pitch, d _fe is back-iron penetration depth, σ_al is conductivity of the secondary aluminum plate, σ_fe is conductivity of the secondary back-iron and g is the effective air gap length.

In general, the secondary leakage reactance is calculated based on the secondary equivalent resistance and slip, if ignored the longitudinal and transverse edge effect. Taking the end effects and aluminum sheet transverse edge shape into account, the secondary leakage reactance can be calculated as: $\begin{eqnarray}\displaystyle L_{2}^{\prime }=R_{2}^{\prime }D\sinh (2d_{1}{\beta})/2{\pi}f, & & \displaystyle\end{eqnarray}$ (8) where 𝛽 = π∕τ.

Figure 4 shows the calculation procedure of SLIM parameters based on the improved EC, where I _m, ω are the magnetizing current and angular frequency respectively. During the calculation, the magnetic resistance R _m is omitted since the influence is small in order to simplify the calculation. The parameters of R ₁, L ₁, L _m0, R ₂ and L ₂ are calculated based on the machine physical structure parameters. In addition, L _m0, R ₂ and L ₂ also need be corrected by transverse edge effect factor and longitudinal effect factor. ϵ is determined by calculation precision. It is assumed 0.5% to circuit parameter, and 1% to force to this SLIM calculation. The excitation inductance L _m is affected by magnetic circuit saturation, so that it is repeatedly amended in the calculation procedure. The difference between initial L _m0 and final L _m1 is the magnetic circuit saturation factor. During the calculation, the magnetic circuit saturation factor need be decided by iteration, and other parameters, $R_{2}^{\prime }$ and $L_{2}^{\prime }$ , need not be repeated.

Figure 5 shows the thrust force and normal force performance curve based on the traditional EC and improved EC under the braking state, respectively. At low velocity, the simple V/F control method is adopted in order to obtain constant thrust force. It is kept until the power supply arrives full voltage, and then the flux is weakened along with the increasing of velocity. During the whole braking state, the slip frequency is also kept 13 Hz. At high velocity, the primary current cannot arrive maximum value, so that the thrust force is low. Along the decrease of velocity, both the primary current and thrust force gradually increase to maximum value. As imaged, the curve of current is same to that of thrust force. As it can be seen, both the thrust force and normal force using traditional EC method are significantly higher than those of improved EC method at low-velocity, because the traditional EC method neglects the transverse edge shape in secondary conducting sheet, which causes the overestimation of secondary resistance. Moreover, the traditional EC method does not consider the eddy current of back-iron and saturation effect, which also causes the calculated results larger due to large slip frequency, especially at low-velocity braking condition with high slip and magnetic flux density.

Fig. 4.

The program flow chart of the improved EC method.

Fig. 5.

The braking thrust force and normal force based on traditional EC and improved EC methods (f ₂ = 13 Hz).

For considering the effects of both secondary sheet and back iron, the equivalent coefficient of transverse edge effect K _TEE is defined as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{TEE}=C_{r}R_{2}^{\prime }/R_{2}\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{2}=R_{2Sheet}R_{2Back}/(R_{2Sheet}+R_{2Back})\end{eqnarray}$ (10)

Figure 6 show the equivalent coefficient K _TEE and longitudinal correction factor K _r under different slip frequency. As can be seen, with the increase of slip frequency, the transverse edge effect increases under traction state, while it decreases under braking state. Moreover, the difference gradually diminishes at high slip frequency. And the operation velocity has an apparent effect on traction state, compared to that of braking state.

Fig. 6.

The end effect equivalent coefficients under braking/traction state (I _a = 300 A).

4. Finite element model and validation

In order to verify the effectiveness of transverse edge effect correct factor in improved EC, the 3D full FE model of this SLIMs including transverse edge and longitudinal end is erected, as shown in Fig. 7. The winding end is simplified for reducing the computing time significantly, but the simplification does not affect the thrust force and normal force. Since the improved EC is amended for improving the accuracy of SLIM with large slip frequency applied in low-speed Maglev train, this validation is carried out from 5 Hz to 16 Hz, which covers the operating area.

Fig. 7.

3D FE full model of the SLIM with simplified winding end.

Fig. 8.

The eddy current distribution in secondary of one pole pair by 3D FE method.

Figure 8 shows the eddy current distributions in secondary of one pole pair. As can be seen, the eddy current flows in the active region of secondary with a closed-loop, which causes the eddy current in secondary have a large transverse direction component that increases the secondary losses. It also has a significant influence on force performances, especially on normal force.

Figure 9(a) calculates the magnetizing inductance by 3D FE model when slip frequency varies from 5 Hz to 16 Hz and the primary current I _a is kept 300 A, and compared with those of improved EC and traditional EC. When the slip frequency increases, the magnetizing inductance increases due to the demagnetization effect of secondary eddy current. Figure 9(b) shows the relationships of magnetizing inductance and current when slip frequency is 13 Hz, and also compared with two EC methods. Apparently, different from the influence of slip frequency, the magnetizing inductance slightly decreases with the increase of current because the iron saturation degree improves. Compared to the results of traditional EC, the calculated results using improved EC method agree well with those of the FE method.

Fig. 9.

The magnetizing inductance comparison of two EC methods and FE method under different slip frequency and current.

Figure 10 calculated the thrust force and normal force based on 3D FE model and compares with the improved EC method. In the 3D FE model, the applied current source is 300 A, and in improved EC method, the current keeps same by adjusting the starting voltage source. Compared with the results of 3D model, both forces employing the improved EC method are bigger. Moreover, the maximum error of EC method is about 9% for thrust force, and 14.6% for normal force. In addition, when the slip frequency is above 8 Hz, the overestimation of forces with improved EC method is mainly caused by overrating the air gap flux density, which also causes the correction errors of relative permeability and secondary equivalent resistance of back-iron. In contrast, when slip frequency is lower than 8 Hz, the secondary equivalent resistance calculated by improved EC method is smaller than the actual value, which causes the predicted thrust force decrease.

From the above comparison, the correction factor of improved EC is more suitable to SLIMs with large slip frequency because the results are much closer to those of 3D model. Therefore, the improved EC method can be used to calculate the force performances.

Fig. 10.

The thrust force and normal force comparison between improved EC method and 3D FE method with different slip frequency (I _a = 300 A).

3D FE model can obtain accurate results, but the computation time is very long due to large size of this SLIM. Therefore, 2D FM model is often used by adopting suitable correction factor for transverse edge effect. The RN transverse edge effect correct factor, K _RN, is commonly adopted in many studies due to its simplicity, which has nothing to do with the slip s and is more suitable to small slip s [13–18]. That is to say, it is not fully taken into consideration the transverse edge effect variation with the slip frequency, so that both the thrust force and normal force have the considerable errors, which are significantly lower than the practical value. To this SLIM of low speed maglev train, the slip frequency is selected above 10 Hz for reducing the normal force, the improved correction factor K _μ of the improved T-type EC of SLIM is as follows: $\begin{eqnarray}\displaystyle K_{{\mu}}=1/{\lambda}C_{r}. & & \displaystyle\end{eqnarray}$ (11)

Figure 11 shows the full 2D FE full model of SLIM which is supplied with current source [19]. If the aluminum sheet resistance is corrected by factor K _μ, the transverse edge effect is considered, called corrected 2D FE model. Otherwise, it is not considered, called no correction model.

Figure 12 shows the thrust force and normal force at different slip frequency with or without K _μ correction under constant current 300A, respectively. Apparently, the transverse edge effect makes a great effect on the thrust force and normal force of the SLIMs. Compared to the results with correction, the results without correction have big errors, especially the normal force. For example, without considering the transverse edge, both the thrust force and normal force are significantly lower when the slip frequency is above 5 H, and the ordinarily attractive normal force even becomes repulsive when the slip frequency is higher than 13 Hz. Therefore, the transverse edge effect should not be neglected in order to obtain correct calculation results.

Fig. 11.

2D FE full model of the SLIM supplied with current source.

Fig. 12.

The thrust force and normal force with or without correction based on 2D FE model under constant current (I _a = 300 A).

Although the thrust force arrives maximum value at slip frequency 5 Hz, the normal force is still so big that the slip frequency need be bigger than 10 Hz in the maglev train. Therefore, the thrust force below 5 Hz is not calculated. Figure 13 compares the thrust force and normal force using K _μ and RNF on the 2D FE model respectively, together with 3D FE model. Apparently, both the thrust force and normal force decrease with the increase of slip frequency, due to the increase of transverse edge effect. Comparing to the results based on 3D model, the maximum errors of the thrust force are about 3.9% and 5.9% for two correction factors, and the normal force has bigger errors, which are 6.1% and 33.3%, respectively. As can be seen, the calculation results by 2D model with RNF correction are close to that with improved correction factor and 3D FE model at low slip frequency, while these have relatively big errors at high slip frequency. This means the RNF is suitable to low slip frequency below 8 Hz, while the correction factor of improved EC is more precise for wide-range slip frequency.

Fig. 13.

Comparison of the thrust force and normal force predicted by different methods under constant current (I _a = 300 A).

5. Force-velocity performances

In order to investigate the force-velocity performances of SLIMs at different slip frequency, the 2D FE model with correction factor of improved EC need be adopted for saving calculation time. The thrust and normal force performance curves at different slip frequency under braking conditions are shown in Fig. 14. As can be seen, the force curves almost keep constant during the low velocity area, and both current and slip frequency keep constant. After the velocity is bigger than certain value which line voltage reaches the maximum value 220 V, the current is reduced with the increase of frequency, which causes the thrust force and normal force decrease. Compared with the thrust force, the normal force and its decrement significantly decreases with the increasing of slip frequency due to the increase of secondary eddy current. By synthetical consideration, the slip frequency near 14 Hz is more suitable to the SLIMs. In addition, the braking normal force curves have a slight fluctuation at low velocity due to the slip changes rapidly.

Fig. 14.

The braking thrust force and normal force performance curves at different slip frequency based on improved EC method.

During former analysis, the slip frequency is selected as integer. In actual application, the slip frequency can have two decimal places. According to the allowed normal force, it is chosen as 13.69 Hz in the application. Based on the 2D FE model with correction factor of improved EC, the thrust force and normal force are calculated and compared with those of 3D FE method under braking and traction conditions, as shown in Fig. 15. Apparently, the force curves almost keep constant during the constant current region at traction state, while the braking force curves have a slight fluctuation at low-velocity. The reason is that the slip and the magnetic flux density are high and have significant variations, especially the normal force. It is a significant factor to braking control and operation. During the constant voltage region, the braking force has larger variation with the velocity than traction, because of the longitudinal end effect at high velocity.

6. Experimental validation

In order to verify the predicted results, the experiments have been carried out. The SLIM prototype and corresponding test platform are built and shown in Fig. 16. The short primary is fixed to the moving bracket and a long cap-shape compound-secondary located on the test base. Due to the size of test platform, in this experiment, the thrust force, normal force and power factor only can be measured at locked conditions for given current and slip frequency. The given currents are 150 A, 200 A, 250 A and 300 A. According to the analysis results in Section 5, the slip frequency is 13.69 Hz, in order to obtain biggest possible thrust force as well as the allowed normal force.

Fig. 15.

The braking/traction thrust force and normal force based on different methods (f ₂ = 13.69 Hz).

Fig. 16.

The prototype of the SLIMs and test platform.

As shown in Fig. 17, the measured thrust and normal forces are compared with the calculated results of 2D FE model with correction factor of improved EC method and 3D FE method under different current from 150 A to 300 A, respectively. With the increase of current, both forces significantly increase, and the difference between predicted and measured results also increases with the current. In addition, the thrust force has smaller errors than the normal force. That is to say, the predicted thrust forces based on improved EC method and 3D FE method are very close to the measured results, but the measured normal force has relatively bigger error, especially at high current, because the secondary temperature and air gap uniform have greater effect on normal force than thrust force. When supplied with the high current, the temperature of secondary improves very quickly, which makes the shape of secondary distorted. Therefore, it is found that the normal force is easy to be affected by air gap length.

Fig. 17.

The thrust force and normal force of the SLIMs by improved EC method, 3D FE method and measurement under different current (f ₂ = 13.69 Hz).

Table 2 compares the power factor based on improved EC method and experimental measurement under different current. The power factor based on improved EC method is only slightly higher than that of measurement, which reaches a maximum error 5.04% at low current, i.e., 150 A. Hence, the predicted results of improved EC are reliable.

Table 2

The power factor comparison of improved EC and measurement under different current

Current (A)	150	200	250	300
Power factor - Improved EC	0.7438	0.7438	0.7439	0.744
Power factor - Measurement	0.7081	0.7087	0.7098	0.711
Error	0.0504	0.0496	0.048	0.0463

7. Conclusion

This paper proposes an improved T-type EC method in order to predict the force performance curves on braking and traction state under different slip frequency. The results show:

(1) The correction factor K _μ for transverse edge effect is proposed to consider the cap-shape conducting sheet and eddy current of back-iron from the improved EC, which can correct the 2D FE model. The simulation results show that compared with 3D FE method, the errors of thrust force and normal force by this corrected 2D model are about 3.9% and 6.1% respectively. Hence, it is more suitable to LIM with wide-range slip frequency than traditional RNF correction.

(2) The results of improved EC method are closer to that of 3D FE method due to considering the magnetic saturation effect of back-iron and magnetic inductance, where the maximum error is about 9% for thrust force, and 14.6% for normal force.

(3) To the braking condition, the braking thrust force has a slight fluctuation at low-velocity, especially the normal force. During the constant voltage region, the braking thrust force has larger variation with the change of velocity than traction one because of the longitudinal end effect at high velocity.

The experiments and 3D FE results validate those of the improved EC method. The power factor using improved EC method reaches a maximum error 5.04%, compare with that of 3D FE method. Hence, the predicted results of improved EC method are reliable. Although FE method is powerful for calculation, the improved EC method is much simpler than FE method, and it can predict the performances curves of SLIMs as quickly as possible. Therefore, it is suitable to calculate the performances of SLIMs and has a good application prospect.

Footnotes

Acknowledgements

This work was supported in part by National Natural Science Foundation of China (NSFC51777190, NSFC51827810), and the Natural Science Foundation of Zhejiang Province (LZ17E070001).

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