1 Introduction

Linear motor is a kind of electrical equipment which can directly convert the electrical energy to linear movement. Compared with traditional rotating machine, the drive system of linear motor works without the intermediate gearing, which simplifies the driver system, and makes its linear velocity unlimited and moving process without mechanical touch. In addition, the noise level of linear motor is very low. For all these characteristics, linear motor is widely applied to high-speed ground transportation [1–4].

When the pole number of a linear motor is not less than six, the values of negative-sequence current and zero-sequence current are much small compared with the positive sequence current; thus their influence on the cascaded linear motor can be ignored [5, 6]. In this case, the asymmetry of three-phase current will not be considered. With three-phase symmetrical current flowing into three-phase winding, linear motor will produce a traveling magnetomotive force (MMF) wave in the air-gap between primary and secondary windings. Since the energy exchange in mechanic-electronics is achieved by the air-gap magnetic field of linear motor [7], the time and space characteristics of the MMF wave directly affect the operating efficiency and energy consumption in linear motor. The motor can work efficiently with the waveform of the MMF close to a sinusoidal traveling wave [8].

The finite element method (FEM) is an effective and accurate method to investigate the linear motor characteristics. Lu et al. [9] analyzed the features of air-gap magnetic field in large air-gap linear induction motor. Selcuk and Kurum [10] built a simplified FEM model of an actual short primary linear induction motor and solved it for air-gap magnetic field distribution. Lu et al. [11] analyzed the two-dimensional transient air-gap magnetic field of long primary induction motor with the help of FEM. Li et al. [12] studied the characteristics of temperature field for tubular linear motor with the FEM. In this paper, we will solve the single-phase winding MMF and three-phase winding-synthesized MMFs in single-side linear induction motor (SLIM) by the classical theory of MMF [13] and adopt the FEM to validate the theoretical results. The work conditions and efficiency of SLIM are then optimized by analyzing MMF characteristics with different three-phase winding arrangement.

2 Theoretical analysis and simulation model for SLIM

2.1 The MMF of A-phase winding

The established theoretical model of SLIM is shown in Fig. 1, in which the leak flux in this model is ignored. In this model, it is assumed that primary and secondary are infinite long, primary iron yoke is not in magnetic saturation, the magnetoconductivity of primary and secondary is infinite, and magnetic induction intensity only contains the component in the y axis direction. It is also assumed that the current flows along the z axis direction, the equivalent air-gap δ between primary and secondary is distributed evenly along the x axis direction, the magnetic potential difference of three-phase winding is distributed evenly along the air-gap, the number of pole pairs is p and pole pitch is three slots pitch long, and the three-phase wingding is a bi-layered full-pitch winding.

Fig. 1
figure 1

The theoretical model of SLIM and A-phase winding MMF distribution diagram

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Without loss of generality, the sinusoidal current as shown in Eq. (1) is assumed flowing into A-phase wingding, i.e.,

$$ i_{\text{A}} = \sqrt 2 I_{\text{A}} \cos \omega t. $$
(1)

With reference to Fig. 1, according to the Ampere’s circuital Law, A-phase winding MMF in air-gap can be expressed as [14]

$$ f_{\text{A}} = H_{\text{y}} \cdot \delta = N \cdot i_{\text{A}} \quad {\text{when}} - \frac{\pi }{2} \pm 2m\pi \le \theta \le \frac{\pi }{2} \pm 2m\pi , $$
(2a)
$$ f_{\text{A}} = - H_{\text{y}} \cdot \delta = - N \cdot i_{\text{A}} \quad {\text{when}}\,\frac{\pi }{2} \pm 2m\pi \le \theta \le \frac{3\pi }{2} \pm 2m\pi , $$
(2b)

where m = 1, 2, 3, …, electrical angle θ = πx/τ, the number of turns in series winding N = 2pNc, and Nc is the number of turns in a single coil.

Since the quantity fA(θ) represents a periodic square wave along the air-gap, it can be represented by the Fourier series as

$$ f_{\text{A}} (\theta ) = f_{{{\text{A}}1}} \cos \theta + f_{{{\text{A}}3}} \cos 3\theta + f_{{{\text{A}}5}} \cos 5\theta + \cdots + f_{{{\text{A}}n}} \cos n\theta , $$
(3)

where fAn is

$$ f_{{{\text{A}}n}} = \frac{4}{T}\int\limits_{0}^{\frac{T}{2}} {f_{\text{A}} (\theta )\cos (n\omega_{0} \theta ){\text{d}}\theta = \frac{1}{n} \cdot \frac{4}{\pi } \cdot Ni_{\text{A}} \sin n\frac{\pi }{2},} $$
(4)

n = 1, 2, 3, …

Substituting Eqs. (1) and (4) into Eq. (3), we have the instantaneous value of A-phase wingding MMF:

$$ f_{\text{A}} (\theta ,t) = \frac{4\sqrt 2 }{\pi }NI_{A} \left[ {\cos \theta - \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta - \cdots + \frac{1}{n}\sin (n\frac{\pi }{2})\cos n\theta } \right]\cos \omega t. $$
(5)

From Eq. (5), we can find that A-phase winding MMF after the Fourier series transformation can be decomposed into fundamental wave and a series of higher harmonics, and the amplitudes of fundamental wave and higher harmonics pulse over time with the current frequency. Because fundamental wave determines the energy conversion of linear motor and its main performance, it is most important and fundamental to analyze the fundamental MMF [15].

2.2 The MMFs of three-phase winding

As shown in Fig. 2, when the B and C-phase windings are respectively d and e slots pitch away from the A-phase winding, we can obtain the MMF of B-phase winding and C-phase winding respectively by moving the MMF of A-phase winding πd/3 and πe/3 along the positive direction of θ axis, respectively.

Fig. 2
figure 2

Three-phase winding distribution diagram

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Let three-phase current flow into three-phase winding, through above analysis we can obtain the expression of the fundamental MMF for A, B, C phase windings, respectively,

$$ F_{\text{A}} (\theta ,t) = \frac{4\sqrt 2 }{\pi }NI_{\text{A}} \cos (\theta )\cos (\omega t), $$
(6a)
$$ F_{\text{B}} (\theta ,t,d) = \frac{4\sqrt 2 }{\pi }NI_{\text{A}} \cos \left(\theta - \frac{1}{3}\pi d\right)\cos \left(\omega t - \frac{2}{3}\pi \right), $$
(6b)
$$ F_{\text{C}} (\theta ,t,e) = \frac{4\sqrt 2 }{\pi }NI_{\text{A}} \cos \left(\theta - \frac{1}{3}\pi e\right)\cos \left(\omega t + \frac{2}{3}\pi \right). $$
(6c)

Adding the instantaneous values of fundamental MMF with A, B, C phase windings, we have the fundamental MMFs:

$$ F(\theta ,t,d,e) = F_{\text{A}} (\theta ,t) + F_{\text{B}} (\theta ,t,d) + F_{\text{C}} (\theta ,t,e). $$
(7)

According to the theoretical model of SLIM, the values of d and e can be taken from 1, 2, 3, 4 and 5 in a cycle of MMF, so that the number of arrangement modes of three-phase winding is 20 with the combination between d and e. The results of fundamental MMFS with different arrangements of three-phase windings will be analyzed later.

2.3 The simulation model for SLIM

The simulation model of SLIM is established by Ansoft Maxwell as shown in Fig. 3. Table 1 presents the specific parameters related to this model. Different simulation data of MMFs will be obtained by changing the relative positions of three-phase winding.

Fig. 3
figure 3

The simulation model of SLIM

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Table 1 SLIM parameters for simulation model
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3 Results and analysis

3.1 The MMFs with d = 1 and e = 2

Referring to Fig. 2, we move B-phase winding and C-phase winding from the position of A-phase winding 1 slot pitch and 2 slots pitch, respectively. Then, substitute d = 1 and e = 2 into Eq. (7) and the expression of fundamental MMFs is calculated by trigonometric formula [16]:

$$ F(\theta ,t,1,2) = \left[ {\frac{2\sqrt 2 }{\pi }NI_{\text{A}} [2\cos (\omega t + \theta ) + \cos (\omega t - \theta ) + \cos (\omega t + \theta - \pi ) + \cos \left( {\omega t - \theta - \frac{1}{3}\pi } \right) + \cos \left( {\omega t - \theta + \frac{4}{3}\pi } \right)} \right] $$
(8)

The MMF is composed of three forward traveling waves and two back traveling waves referring to Eq. (8), and the results are shown in Fig. 4. As can be seen in Fig. 4a, when the spatial MMF waveform is at the transient time of ωt = 0, π/2, π, 3π/2, respectively, and with a scaling factor \( \beta = 2\sqrt 2 NI_{\text{A}} /\pi \), the MMF pushes to the left in space and exists obvious vibration. The movement with variable velocity is shown in Fig. 4c, d. These features will reduce the efficiency of linear motor thrust in the horizontal direction, and at the same time also bring high energy loss.

Fig. 4
figure 4

The MMFs with d = 1 and e = 2 and its amplitude and velocity characteristics. a theoretical fundamental MMFs. b simulated MMFs. c amplitude of theoretical fundamental MMF. d velocity of theoretical fundamental MMF

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According to the simulation model for SLIM, when arranging three-phase winding to be d = 1 and e = 2, the simulation transient waveform of the MMF is shown in Fig. 4b, which further validates the characteristics of MMF with d = 1 and e = 2.

3.2 The MMFs with d = 2, e = 4, and d = 4, e = 2

In order to keep the amplitude and wave velocity of MMF invariant over time and avoid the pulse vibration and movement with variable velocity of MMF, we solved the MMFs under different arrangements of three-phase winding. The result indicates that the MMFs in the condition of d = 2, e = 4 or d = 4, e = 2 are a traveling wave with its amplitude and velocity being constant as shown in Fig. 5a, c, which are expressed, respectively, by Eqs. (9a) and (9b):

$$ F(\theta ,t,2,4) = \frac{6\sqrt 2 }{\pi }NI_{\text{A}} \cos \left( {\omega t - \theta } \right), $$
(9a)
$$ F(\theta ,t,4,2) = \frac{6\sqrt 2 }{\pi }NI_{A} \cos \left( {\omega t + \theta } \right), $$
(9b)

where \( \theta = k \cdot x = \frac{\pi }{\tau } \cdot x \).

Fig. 5
figure 5

The MMFs with d = 2, e = 4 and d = 4, e = 2. a theoretical fundamental MMFs with d = 2 and e = 4. b simulated MMFs with d = 2 and e = 4. c theoretical fundamental MMFs with d = 4 and e = 2. d simulated MMFs with d = 4 and e = 2

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From Fig. 5a, c, we can see that F(θ, t, 2, 4) is a forward traveling wave and F(θ, t, 4, 2) is a backward traveling wave. Their amplitude and wave velocity can be expressed as

$$ F_{\text{m}} (\theta ,t,2,4) = F_{\text{m}} (\theta ,t,4,2) = \frac{6\sqrt 2 }{\pi }NI_{\text{A}} , $$
(10a)
$$ \nu (\theta ,t,2,4) = \nu (\theta ,t,4,2) = \frac{\omega }{k} = \frac{\tau \cdot \omega }{\pi }. $$
(10b)

Based on the simulation model for SLIM, we also changed the arrangement mode of three-phase winding to d = 2, e = 4 and d = 4, e = 2, respectively, and obtained the MMFs simulated by Ansoft Maxwell. The results are shown in Fig. 5b, d. Since the simulation model for SLIM is not fully ideal and its MMFs contain other higher harmonics, the MMFs are close to sinusoidal traveling wave with constant amplitude and velocity.

4 Conclusion

Based on the classical MMF theory of rotating machine, we established the theoretical model and simulation model for SLIM, in which three-phase winding has 20 arrangement modes in a cycle of MMF, and the arrangement mode determines the time and space characteristics of MMFs. Through the calculation with electromagnetic theory and finite element software simulation, we solved and discussed the MMFs under 20 arrangement modes of three-phase winding, respectively. On the basis of the above analysis results, the conclusions are drawn as follows:

  1. (1)

    The MMFs with d = 2 and e = 4 are close to sinusoidal wave, and it travels toward the positive direction with the constant amplitude and velocity. In this case, the motor can work normally with a high efficiency.

  2. (2)

    The MMFs with d = 4 and e = 2 are close to sinusoidal wave, and travel toward the negative direction with the constant amplitude and velocity. Also the motor can work normally with a high efficiency.

  3. (3)

    In other 18 conditions, the MMFs show obvious pulse vibration and movement with variable velocity, such as the MMFs under d = 1, e = 2. This means that the motor does not work normally, and instead it has high energy loss.