Analysis of core loss of three-phase induction motors equipped with concentric single-double layer star-delta winding

Abstract

Although star-delta winding can improve the efficiency of motors because of a higher fundamental winding coefficient and lower harmonic winding coefficient. The requirement of turns ratio is very strict. If the turns ratio is not ideal, the core loss of the motor cannot be reduced, thus the efficiency will not be improved. In this paper, a design method for converting a star-delta winding of stator in small- and medium-sized motor into the concentric single-double layer star-delta (CSDLSD) winding is introduced and compared with a traditional delta winding and a star-delta winding. Besides, a method is provided to calculate the winding coefficients of the motors with CSDLSD winding according to the turns number per slot. Harmonic component of magnetic flux density in stator tooth and yoke is consequently reduced, combined with partial core loss calculation method, the core loss is decreased. In addition, the core loss in three-phase induction machines equipped with front three winding types is also been estimated and analyzed. To check the analysis validity, two-dimension (2-D) Finite element method (FEM) is applied on three cage induction motors equipped with different winding types but the same prototypes.

Keywords

Induction machine CSDLSD winding winding coefficient core loss finite element method

1. Introduction

Many techniques for improving efficiency exist in the process of motor manufacturing, such as changing the thickness of the magnetic materials, changing the axial length of the motor and increasing the external diameter of the stator [1,2]. Reducing loss is the most widely applied and direct method. The concept of sinusoidal winding was proposed in [3] for the purpose to decrease harmonic content, so that a greater output can be obtained from a machine with a given frame. The theory of sinusoidal winding departed into two types from the 1990s [4–6], one type is combined star-delta three phases winding, the other is concentric low harmonic winding. The first reference on star-delta winding is the patent of Kothals issued in 1918. However, compared to an equivalent six phase winding, additional spatial harmonics are generated due to the different MMF in the star and delta part of the winding caused by the restriction of choosing turns number [7,8]. The theory of concentric low harmonic winding is that the distribution of conductors along the inner surface of the stator is sinusoidal, which means the current wave form will be closer to sine curve [9]. The star-delta winding gradually attracts increased attention. The influence of slots per pole pair was given in [10]. In [11], the authors discussed the losses especially copper loss and efficiency for the three-phase induction motors equipped with combined star-delta winding. A factor named harmonic leakage factor was proposed to illustrate how far away the winding is from the ideal sinusoidal winding. The waveform of phase current in this type of winding is not very closed to sinusoidal type, because the turns number can hardly obtain the exact value. The paper [12] analyses the influence of different spatial displacement and gives the conclusion that the harmonic components are the lowest when the displacement between the two parts is 30°. And in [13], the authors proposed a theory to create one layer combined windings with only one sort of coils by increasing the number of parallel branches. The theory and advantage of double layer concentric low harmonic winding were given in [9,14]. The influence of pole pairs was discussed in [12]. The different slot pitch was compared and the winding coefficient of different orders harmonic was showed in [9]. In [15], the authors provide a utility and accurate electrical loss model to calculate each type of loss, and with the help of 2-D FEM to affirm the conclusion.

In this paper, the authors proposed an improved theory by changing the way of connection and proportion of winding distributed per phase and per slots of both star-part and delta-part. To achieve the goal that the star-delta winding and the concentric low harmonics winding is combined together to reach a better performance. The air-gap flux density is much more closed to sine curve and the harmonics of electromotive force (EMF) and MMF are reduced. Then a quantificational analytical method is proposed to estimate the amplitude of core loss in three-phase induction motors equipped with three types of winding separately. 2-D FEM is also used to verify the correctness of the method proposed.

2. Analysis of concentric single-double layer star-delta winding characteristics

2.1. Basic characteristics of concentric single-double layer star-delta winding

Concentric single-double layer star-delta winding is based on the combined star-delta winding, so that the basic characteristics are similar to star-delta winding. Figure 1 gives the equivalent electrical circuit of a CSDLSD winding with associated current vectors and voltage vectors. In Fig. 1, E is the vector of an induced voltage and i is the vector of a phase current. The circuit is departed into two parts, the outer part of the winding is connected in star and the inner part of the winding is connected in delta. For the purpose to avoid more circulation, the two parts are connected in series instead of in parallel. There is a phase shift of 30 degrees between the star and delta part in the space vector. And combined star-delta winding is also called pseudo six-phase winding. With the increase of phase number, the lower harmonics such as 5th and 7th are suppressed in theory [16].

Fig. 1.

Equivalent circuit diagram of the CSDLSD winding.

Fig. 2.

Vector diagram of the CSDLSD winding.

Figure 2(a) illustrates the current vector diagram of the combined star-delta winding. The phase current in delta part is 30° advanced the phase current in star part. But the ratio between the amplitude of the phase current in two parts (I _Y∕I _Δ) is $\sqrt{3}$ . The formula of phase current of star part is: $\begin{eqnarray}\displaystyle i_{A_{Y}}={\dot{I}}\cdot \cos ({\omega}t+{\varphi}_{i});\quad i_{B_{Y}}={\dot{I}}\cdot \cos \left({\omega}t+{\varphi}_{i}-{\displaystyle \frac{2{\pi}}{3}}\right);\quad i_{C_{Y}}={\dot{I}}\cdot \cos \left({\omega}t+{\varphi}_{i}-{\displaystyle \frac{4{\pi}}{3}}\right), & & \displaystyle\end{eqnarray}$ (1) where I is the current amplitude, ω is the current angular frequency, t is the time and 𝜑_i is the initial phase angle. Applying the Kirchhoff’s current theorem to the connection points between star part and delta part, the phase current in the delta part are: $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}i_{A_{{\Delta}}}=\frac{{\dot{I}}}{\sqrt{3}}\cdot \cos \left({\omega}t+{\varphi}_{i}+{\displaystyle \frac{{\pi}}{6}}\right);\quad i_{B_{{\Delta}}}={\displaystyle \frac{{\dot{I}}}{\sqrt{3}}}\cdot \cos \left({\omega}t+{\varphi}_{i}-{\displaystyle \frac{2{\pi}}{3}}+{\displaystyle \frac{{\pi}}{6}}\right);\\ i_{C_{{\Delta}}}={\displaystyle \frac{{\dot{I}}}{\sqrt{3}}}\cdot \cos \left({\omega}t+{\varphi}_{i}-{\displaystyle \frac{4{\pi}}{3}}+{\displaystyle \frac{{\pi}}{6}}\right).\end{array} & & \displaystyle\end{eqnarray}$ (2)

Because the MMF is proportional to multiple phase current and turns number in series per phase I _ph N _ph, in order to guarantee the machine working regularly, the ampere turns of each phase must be balanced as presented by Fig. 2(b). Which means the ratio between the turns number per phase of two parts (N _Δ∕N _Y) should be $\sqrt{3}$ .

Hypothesizing that the distributed winding is symmetrical, the equivalent electrical parameters in delta connection are calculated in [17]: $\begin{eqnarray}\displaystyle R_{{\Delta}ph}={\displaystyle \frac{R_{Yph}}{3}},\quad X_{{\Delta}{\sigma},ph}={\displaystyle \frac{X_{Y{\sigma},ph}}{3}},\quad X_{{\Delta}m,ph}={\displaystyle \frac{X_{Ym,ph}}{3}}. & & \displaystyle\end{eqnarray}$ (3)

According to (3), the resistances and reactance in delta part is 1/3 times as large as them in star part. The copper wire diameter has relationship with the value of resistances and reactance. The rate of wire diameter in two parts (D _Y∕D _Δ) is $\sqrt{3}$ . Suitable line diameter and conductors per slot could control the spacer factor and thermal load inside the security perimeter.

The voltage of delta-part would be calculated according to star-part and the above parameters. $\begin{eqnarray}\displaystyle |\sqrt{3}{\dot{I}}_{BY}\text{e}^{\text{i}30^{\circ }}\cdot Z^{\ast }+\dot{U}_{B{\Delta}}+\sqrt{3}{\dot{I}}_{CY}\text{e}^{\text{i}150^{\circ }}\cdot Z^{\ast }|=U, & & \displaystyle\end{eqnarray}$ (4) where U is the amplitude of rated voltage; Ů _BΔ is the voltage of phase B; ${\dot{I}}_{BY}$ and ${\dot{I}}_{CY}$ are the amplitude of phase current in star part; e^iθ is the angle of voltage vector; Z ^∗ is the impedance of phase B in star part, which is calculated as: $\begin{eqnarray}\displaystyle Z^{\ast }=\sqrt{R_{BY}^{2}+(X_{m,BY}+X_{{\sigma},BY})^{2}}. & & \displaystyle\end{eqnarray}$ (5)

Assuming that the circuit is three-phase symmetric, both the amplitude of phase current, the impedance and the voltage are almost equal per phase. ${\dot{I}}_{CY}$ can be substituted by ${\dot{I}}_{BY}$ , and Z ₁ can represent impedance of phase B or phase C. On the basis of (4) and (5), the amplitude of voltage of phase B would be calculated. The value of amplitude of voltage per phase is calculated as $\sqrt{3}/2$ times as much as U.

Based on the phase voltage calculated above, the EMF can be calculated as: $\begin{eqnarray}\displaystyle E_{1}=(1-{\varepsilon}_{L}^{\prime })U_{N{\phi}}, & & \displaystyle\end{eqnarray}$ (6) where ϵ^′ _L is the estimation of the ratio of the impedance voltage drop to the rated phase voltage at the rated load of the stator winding. The factor (1-ϵ^′ _L) is usually ranged between 0.85 to 0.95 in the medium and small induction motors.

2.2. Unique characteristics of concentric single-double layer star-delta winding

The coil turns ratio in two parts acquiring $\sqrt{3}$ is the ideal situation. Once turns number per phase of one part be confirmed, of the other part could be calculated. In reality, the turns number cannot be accorded with sinusoidal law accurately, so the turns number can be only close to the value we calculated. If the value is accurate, in accordance with the rotating field theory, the higher harmonic orders of 𝜐 = 6k ± 1 with k = 0,1,2,3, … would be eliminated in star-delta winding. The approximation would cause series problems. The higher orders harmonic cannot be eliminated, they can only be weakened. Meanwhile the unbalanced ampere turns may cause harmonics of other orders [11]. In regard to these problems, the concentric low harmonics winding is proposed. By distributing the conductors around the inner surface of stator according to sinusoidal law, the harmonic content produced by this type of winding could be further weakened. However, this type of winding would reduce the fundamental wave simultaneously, so that the fundamental winding coefficient would decrease.

CSDLSD winding is proposed to combine the advantages of two front types of winding and reduce their disadvantages.

Fig. 3.

CSDLSD winding expansion with two poles.

Figure 3 displays the winding expansion of CSDLSD winding. Two types of line on the middle part in the figure, the active lines indicate they are single layer and the upper layer of double layer, and the dotted lines show they are lower layer of double layer. The winding of each phase is followed the concentric method, the conductors of each slot is calculated to satisfy the conditions of both two types of winding.

Fig. 4.

Winding layouts of (a) Star-delta winding. (b) CSDLSD winding.

In Fig. 4, the different distributions between the two types of winding are displayed clearly. Different kinds of colors indicate different phases, phase A in orange, phase B in green and phase C in purple. Different depths of color show different parts, the star delta displayed by deeper color. Different shapes make clear different layers, the round shape expresses single layer and the oval shape indicates double layer. According to the above two figures, the winding coefficient could be reckoned.

Let a _Y = a _Δ and $N_{{\phi}{\rm\Delta}}=\sqrt{3}N_{{\phi}Y}$ . The most ideal case for single-delta layer star-delta winding is: $\begin{eqnarray}\displaystyle q=2(2k+1),\quad k=1,2,3,\ldots , & & \displaystyle\end{eqnarray}$ (7) where q is the number of slots per pole and phase calculated by Z ₁∕2pm; m is the number of phase; Z ₁ is the number of slots in stator and p is the number of pole pairs. $\begin{eqnarray}\displaystyle q_{{\Delta}}=q_{Y}={\displaystyle \frac{q}{2}}=2k+1,\quad k=1,2,3,\ldots . & & \displaystyle\end{eqnarray}$ (8)

Because q is even, the number of slots per pole and phase of star part and delta part is equal to the half of q. $\begin{eqnarray}\displaystyle {\alpha}={\displaystyle \frac{p\cdot 360^{\circ }}{Z_{1}}}, & & \displaystyle\end{eqnarray}$ (9) where 𝛼 is the pitch, the axis of two adjacent slots in stator. $\begin{eqnarray}\displaystyle \displaystyle k_{dpY}={\displaystyle \frac{1}{\displaystyle \mathop{\sum }_{i=1}^{\frac{q}{2}-1}N_{Y}+{\displaystyle \frac{N_{Y}}{2}}}}\left[\mathop{\sum }_{i=1}^{\frac{q}{2}-1}N_{Y}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }\right)+{\displaystyle \frac{N_{Y}}{2}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }+{\alpha}\right)\right] & & \displaystyle\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \displaystyle k_{dp{\Delta}}={\displaystyle \frac{1}{\displaystyle \mathop{\sum }_{i=1}^{\frac{q}{2}-1}N_{{\Delta}}+{\displaystyle \frac{N_{{\Delta}}}{2}}}}\left[\mathop{\sum }_{i=1}^{\frac{q}{2}-1}N_{{\Delta}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }\right)+{\displaystyle \frac{N_{{\Delta}}}{2}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }-{\alpha}\right)\right], & & \displaystyle\end{eqnarray}$ (11) where k _dpY and k _dpΔ are the winding coefficient of star part and delta part, y ₁ is the pitch and N _Y is the turn number per slots. N _Y is proportional to the cosine value of the angle between the axis of each phase and the slots. According to the theory of MMF, turn number per phase in delta part N _𝜙Δ is $\sqrt{3}$ times of that in star part N _𝜙Y. However, limited by the reality, N _𝜙Δ is the closest number to the value $\sqrt{3}N_{{\phi}Y}$ . The winding coefficient for the machine we obtained are: 0.9861 for the star part and 0.9816 for the delta part.

While q is odd, q _Y and q _Δ are made as equal as possible, i.e.: $\begin{eqnarray}\displaystyle q_{{\Delta}}=(q\pm 1)/2;\quad q_{Y}=(q\mp 1)/2. & & \displaystyle\end{eqnarray}$ (12)

The winding coefficient can be calculated as: $\begin{eqnarray}\displaystyle \displaystyle k_{dpY} & = & \displaystyle {\displaystyle \frac{1}{\mathop{\sum }_{i=1}^{\frac{q}{2}-2}N_{Y}+N_{Y}}}\left[\mathop{\sum }_{i=1}^{\frac{q}{2}-2}N_{Y}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }\right)+{\displaystyle \frac{N_{Y}}{2}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }-{\alpha}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.+∼{\displaystyle \frac{N_{Y}}{2}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }+{\alpha}\right)\right],\end{eqnarray}$ (13) and (11) when q _Y is odd and q _Δ is even; or: $\begin{eqnarray}\displaystyle k_{dp{\Delta}} & = & \displaystyle {\displaystyle \frac{1}{\mathop{\sum }_{i=1}^{\frac{q}{2}-2}N_{{\Delta}}+N_{{\Delta}}}}\left[\mathop{\sum }_{i=1}^{\frac{q}{2}-2}N_{{\Delta}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }\right)+{\displaystyle \frac{N_{{\Delta}}}{2}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }-{\alpha}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.+∼{\displaystyle \frac{N_{{\Delta}}}{2}}\sin \left({\displaystyle \frac{2py_{1}}{Z_{1}}}\cdot 90^{\circ }+{\alpha}\right)\right],\end{eqnarray}$ (14) and (10) when q _Δ is odd q _Y and is even.

While q =2(2k), k =1,2,3, …, the coefficient would be (13) and (14).

The whole winding combination is excited by a three-phase symmetrical current. Assume that phase A is located at a spatial angle of θ, so that the MMF of a 𝜐th harmonic in the Y - and Δ-connected windings are: $\begin{eqnarray}\displaystyle f_{Y{\upsilon}}({\theta},t) & = & \displaystyle F_{{\phi}Y{\upsilon}}\cos {\upsilon}{\theta}\cos {\omega}t+F_{{\phi}Y{\upsilon}}\cos {\upsilon}\left({\theta}-{\displaystyle \frac{2{\pi}}{3}}\right)\cos \left({\omega}t-{\displaystyle \frac{2{\pi}}{3}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼F_{{\phi}Y{\upsilon}}\cos {\upsilon}\left({\theta}+{\displaystyle \frac{2{\pi}}{3}}\right)\cos \left({\omega}t+{\displaystyle \frac{2{\pi}}{3}}\right)\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle f_{{\Delta}{\upsilon}}({\theta},t) & = & \displaystyle F_{{\phi}{\Delta}{\upsilon}}\cos {\upsilon}\left({\theta}+{\displaystyle \frac{{\pi}}{6}}\right)\cos \left({\omega}t+{\displaystyle \frac{{\pi}}{6}}\right)+F_{{\phi}{\Delta}{\upsilon}}\cos {\upsilon}\left({\theta}-\frac{{\pi}}{2}\right)\cos \left({\omega}t-{\displaystyle \frac{{\pi}}{2}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼F_{{\phi}{\Delta}{\upsilon}}\cos {\upsilon}\left({\theta}+{\displaystyle \frac{5{\pi}}{6}}\right)\cos \left({\omega}t+{\displaystyle \frac{5{\pi}}{6}}\right).\end{eqnarray}$ (16) According to [18], in ((11)) and ((12)), the MMF coefficients can be defined by: $\begin{eqnarray}\displaystyle F_{{\phi}Y{\upsilon}}={\displaystyle \frac{2\sqrt{2}}{{\pi}}}{\displaystyle \frac{Sq_{Y}N_{Y}}{\sqrt{3}{\upsilon}a_{Y}}}Ik_{dpY{\upsilon}} & & \displaystyle\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle F_{{\phi}{\Delta}{\upsilon}}={\displaystyle \frac{2\sqrt{2}}{{\pi}}}{\displaystyle \frac{Sq_{{\Delta}}N_{{\Delta}}}{\sqrt{3}{\upsilon}a_{{\Delta}}}}Ik_{dp{\Delta}{\upsilon}}. & & \displaystyle\end{eqnarray}$ (18) The above expression points out that F _𝜙𝜐 varies directly with k _dp _𝜐. That is the reason why the harmonic MMF can be reduced if the harmonic winding coefficient decreases.

The magnetic flux can be calculated by E and K _dp. $\begin{eqnarray}\displaystyle {\Phi}={\displaystyle \frac{E}{4K_{Nm}K_{dp}fN}}, & & \displaystyle\end{eqnarray}$ (19) where K _Nm is the magnetic field waveform factor, which is equal to 1.11 and not decreased until the magnetic circuit reaching saturation. N is the coil turns number in per phase. And f is the power frequency, the value of f is 50 Hz.

The MMF is added by magnetic pressure drop on each magnetic circuit. The magnetic pressure drop of the air-gap F _δ is the largest proportion in each magnetic circuit, which can be used to calculate the magnetic flux density in air-gap B _δ. $\begin{eqnarray}\displaystyle B_{{\delta}}={\displaystyle \frac{{\Phi}}{{\alpha}_{p}^{\prime }{\tau}l_{ef}}}={\displaystyle \frac{{\mu}_{0}F_{{\delta}}}{K_{{\delta}}{\delta}}}. & & \displaystyle\end{eqnarray}$ (20)

The parameter 𝜇₀ is the air permeability, which is usually equal to 0.4πE-6. K _δ is the air-gap factor. And δ is the air-gap length. Where τ is the pole pitch, l _ef is the armature calculated length, which is equal to the core length if the edge effect is not been considered. $\begin{eqnarray}\displaystyle {\alpha}_{p}^{\prime }={\displaystyle \frac{{\displaystyle \frac{1}{{\tau}}}\displaystyle \int _{-\frac{{\tau}}{2}}^{\frac{{\tau}}{2}}B(x)dx}{B_{{\delta}}}}={\displaystyle \frac{B_{{\delta}av}}{B_{{\delta}}}}={\displaystyle \frac{1}{F_{s}}}. & & \displaystyle\end{eqnarray}$ (21)

The factor 𝛼^′ _p is the calculated pole embrace, which is the ratio of average and the maximum flux density when the motor is not slotted. 𝛼^′ _p is decided by the shape of MMF distribution curve, air-gap uniformity and the magnetic saturation degree. Its reciprocal is the amplitude coefficient. $\begin{eqnarray}\displaystyle B_{x}={\displaystyle \frac{F_{s}{\Phi}}{A_{x}}}. & & \displaystyle\end{eqnarray}$ (22)

The variable x can be chosen from stator yoke, stator tooth, rotor yoke and rotor tooth. The flux density B _x is changed following the selected area A _x.

3. Calculation of core losses in the stator

A 380 V, 2 poles three induction motor with rated power 22 kW is analyzed in this paper. Table 1 gives the data of the prototype squirrel cage induction motor.

Table 1
Main characters of prototype motor

Parameters Value

Rated power (kW) 22

Rated voltage (V) 380

Rated frequency (Hz) 50

Number of poles 2

Core length (mm) 175

Air-gap length (mm) 0.8

Stator slot number 48

Rotor slots number 44

Stator outer diameter (mm) 290

Rotor outer diameter (mm) 158.4

Parameters	Value
Rated power (kW)	22
Rated voltage (V)	380
Rated frequency (Hz)	50
Number of poles	2
Core length (mm)	175
Air-gap length (mm)	0.8
Stator slot number	48
Rotor slots number	44
Stator outer diameter (mm)	290
Rotor outer diameter (mm)	158.4

The core loss of the silicon steel mainly results from the hysteresis effect and eddy current effect.

The circumstances for flux density in yoke and tooth are not the same, so that the following formula is applied in analytical method to obtain more accurate results. $\begin{eqnarray}\displaystyle P_{Fe}=k_{a}\cdot G\cdot ({\sigma}_{h}fB^{2}+{\sigma}_{c}f^{2}B^{2}), & & \displaystyle\end{eqnarray}$ (23) where k _a is an experience coefficient which is content the error caused by processing, unevenly distributing of flux density, variating with time, rotation magnetizing etc. The value of coefficient k _a is 1.5 of yoke and 1.8 of tooth in induction motor. G is the weight of yoke or tooth, considering the motor we simulated, the value of G is given by Table 2. The factors σ_h and σ_c are the hysteresis loss coefficient and eddy current loss coefficient per unit of weight, generally they are given with silicon steel sheet type. B means magnetic flux density. When calculate B of yoke part, the maximum value is chosen, while of tooth part, the average value of flux density along the tooth path is put to use.

Table 2

Magnetic density corresponding to the weight of each part

	Stator yoke	Stator tooth	Rotor yoke	Rotor tooth
G (kg)	40.92	7.32	11.37	4.96
B (T)	1.396	1.718	1.342	1.65

While in FEM, the method to calculate the core losses using the Bertotti model: $\begin{eqnarray}\displaystyle P=k_{h}fB_{m}^{2}+k_{c}(fB_{m})^{2}+k_{e}(fB_{m})^{1.5}, & & \displaystyle\end{eqnarray}$ (24) where B _m is the flux density of order m, k _h is the coefficient of hysteresis loss caused by alternating magnetic field, k _c is the coefficient of eddy current loss rose from the eddy effect and k _e is the coefficient of excessive loss. Usually the loss coefficients can be obtained from the B-P curves of the material. In this paper, the type of silicon steel is DW540-50. The coefficients k _h, k _c, k _e are 268 w/m³, 0.822 w/m³ and 0 w/m³ respectively, given by the software Maxwell straightly. So decreasing harmonic magnetic flux density is an effective way to reduce the core loss in stator.

In order to calculate the core loss of an induction machine in rated condition s ≈ 0.036, a method combined FEM with analytical method is used to figure out the core losses of different harmonics. On the one hand, Fig. 5 shows the stator tooth distribution of the whole machine. In order to calculate the specific core loss for each part, the stator is divided into 38 units. Firstly, divide the whole stator into two areas on the basis of yoke and tooth as the magnetic paths in them are different. We can consider the core loss in half stator for the reason that the absolute value of magnetic flux density under both poles is almost equal.

Fig. 5.

Distribution of flux density: (a) traditional winding, (b) CSDLSD winding.

Then separate the upper half area into units by each slot centerline to simulate the core loss more detailed. The flux density inside both areas of the stator is considered to be at the same level. On the other hand, Fig. 6 compares the flux density in stator between the traditional winding and the CSDLSD winding. According to the figure, the flux density in CSDLSD winding is more balanced than in traditional winding and the risk of high thermal load is reduced. The optimized thermal load indicates the lifetime of machine could be extended.

Fig. 6.

Comparison of air-gap induced voltage of three types of winding.

The induced voltage or EMF is an evaluation criterion for the harmonic content. As displayed in Fig. 6, the induced voltage of machine with three types of winding is near. But it is clear from the figure that the EMF of the traditional winding is the lowest, and of the star-delta winding is the highest. Nevertheless, the EMF of the CSDLSD winding is the smoothest curve of the three. The closer the EMF curve to sine curve, the less harmonic content it contains, which means the better performance it would achieve.

With the help of method named fast Fourier transform (FFT), the spectrum of the EMF is given by Fig. 7. A histogram is more intuitive than a graph to show the advantage of CSDLSD winding. Considering the distribution of the value, the vertical axis is chosen to follow logarithmic law. From the figure we can observe that the main content is almost the same when time comes to 50 ms. However, harmonic content is the least in CSDLSD winding. Because of the limitation of the turns number in realistic winding, the value of star-delta winding is not very steady, sometimes it is less than that of traditional winding, but more in other times. The spectrum of the EMF gives a reference before we consider the situation of magnetic flux density.

Fig. 7.

Spectrum of the EMF of a stator with 6 slots per pole pair and partial enlargement.

The advantage of flux density of CSDLSD winding on another aspect is displayed in Fig. 9. Among them, Fig. 9 shows the magnetic flux density in air-gap distributed by circumference. Comparing the three types of winding, the curve of traditional winding is the highest, the star-delta winding is lower than the front one, but at some point appears an uprush value as a result of its unbalanced MMF, CSDLSD winding is much closer to sine curve than the front two types of winding. In the transition between the two poles, the magnetic density of the CSDLSD winding is also relatively stable, without too much mutation.

Fig. 8.

Comparison of air-gap magnetic flux density of three types of winding.

The sinusoidal degree expresses the content of higher harmonic content in the flux density such as 5th, 7th, and so on. More details are provided by Fig. 8, the width of flux density curve explains the magnitude of harmonic content. The traditional winding is the widest. The star-delta winding is a little narrower than traditional one on most points, but it is wider on some points especially near the junction of the two poles. Compared with the other two types of winding, the curve of CSDLSD winding is the narrowest, which illustrates the harmonic content is the least among the three winding types.

Fig. 9.

Comparison of spectrum of the flux density of a stator with 6 slots per pole pair with different winding types.

By the aid of FFT, harmonic components can be separated out from flux density as given in Fig. 9, and the data in Table 2. describe the harmonic factors in detail. The ideal situation is higher fundamental content and lower harmonic content. According to the figure, the components of harmonic in CSDLSD winding are ore significant decrease compared with star-delta winding on the basis of traditional winding. However, the change of winding type has little effect on tooth harmonics. The main effect of the change is on the stator magnetic potential phase band harmonics. The terms with 𝜈 = kZ∕p ±1 (k = 1, 2, 3, …) orders are named as the tooth harmonics, which is almost only related to the tooth slot parameters of stator and rotor.

Table 3

Amplitude of three types winding

Orders	Traditional	Star-delta	Single-double layer Sd
1	0.887	0.929	0.959
5	0.063	0.040	0.031
7	0.049	0.035	0.031
11	0.012	0.010	0.010
13	0.015	0.008	0.006
17	0.010	0.022	0.012
19	0.016	0.012	0.002
23	0.010	0.008	0.005
25	0.051	0.043	0.029
29	0.142	0.122	0.118
31	0.037	0.033	0.027
35	0.228	0.227	0.226
37	0.213	0.210	0.210

The curve shown by Fig. 10 is the comparison among core loss of three types of winding when the motor is stable. During the starting stage, the amplitude of core loss could reach 1.25 kW. However, as the motor operation is tending to stationary, the amplitude of the core loss is limited between 0.25 kW to 0.385 kW. From the figure, we can catch that the core loss when the motor running safely and stably is fluctuated up and down near a value. So partial enlargement is used for total core loss from 200 ms to 500 ms to explore the details. By means of smoothing analysis, obvious results among the three types of winding are given. The average values of core loss of traditional, star-delta and CSDLSD winding are 0.28869 kW, 0.32574 kW and 0.36215 kW respectively. The least core loss curve belongs to the proposed CSDLSD winding.

Fig. 10.

Comparison of total core loss of three types of winding.

Total core loss is the summation of each unit. The difference between the core loss amplitude of yoke and tooth under each tooth is about 5W. Compared to traditional winding, although total core loss of star-delta winding is diminished, the value might not be very steady. Figure 11 reveals that when focus on particular unit, the core loss amplitude of star-delta winding would uprush to even 2 times of normal value. But the problem is solved in CSDLSD winding, there are little spines on the wave. What is more the amplitude of core loss of each yoke or tooth of the proposed winding is the lowest among the three types of winding.

Fig. 11.

Comparison of core loss of yoke 4 of three types of winding and partial enlargement with smooth analysis.

To explore the core loss of each yoke unit and tooth unit, the simulation of the motor on 2-D FEM has been repeated 108 times to get the data. Each time we select one of the 36 units on the upper side of the motor stator to calculate the core loss, and select the maximum point after it is stable. It is shown by Fig. 12(a),(b) that the core loss of stator yoke and tooth varies with the uniform sampling position along the inner circumference of the stator under the same pole. Hypothesis that the magnetic flux density of induction motor is symmetrical under a pair of poles, so only half side of motor stator is chosen in this simulation. The value of core loss of stator yoke is taken the average, whereas the value of core loss of stator tooth is taken the maximum within the simulation time.

Fig. 12.

Comparison of core loss of three types of winding.

Compare with the core loss of tooth in each motor, the change of core loss of yoke is relatively mild. The average value of core loss of unit yoke is 15.512 W, 14.089 W and 13.116 W of three different types of winding respectively. When equipping star-delta and CSDLSD winding, compared with the traditional double layer short distance winding, the reduction of iron bar is 9.17% and 15.44%.

Fig. 13.

Comparison of core loss of three types of winding under different load torque.

The variable range of core loss of stator tooth is rather wide, the difference between the maximum and the minimum is about 1 W. The general trend of Fig. 12 is that the core loss on each unit of CSDLSD winding is the lowest in three types of winding, and traditional winding is the highest.

Another message passed by Fig. 13 is that the variation of core loss takes about six units as a period, which is exactly equal to q (the number of slots per phase per pole). The change of core loss on tooth unit is larger than that on yoke unit. With the aid of 2-D FEM, the regulation on the way the core loss of stator varies with the load torque is studied. Measurement of stator core loss of 2-D models of three winding types from no-load to full-load each additional 10NM as load steps is displayed in Fig. 14. As the load increases, the difference of stator core loss of the three winding types is almost unchanged. Although the value of core loss of stator is increased with the raise of load torque and the trend of change slowed down with the load torque gradually approaching to full load. In the process of increasing the load torque, the stator core loss of proposed CSDLSD winding is always the smallest one.

Fig. 14.

Flux density of stator of three types of winding under different load torque.

To explore the relationship between flux density of stator and load torque, a simulation is modeled by 2-D FEM. The result is described by Fig. 14 that the flux density of each area is positive correlation with the load torque. Either the average value of the yoke flux density or the maximum value of the tooth flux density, the larger the winding coefficient is, the smaller the harmonic flux density resulted in. The magnetic flux density harmonics in the middle position of the stator teeth are the largest, followed by the stator toothtip, and the minimum amplitude of the magnetic density harmonics in the stator yoke.

Table 4

Parameters of prototype motor

Loss (W)	Traditional	Star-delta	CSDLSD
Copper Loss S	575.783	637.293	602.066
Copper Loss R	697.285	777.702	736.549
Core loss	518.362	446.906	418.202
Fric. & Wind.	50.3059	49. 6344	49. 4394
Stray Loss	220	220	220
Efficiency (%)	90.432	91.8996	92.115
Power factor	0.9153	0.9329	0.9653

The results we got from 2-D FEM support the analytical in [9], that the copper loss of star-delta winding is lower than that of traditional winding with the same prototype motor. The copper loss of CSDLSD winding in the simulation is about similar level to the star-delta winding. However, with the lower core loss the efficiency and power factor of proposed winding are the best in three types of winding as shown in Table 3. The efficiency and power factor of proposed winding are promoted 0.683% and 0.276 compared to traditional winding. The date would be 0.2454% and 0.0024 when compared to star-delta winding.

4. Conclusion

This paper proposes an improved type of motor winding named concentric single-double layer star-delta (CSDLSD) winding. And presents an analysis method for the analytical calculation of the winding coefficients of the promoted winding according to the turns number and the end connection. The amplitudes of the harmonics in the stator tooth and the yoke are estimated by analytical method. The calculation of the winding coefficients shows that additional harmonics (5p, 7p, 17p, 19p) of CSDLSD windings are decreased by 35%–40% compared with traditional windings, and 10–25% compared with star-delta winding. Based on the winding coefficients of each phase, the electrical parameters in each phase of the star-delta winding are determined. The stator is divided into small units by tooth and yoke under each tooth area. The tendency reveals that because of the lowest harmonic component of air-gap magnetic field, the lowest value of stator core loss belongs to CSDLSD winding, and the highest one pertains to traditional winding. The 2-D FEM simulation results verify the correctness of the method proposed. The result reveals that as the flux density approaching to sine curve, the core loss is diminished. And with the comparison of the motors equipped with three types of winding under different load torque, the core losses are increased according to the increasing load, both the tooth part and the yoke part. It lays a foundation for further investigation for stray losses in motors equipped with proposed winding.

Footnotes

Acknowledgements

Supported by the National Nature Science Foundation of China (51677051).

References

Dems

Komeza

and Rodriguez

H.G.

, Methods for increasing the efficiency of an asynchronous motor with increased speed fed from the PWM inverter, International Journal of Applied Electromagnetics and Mechanics (IJAEM) 57(S1) (2018), 61–71.

Nakazaki

Kai

Todaka

and Enokizono

, Iron loss properties of a practical rotating machine stator core at each manufacturing stage, International Journal of Applied Electromagnetics and Mechanics (IJAEM) 33(1-2) (2010), 79–86.

Huges

, B.Sc, Ph.D. New 3-phase winding of low mmf harmonic content, Proc. of IEE 117(8) (1970), 1657–1666.

Auinger

, Combined star-polygon plurl phase winding for an electrical machine. European Patent EP0 557 809 BI, September 1993.

Chen

J.Y.

and Chen

C.Z.

, A low harmonic, high spread factor induction motor, in: International Conference on Power Electronic Drives and Energy Systems for Industrial Growth, 1998. Proceedings, Vol. 1, 1998, pp. 129–134.

Kasten

and Hofmann

, Electrical machines with higher efficiency through combined star-delta windings, in: 2011 IEEE International Electric Machines Drives Conference (IEMDC), 2011, pp. 1374–1379.

Kothals-Altes

, Motor winding, U.S. Patent 1 267 232, May 21, 1918.

Cistelecan

M.V.

Ferreira

F.J.T.E.

and Cosan

H.B.

, Generalized MMF space harmonica and performance analysis of combined multiple-step, star-delta, three-phase windings applied on induction motors, in: 2008 18th International Conference on Electrical Machines. IEEE Conference, 2008, pp. 1–5.

Zhang

Huang

Y.K.

Dong

J.N.

Guo

B.C.

and Zhou

, Stator Winding Design of Induction Motors for High Efficiency, in: Electrical Machines and Systems (ICEMS) 2014, 17th International Conference on, 2014, pp. 130–134.

10.

Kasten

and Hofmann

, Electrical machines with higher efficiency through combined star-delta windings, in: 2011 IEEE International Electric Machines Drives Conference (IEMDC), 2011, pp. 1374–1379.

11.

Misir

Raziee

S.M.

Hammouche

Klaus

Kluge

and Ponick

, Prediction of losses and efficiency for three-phase induction machines equipped with combined star-delta windings, IEEE Trans. Ind. Appl., IEEE Journals Magazines 53(4) (2017), 3579–3587.

12.

Cistelecan

M.V.

Ferreira

F.J.T.E.

and Popescu

, Adjustable flux three-phase AC machines with combined multiple-step star-delta winding connections, IEEE Trans. Energy Convers. 25(2) (2010), 348–355.

13.

Kasten

and Hofmann

, Combined stator windings in electric machines with same coils, in: Electrical Machines 2012 12th International Conference on IEEE, 2012, pp. 103–108.

14.

Xiong

and Wang

X.F.

, Design of a low-harmonic-content wound rotor for the brushless doubly fed generator, IEEE Trans. Energy Convers. 29(1) (2014), 158–168.

15.

Yuan

L.Q.

Liu

Zhao

Z.M.

and Lu

, A utility and accurate electrical loss model and application for induction motors utilizing 2-D finite element analysis, in: IEEE Energy Conversion Congress and Exposition (ECCE), 2015, pp. 285–291.

16.

Misir

and Ponick

, Analysis of three-phase induction machines with combined star-delta windings, in: 2014 IEEE 23rd International Symposium on Industrial Electronics (ISIE), 2014, pp. 756–761.

17.

Misir

Raziee

S.M.

Hammouche

Klaus

Kluge

and Ponick

, Calculation method of three-phase induction machines equipped with combined star-delta windings, in: Electrical Machines (ICEM), 2016 16th International Conference on, 2016, pp. 166–172.

18.

Lei

Zhao

Z.M.

Wang

S.P.

Dorrell

D.G.

and Xu

, Design and analysis of star-delta hybrid windings for high-voltage induction motors, IEEE Trans. Ind. Electron. 58(9) (2011), 3758–3767.

Analysis of core loss of three-phase induction motors equipped with concentric single-double layer star-delta winding

Abstract

Keywords

1. Introduction

2. Analysis of concentric single-double layer star-delta winding characteristics

2.1. Basic characteristics of concentric single-double layer star-delta winding

Table 1 Main characters of prototype motor Parameters Value Rated power (kW) 22 Rated voltage (V) 380 Rated frequency (Hz) 50 Number of poles 2 Core length (mm) 175 Air-gap length (mm) 0.8 Stator slot number 48 Rotor slots number 44 Stator outer diameter (mm) 290 Rotor outer diameter (mm) 158.4

Footnotes

Acknowledgements

References