Comparative analysis of current under static and dynamic eccentricity in 3-phase induction motors on parallel branches

Abstract

Eccentricity problem is always ineluctable in rotating electric machine, especially in large electric machine. Hence, the influence of eccentricity in operating machine is necessary to be investigated. According to the investigation, current of different parallel branches are varied under the same eccentricity. This paper focuses on comparative current analysis on two types of parallel branches in consideration of static and dynamic eccentricities. Parallel branch model in circumferential position is established for analysis and explanation of branch current due to eccentricity. Stator winding current and rotor wingding current are worked out under static and dynamic eccentricity at two types of parallel branches. Their winding losses are calculated for comparison, respectively. Finite element model simulations are used for validation.

Keywords

Static and dynamic eccentricity parallel branch current analysis winding loss

1. Introduction

Eccentricity problem usually exists in large motors, especially in slender structures. It is necessary to make clear what harmful eccentricity brings on operating machines. According to previous investigations, much work has been done on the analysis of eccentricity.

2D eccentricity, which means the center of rotor is aligned in axial direction, is the mostly studied. 2D eccentricity could be static, dynamic, or even mixed (combination of static and dynamic eccentricity). In several types of motors, the eccentricity could have some negative effects on parameters such as the output torque, the stator current, the core loss, the damper bar current and back-EMF, as well as leading to an unbalance of magnetic force, etc. Previous work on this topic has been published by Zhu et al. [1], who investigated the influence of slot/pole number combination and winding configuration on the electromagnetic performance, such as flux density, back-EMF and electromagnetic torque of permanent magnet machines with static or rotating rotor eccentricities. Keller et al. [2], Dorrell [3] and Dorrell and Kayani [4], Jiang et al. [5], Kim et al. [6], Li et al. [7] have done the calculation of axial unbalanced magnetic force due to eccentricity in hydro-generator, induction motor, interior permanent magnet synchronous machine, BLDC motor and permanent magnet machine, respectively. This paper also shows a study the effect of series and parallel windings on stator current with eccentricity. Rezaee-Alam et al. [8] introduced an improved conformal mapping (ICM) method for the calculation and analysis of unbalanced magnetic force (UMF) in eccentric surface-mounted permanent magnet (SMPM) motors considering the static, dynamic, and mixed rotor eccentricities. Belahcen and Arkkio [9] investigated the effect of eccentric rotor on the power losses in an induction motor with PWM-voltage supply. Bangura and Demerdash [10] investigated the effect of air gap eccentricities on Ohmic and core losses of induction motors in ASDs using a coupled finite element-state space method. Wang et al. [11] and Bao et al. [12, 13, 14] monitored the magnetic field in consideration of slotting effect and current analysis under eccentricity fault.

Figure 1.

Spatial position of stator geometric center, rotor geometric center and rotation center. a) Healthy; b) Static eccentricity; c) Dynamic eccentricity.

This paper presents comparative current analysis of static and dynamic eccentricity on parallel branch a $=$ 1 and a $=$ 6 in induction motors. Relationship between branch current and air gap length is described. Magnetic motive force (MMFs) of stator poles and rotor poles are calculated with the help of finite element method. The stator phase current with different parallel branches is analytically proved to be reduced by eccentricity. Stator and rotor winding losses are calculated for measuring the change of stator and rotor current.

Figure 2.

Magnetic path of a 6-pole induction motor.

Figure 3.

Magnetic circuit of one pole in induction motor.

Figure 4.

Computing regions of stator poles and rotor poles.

Figure 5.

Flux linkage at different parallel branches and at different loads under healthy condition and 60% static eccentricity.

Figure 6.

Spatial position of coil groups of three phases. a) Phase A; b) Phase B; c) Phase C.

2. Types of eccentricities

Mixed eccentricity is the combination of static and dynamic eccentricity. To accomplish the analysis of mixed eccentricity, the static and dynamic eccentricities are individually considered in this paper. In this paper only the two-dimensional (2D) eccentricity is considered. This means that the relative position between $O_{1}$ (stator geometric center) and $O_{2}$ (rotor geometric center) or between $O_{1}$ and $O_{r}$ (rotation center) does not vary along the axial direction. It is noteworthy that these two types of eccentricities are the principal causes of the stator-rotor rub failure type. Relative positions between stator geometric center, rotor geometric center and rotation center under static and dynamic eccentricity are shown on Fig. 1.

2.1 Static eccentricity

For static eccentricity, rotor geometric center and rotation center overlap but there is deviation between stator geometric center and rotation center as shown in Fig. 1b. Relatively, $O_{1}$ (stator geometric center), $O_{2}$ (rotor geometric center) and $O_{r}$ (rotation center) are concentric in healthy motor shown in Fig. 1a. Since there is no deviation between rotor geometric center and rotation center, air gap lengths between stator inner circle and rotor outer circle are not varied with time when the rotor rotates, although the air gap varies in the azimuthal direction.

A relative polar coordinate system is defined starting from the position of minimum air gap length. As shown on Table 1, the deviation of $O_{1}$ and $O_{r}$ is named $e_{1}$ in static eccentricity. The radial distance between the rotor outer diameter and the stator inner diameter can be expressed in polar coordinates as follows: In reality, since $R_{2}$ (outer radius of rotor) is much larger than $\delta$ (healthy air gap length), air gap length at the circumference can be expressed as

$\displaystyle d=\delta-e_{1}\cos\theta$ (1)

Table 1

Types of eccentricities and their relative positions

Type of eccentricity	$O_{1}-O_{r}$	$O_{2}-O_{r}$
Static eccentricity	$e_{1}$	0
Dynamic eccentricity	0	$e_{2}$

Figure 7.

Phase current at a $=$ 1, a $=$ 6 and at no load, full load under different static eccentricity degrees. (a) at no load; (b) at full load.

Figure 8.

Branch currents of three phases under static eccentricity at no load. (a) branch currents of phase A under static eccentricity; (b) branch currents of phase B under static eccentricity; (c) branch currents of phase C under static eccentricity.

Figure 9.

Branch currents of three phases under static eccentricity at full load. (a) branch currents of phase A under static eccentricity; (b) branch currents of phase B under static eccentricity; (c) branch currents of phase A under static eccentricity.

where: $\theta=$ angle between the direction corresponding to the minimum air gap and a generic air gap along the azimuthal direction in the polar system, as shown on Fig. 1b; $\delta=$ ideal air gap corresponding to the healthy condition.

2.2 Dynamic eccentricity

For dynamic eccentricity, stator geometric center and rotation center overlap but there is deviation between rotor geometric center and rotation center as shown in Fig. 1c. Since rotor rotates in eccentric case with respect to the deviation between $O_{2}$ and $O_{r}$ , air gap lengths between stator inner circle and rotor outer circle are varied with time when rotor rotates.

The deviation of $O_{2}$ and $O_{r}$ is named $e_{2}$ in dynamic eccentricity. Air gap length expression of dynamic eccentricity is quite similar to that of static eccentricity because for every transient time instant, air gap situation of dynamic eccentricity is equivalent to that of static eccentricity with the same displacement, as shown on Fig. 1. The only difference is the minimum air gap position (relative to stator). The dynamic eccentricity in fact varies when rotor rotates, therefore a reference $\rho$ axis is defined. Azimuthal distribution of air gap length for dynamic eccentricity can be therefore expressed as

$\displaystyle d=\delta-e_{2}\cos(\omega_{r}t+\theta_{0})$ (2)

where: $\omega_{r}$ is angular velocity of rotor; $\theta_{0}$ is the included angle between reference $\rho$ axis and the position of minimum air gap length; $t$ is the time.

The eccentric degree can also be defined as:

$\displaystyle\xi=\frac{d_{\max}-\delta}{\delta}$ (3)

where $d_{\max}$ refers to the maximum air gap length.

3. Comparative analysis of different parallel branches under eccentricities

Table 2
Parameters of simulated motor

Quality	Simulated motor
Nominal power	75 kW
Supply voltage	380 V
Connection	Delta
Frequency	50 Hz
Number of rotor slots	66
Number of stator slots	72
Number of poles	6
Winding layer	2
Air gap length	2.5 mm

Table 3

Parameters of rotor resistances

Resistance	$\times$ 10 ${}^{-5}$ ( $\Omega$ )
RB	4.16
RR	0.016
R2	4.555

Figure 10.

Comparison of stator winding loss under static eccentricity. (a) at no load; (b) at full load.

Figure 11.

RMS values of Rotor MMF. (a) a $=$ 1 at no load; (b) a $=$ 6 at no load; (c) a $=$ 1 at full load; (d) a $=$ 6 at full load.

Figure 12.

Comparison of rotor winding loss under static eccentricity. (a) at no load; (b) at full load.

Figure 13.

Comparison of winding loss under static eccentricity. (a) at no load; (b) at full load.

Eccentricity breaks the even distribution of air gap length between stator and rotor. For rotating magnetic pole in induction motor, air gap reluctance in the magnetic circuit shown on Fig. 2 becomes varied due to eccentricity. However, to satisfy constant load operation, main magnetic flux must not change and properly defined by the motor design parameters, whereas the variation of magnetic reluctance induces a variation of excitation magnetic motive force (MMF). For induction motor, excitation MMF is affected by magnetic potential from stator winding current and magnetic potential from rotor bar current, therefore eccentricity could have an impact on stator current and rotor current. According to the investigation of Jiang et al. [5], currents of stator coil groups remain unchanged when stator windings are connected in series. Currents of stator coil groups are a little different due to slight inductances change (produced by uneven spatial air gap distribution) of each coil group when stator windings are parallel connected. On the other end, eccentricity would make no difference to stator phase current. When stator current remains unchanged, rotor current only could induce some distortion of the magnetic reluctance because of eccentricity. In this paper, current analysis of various 2D eccentricities is taken as the main focus.

With the support of finite element simulation, a 2D model of a 6-pole induction motor is created and computed in Maxwell 2D transient module. Parameters of the simulated motor are listed in Table 2. Figure 2 shows the magnetic path of flux in the induction motor as computed by the code. It is noteworthy that the outer loop of one-pole flux is the largest. For effective and convenient investigation, this paper focuses only on the outer loop of one-pole flux. Accordingly, magnetic circuit model of the outer loop is shown on Fig. 3. This figure illustrates the reluctances of silicon steel material in the yellow frames and and reluctances of air in the green frames. Subscript numbers 1, 2 of reluctances in Fig. 3 mean stator and rotor, respectively; Subscript $t$ , $y$ and $a$ of reluctances respectively denote tooth, yoke and air gap; $F_{1}$ and $F_{2}$ are MMFs of stator pole and rotor pole. In addition, in consideration that it is unnecessary to work out various reluctances in silicon steel sheet, total reluctance $R_{s}$ in silicon steel sheet is substituted in the formula of the magnetic flux:

$\displaystyle\Phi=\frac{F_{1}+F_{2}}{R_{s}+2R_{a}}=\frac{F_{m}}{R_{s}+2R_{a}}$ (4)

In Fig. 4, the motor is divided equally into six-pole regions. Regions of six different colors contain six rotor poles. These pole regions do not move along with rotating rotor. Each of these regions are associated to a MMF by Eq. (5).

$\displaystyle F_{p}=\int\!\!\!\int\limits_{S}{J_{z}dS}$ (5)

where: $J=$ current density; $S=$ color region.

MMFs of six stator poles and six rotor poles under two different types of eccentricity are calculated to present how eccentricity influences the currents in running induction motor. Since eccentricity has a big impact on rotor current according to the investigation, rotor phase resistance is brought in to measure rotor bar loss under eccentricities.

Rotor bar current and end ring current satisfy Eq. (6), where $\alpha$ is electrical angle between two adjacent slots. Based on rotor bar current, rotor phase resistance Eq. (8) is calculated from Eqs (6) and (7). Coefficient $k$ Eq. (9) is introduced for the calculation of single rotor bar current. Results of resistances of the simulated motor are listed in Table 3. Rotor electrical loss can be worked out by Eq. (10), in which $p$ is pole pairs and $Z_{2}$ is rotor slot numbers, $F_{2j}$ is MMF of the $j$ -th rotor pole.

$\displaystyle I_{R}=\frac{I_{B}}{2\sin(\textstyle{\alpha\over 2})}$ (6) $\displaystyle I_{B}^{2}R_{2}=I_{B}^{2}R_{B}+2I_{R}^{2}R_{R}$ (7) $\displaystyle R_{2}=R_{B}+2R_{R}\left({\frac{I_{R}}{I_{B}}}\right)^{2}$ (8) $\displaystyle\begin{array}[]{l}k=1+2\cos(p\alpha)+2\cos(2p\alpha)+2\cos(3p% \alpha)+2\cos(4p\alpha)+2\cos(5p\alpha)\\ \end{array}$ (9) $\displaystyle P_{\textit{loss}}=\frac{1}{T}\left({\int_{0}^{T}{\sum\limits_{j=% 1}^{2p}{\left({\frac{F_{2j}}{k}}\right)^{2}}}\cdot dt}\right)\cdot R_{2}\cdot% \frac{Z_{2}}{2p}$ (10)

Figure 14.

Phase current at a $=$ 1, a $=$ 6 and at no load, full load under different dynamic eccentricity degrees. (a) at no load; (b) at full load.

Table 4

Comparative parameters of stator winding

	Parallel branch
	a $=$ 1	a $=$ 6
Conductors per slot	6	36
Number of strands	18	3
Wire diameter	1.4 mm	1.4 mm
Coil pitch (teeth)	10	10
Phase resistance	0.0656677 $\Omega$	0.0656677 $\Omega$

Figure 15.

FFT analysis of phase current at no load under dynamic eccentricity.

Figure 16.

FFT analysis of phase current at full load under dynamic eccentricity.

According to the investigation, different parallel branches under eccentricities show varied influences on the induction motor. For fair comparison, serial number of turns per phase $N$ and slot fill factor are set the same in two machines, comparative parameters of stator winding are listed in Table 4.

3.1 Current comparison of a

=

1 and a

=

6 under static eccentricity

As an example of current circulation in parallel branches under static eccentricity, flux linkage of serial coil group per phase $\Psi_{m1}$ is calculated as shown on Eq. (11), where $m$ is phase number, $N$ is serial number of turns per phase, $K_{dp1}$ is winding factor, $l_{ef}$ is effective axial length, $\tau$ is polar pitch; $I_{c}$ is current of coil group, $\Lambda_{ef}$ Eq. (12) is effective air-gap magnetic permeance; $\mu_{0}$ is air gap permeability.

$\displaystyle\psi_{m1}=\frac{2\sqrt{2}m}{\pi^{2}p}(NK_{dp1})^{2}l_{ef}\tau aI_% {c}\cdot\Lambda_{ef}$ (11) $\displaystyle\Lambda_{ef}=\mu_{0}\frac{1}{d}$ (12)

As shown on Fig. 5, flux linkage of serial coil group per phase $\Psi_{m1}$ at parallel branch a $=$ 1 or a $=$ 6 under static eccentricity is almost the same whether at no load or full load.

In Eq. (11), parallel branch only changes $I_{c}$ and $\Lambda_{ef}$ . The other parameters don’t change. At parallel branch a $=$ 1, substitute Eq. (1) in Eq. (12). Since stator winding at parallel branch a $=$ 1 encompasses air gap in whole directions and in consideration of initial position angle $\alpha$ , air gap permeance of a $=$ 1 is calculated as Eq. (13) and according to integral table, result is worked out as Eq. (13). It can be seen that the initial position angle $\alpha$ has no influence and static eccentricity adds air gap permeance at a $=$ 1. As a consequence in the case of the same flux linkage, phase current will decrease compared with that in healthy condition due to static eccentricity.

Table 5

Analytical relationship between air gap permeance and static eccentricity degree

$\xi$ (SE)	0%	20%	40%	60%	80%
$\Lambda_{a1}$ / $\Lambda_{0}$	1	1.0206	1.0910	1.25	1.6667
$\Lambda_{a6}$ / $\Lambda_{0}$	1	1.0021	1.0086	1.0208	1.0419

At parallel branch a $=$ 6, substitute Eq. (1) in Eq. (12). Each parallel branch takes up 65 ${}^{\circ}$ angle as shown on Fig. 6. Equation (15) is the expression of effective air gap permeance of each parallel branch at a $=$ 6. Phase current of a $=$ 6 Eq. (16) is the sum of six-branch currents. For fair comparison, average air gap permeance of six branches is expressed as Eq. (17). Similarly, initial position angle $\beta$ does not work in Eq. (16). Table 5 lists air gap permeances of parallel branch a $=$ 1, a $=$ 6 under healthy condition and 20%, 40%, 60%, 80% static eccentricity. It can be seen that air gap permeance of parallel branch a $=$ 1 grows faster with static eccentricity degree than that of parallel branch a $=$ 6.

$\displaystyle\Lambda_{a1}=\mu_{0}\frac{1}{\delta}\cdot\frac{1}{2\pi}\int_{% \alpha}^{\alpha+2\pi}{\frac{1}{1-\xi\cos(\theta)}}d\theta$ (13) $\displaystyle\Lambda_{a1}=\mu_{0}\frac{1}{\delta}\cdot\frac{1}{\sqrt{1-\xi^{2}}}$ (14) $\displaystyle\Lambda_{a6i}=\mu_{0}\frac{1}{\delta}\cdot\frac{36}{13\pi}\int_{% \beta+(i-1)\pi/3}^{\beta+(i-1)\pi/3+13\pi/36}{\frac{1}{1-\xi\cos(\theta)}}d\theta$ (15) $\displaystyle I_{p}=\sum\limits_{j=1}^{a}{I_{cj}}$ (16) $\displaystyle\Lambda_{a6}=\frac{a}{\sum\limits_{i=1}^{a}{\frac{1}{\Lambda_{a6i% }}}}$ (17)

Figure 17.

Current of branch A1 at no load under 60% dynamic eccentricity.

Figure 18.

Comparison of stator winding loss under dynamic eccentricity. (a) at no load; (b) at full load.

Figure 19.

Comparison of rotor winding loss under dynamic eccentricity. (a) at no load; (b) at full load.

Figure 20.

Comparison of winding loss under dynamic eccentricity. (a) at no load; (b) at full load.

Table 6

Fft analysis of phase current of varied parallel branches at no load under static eccentricity

Mag (A)	50 Hz		1150 Hz
	a $=$ 1	a $=$ 6	a $=$ 1	a $=$ 6
Healthy	59.76	59.76	3.75	3.75
SE20	59.82	59.9	3.8	3.77
SE40	59.54	59.85	4.03	3.91
SE60	59.04	59.75	4.55	4.17

Table 7

Fft analysis of phase current of varied parallel branches at full load under static eccentricity

Mag (A)	50 Hz		1140 Hz
	a $=$ 1	a $=$ 6	a $=$ 1	a $=$ 6
Healthy	127.5	127.5	6.48	6.48
SE20	127.7	127.7	6.56	6.54
SE40	127.6	127.8	6.92	6.82
SE60	127.8	128.3	7.81	7.48

Figure 6 shows the spatial position of coil groups of phase A, phase B and phase C. Figure 7 shows the comparison of phase current at different loads and parallel branches under static eccentricity. Tables 6 and 7 list FFT analysis of phase current in Fig. 7 at no load and at full load, respectively. In these tables, only two main frequencies (fundamental harmonic and stator slot harmonic) are mentioned. From Table 6, it can be seen that at 50 Hz, no load, phase current a $=$ 1 has declining tendency with growing static eccentricity and phase current a $=$ 6 shows no much difference, which is in line with analytical analysis in Table 5. If the bucking effect on big rotor MMF at full load, at 50 Hz and full load, the results are not the same. Phase current at a $=$ 1 in fact does not decrease with the increase of static eccentricity, phase at a $=$ 6 increases a little bit with rising static eccentricity. In addition, regardless whether this is a no-load or full load condition, as shown on Tables 6 and 7 where the current magnitudes are reported for the slot harmonics at 1150 Hz and 1140 Hz.

Besides, static eccentricity causes unbalanced branch currents in a $=$ 6. Furthermore, due to different spatial positions of three phases as shown on Fig. 6, amplitude of branch currents is varied in branches of three phases. So branch current of a $=$ 6 is investigated.

As shown on Fig. 6, all static eccentricities move toward angle 0 ${}^{\circ}$ . Due to stagger arrangement of branches of three phases, branch currents are different in the various spatial positions. Figures 8 and 9 show branch currents (root mean square) of three phases under static eccentricity at no load and at full load, respectively. It can be seen that hexagon composed of six branch current points at no load are more asymmetric than those at full load. For induction motor, rotor MMF at full load becomes very big and shares the impact of static eccentricity with stator MMF. Hence, branch current at full load changes slower with eccentricity than that at no load.

From Fig. 10, stator winding loss of a $=$ 1 shows no much difference with growing eccentricity degree whether at no load or at full load but there is a tiny decline at no load under static eccentricity. Stator winding loss of a $=$ 6 presents obvious escalating trend with increasing static eccentricity degree both at full load and at no load.

Figure 11 shows rotor MMFs (RMS value) calculated in six regions (as seen in Fig. 4) under static eccentricity. It is interesting to see that variation of a $=$ 1 is larger than that of a $=$ 6 at no load but variation of a $=$ 1 is smaller than that of a $=$ 6 at full load. In addition, changing rules of a $=$ 1 and a $=$ 6 are different as seen in Fig. 11. According to expression Eq. (10), rotor winding loss under static eccentricity is shown on Fig. 12. Besides, winding loss is worked out in Fig. 13 from the sum of stator winding loss and rotor winding loss. Overall, in fair comparison, winding loss of a $=$ 6 rises faster with static eccentricity than that of a $=$ 1.

3.2 Current comparison of a

=

1 and a

=

6 under dynamic eccentricity

Every moment of dynamic eccentricity is static eccentricity. The only difference is that their deviation positions are varied with time. Figure 14 gives comparison between phase current of a $=$ 1 and a $=$ 6. This shows that phase current of a $=$ 1 at no load condition under dynamic eccentricity is slightly reduced. Figure 15 gives and Fig. 16 shows magnitude of main frequencies.

Magnitudes of fundamental frequency (50 Hz) fall off a little bit due to dynamic eccentricity, which is in line with the analysis above. Besides, the other harmonics all go up with dynamic eccentricity and slope of a $=$ 1 is bigger than that of a $=$ 6.

For a $=$ 6 under dynamic eccentricity, due to varied air gap with time below parallel branches, amplitude of each branch current has an oscillating behavior shown on Fig. 17. It can be seen that dynamic eccentricity may lead to overcurrent in some branch.

Figures 18–20 show stator winding loss, rotor winding loss and total winding loss under dynamic eccentricity, respectively. Stator winding loss goes down a little with dynamic eccentricity at a $=$ 1 and goes up at a $=$ 6. At no load rotor winding loss of a $=$ 1 and a $=$ 6 both goes up with dynamic eccentricity but at full load rotor winding loss of a $=$ 1 goes down with dynamic eccentricity, a $=$ 6 makes no much difference.

Overall, winding loss of a $=$ 1 gets down with dynamic eccentricity while winding loss of a $=$ 6 increases with dynamic eccentricity.

4. Conclusion

This paper presents comparative analysis of current under static and dynamic eccentricity in induction motor on parallel branches. According to flux path, magnetic circuit of one pole is calculated for studying the relationship between air gap length and current of stator or rotor. In this investigation, branch currents and phase currents with air gap length are analyzed. According to the results from finite element simulation, stator branch currents and phase currents of two types of parallel branches are validated, and both static and dynamic eccentricities are compared.

In conclusion, total winding loss of parallel branch a $=$ 1 grows slower than that of parallel branch a $=$ 6 under static eccentricity. Under dynamic eccentricity, total winding loss of parallel branch a $=$ 1 goes down a little bit with rising dynamic eccentricity while winding loss of parallel branch a $=$ 6 keeps increasing. For induction motor, over one parallel branch means that stator winding shares the variation of air gap length with rotor winding. That’s to say, over one parallel branch does more work to face air gap length change since stator MMF is generally larger than rotor MMF. For this reason, winding loss of the motor with over one parallel branch is higher than that of the motor with one parallel branch under 2D eccentricity.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 51677051 and No. 51377039), Anhui Province key laboratory of Large-scale Submersible Electric Pump and Accoutrements.

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Comparative analysis of current under static and dynamic eccentricity in 3-phase induction motors on parallel branches

Abstract

Keywords

1. Introduction

2.1 Static eccentricity

Table 2 Parameters of simulated motor

4. Conclusion

Footnotes

Acknowledgments

References

Table 2
Parameters of simulated motor