INVESTIGATION ON ELECTROMAGNETIC FORCES AND EXPERIMENTAL VIBRATIONCHARACTERISTICS OF STATOR END WINDINGS IN GENERATOR

Shuting Wan, Changgeng Zhan, Xiaofang Yao, Yanmin Deng and Guowei ZhouDepartment of Mechanical Engineering, North China Electric Power University, Baoding 071003, China

E-mail: 13582996591@139.com; Zhanchanggeng@126.com

Received May 2013, Accepted June 2014No. 13-CSME-138, E.I.C. Accession 3595

ABSTRACTThis paper investigates the electromagnetic forces acting on the stator end windings, by using the stator endwindings structure of a 600 MW large-capacity turbo-generator as the analysis object. Firstly, based onthe method of images, the Biot–Savart Law and superposition theorem, the calculation model of the statorend windings is proposed. Then, the average electromagnetic forces acted on the end windings and therelationship between radial and tangential electromagnetic force densities are analyzed. The result showsthat the tangential electromagnetic force should be treated as radial one and should not be ignored. Finally,the radial and tangential vibrations are measured to verify the theoretical analysis. The investigation resultswill be beneficial to the binding and fault diagnosis of stator end windings.

Keywords: turbo-generator; stator end windings; tangential electromagnetic force density; radial electro-magnetic force density.

RECHERCHE PORTANT SUR LES FORCES ÉLECTROMAGNÉTIQUES ET LESCARACTÉRISTIQUES EXPÉRIMENTALES DE VIBRATION DE LA BORNE

D’ENROULEMENT DU STATOR D’UN GÉNÉRATEUR

RÉSUMÉCette recherche s’intéresse aux forces électromagnétiques agissant sur la borne d’enroulement du stator,en se servant de la structure de la borne d’enroulement d’un turbogénérateur d’une capacité de 600 MWcomme sujet d’analyse. En se basant sur la méthode d’images, loi de Biot–Savart et le théorème de lasuperposition, on propose un modèle de calcul de la borne d’enroulement du stator. Par la suite, on analyseles forces électromagnétiques moyennes agissant sur la borne d’enroulement et la relation entre la densité desforces électromagnétiques radiales et tangentielles. Les résultats démontrent que la force électromagnétiquetangentielle devrait être traitée comme une force radiale. Finalement, les valeurs de vibration réelle radialeet tangentielle sont mesurées pour vérifier l’analyse théorique. Les résultats de cette recherche seront utilespour le diagnostic de défaillance de la borne d’enroulement du stator.

Mots-clés : turbogénérateur; borne d’enroulement du stator; densité des forces électromagnétiquestangentielles; densité des forces électromagnétiques radiales.

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NOMENCLATURE

B the magnetic flux density vector (T = Wb/m2)Z number of stator slotsq number of slots per pole per phaseβ pitch-short factora number of parallel branches per phaseIN rated current of the stator (A)IA current of the A phase (A)IB current of the B phase (A)IC current of the C phase (A)Im the maximum current (A)Fr the radial electromagnetic force density (N/m)Fz the axial electromagnetic force density (N/m)Fh the resultant electromagnetic force density (N/m)Fθ the tangential electromagnetic force density (N/m)F the electromagnetic force density vector (N/m)

1. INTRODUCTION

Generator stator end windings is one of the highest frequency vibration component, because of the influencecaused by the alternating electromagnetic force in normal operation and the sudden huge electromagneticforce in short circuit condition. The fault analysis of generator showed that long-term excessive vibrationwould cause the fastening structure loose, insulation wear, mechanical fatigue and fracture of the end wind-ings, serious fault even cause interphase short circuit in the end [1–3]. In order to find the fault and ensurethe safety of generator operation, it is necessary to analyze the electromagnetic force and the dynamic char-acteristics of stator end windings.

As an important part of generator, the electromagnetic forces acted on the end windings have always beennoticed by engineering designers and testing specialists [4–9]. There are two directions for the analysis ofthe end windings. The first one is numerical calculation based on the simplified model of end windings andthe other is analog simulation based on the finite element method (FEM).

Hu et al. [4] gave the uniform expressions of magnetic field distribution based on the basic equations ofelectromagnetic field, and deduced the calculation formula of electromagnetic force in end windings. And itcould lead to the inaccuracy by using the simple line to replace the involute part. Based on Hu et al. [4], Liuet al. [5] used several pieces of line segments to fit the involute part, and discussed the vibration of the endwindings by numeral methods. D. Mori and Takeo Ishikawa [6] calculated the radial electromagnetic forceof induction motor based on 2D nonlinear FEM, and analyzed the radial electromagnetic force at differentpositions in two motors. Stancheva and Iatcheva [7] analyzed the electromagnetic field by using the 3DFEM computation, and studied the dynamic behavior of the local electromagnetic force densities. Kim [8]analyzed the distribution of electromagnetic force by using the FEM and the Biot–Savart Law, comparedthe two methods and drew a conclusion that the result based on the Biot–Savart Law has some validityat the end winding region, but there is no research on the tangential electromagnetic force. Schlegl et al.[9] developed a full parameterized FE-modeling tool for efficient investigations on generator end windingsduring the design phase of turbo- and hydro-generators.

Meanwhile, actual vibrations of the stator end windings were tested [1–3, 10]. Liu et al. [1], Cao et al. [2],and Zhang et al. [10] introduced the testing process including testing principles and methods, acquired thevibration characteristic of the stator end windings, and pointed out that the vibrations are mainly composedby the component of rotating frequency (50 Hz) and other frequency multiplications (as 100 Hz, 150 Hz,etc.). Liu et al. [3] modeled the stator end windings, analyzed the inherent frequency, and obtained the

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(a) (b)

Fig. 1. Stator end windings structure. (a) General outlook. (b) Supporting and binding structure.

conclusion that the main factor affecting the stator end windings vibration is the electromagnetic forcedensity as well as the position and fastness level of the winding bind.

All of the above mentioned papers put forward important findings and they are the basis of the vibra-tion analysis of the stator end windings. However, none of them studies the specific positions where theyneed to be bound more securely, and they paid no attention to the influence of the vibration caused by thetangential and axial electromagnetic forces. As an improvement, this paper analyzes the specific positionswhere the maximum electromagnetic force appeared, researches the influence of axial, radial and tangentialelectromagnetic forces, and verifies the result by testing the vibration.

2. CALCULATION MODEL OF STATOR END WINDINGS

This paper selects the end structure of a 600 MW large-capacity turbo-generator as analysis object, and thestructure is shown in Fig. 1. There are two reasons for choosing this type. On the one hand, the 600 MWturbo-generator is a typical type in Chinese power plants, and using this type for research could make animportance value for maintenance and diagnosis. On the other hand, the vibration of the end windings couldbe tested personally and this data can make research more reliable. In order to simplify the analysis, somenecessary assumptions are made:

1. The permeability of ferromagnetic medium is infinity, and the influences which are caused by theretaining rings, central rings, end press finger, pressure washer, press finger, magnetic shielding, shaftand the air gap are ignored [11].

2. The higher order components of stator current harmonics and the displacement current are neglected[12].

3. All material properties are isotropic, the eddy current in the stator core is not considered, and thehysteresis effect is ignored [12].

4. The influence of the rotor windings is ignored because of the large distance between the rotor and theend windings of stator.

5. The winding bar in the same layer has the same shape. The actual bar is replaced by the center lineof it, and the circular area in the bar is replaced by straight line segments. This point makes thecalculation easy and has little influence for results.

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Fig. 2. Physical model of stator end windings.

Figure 2 shows the physical model of the stator end windings. In this model, the involute part is fittedby several pieces of line segments, the upper and lower end windings are divided into 28 units respectively,where Nos. 1–4 called the exit part of slot indicates the straight part, No. 25–28 called the nasal tip areaindicates the connection part between the upper and the lower, No. 5–24 called the involute part indicatesthe elevation part. The generator parameters used in the model are as follows: Number of stator slotsZ = 42, number of slots per pole per phase q = 7, pitch-short factor β = 17/21, number of parallel branchesper phase a = 2, rated current of the stator IN = 19245 A.

The three-phase stator currents are IA = 0.5× Im sin(ωt)

IB = 0.5× Im sin(ωt−2π/3)

IA = 0.5× Im sin(ωt +2π/3)

(1)

where Im is the maximum current, Im =√

2 I, I is the effective value of current.Under normal conditions, the distribution of three-phase currents is symmetrical. In this case, the distri-

bution of B phase winding is used as an example, and the developed winding diagram is shown in Fig. 3.In Fig. 3, the solid line indicates the upper bar, and the dash line indicates the lower bar. The parameters

Si and Xi are used to indicate the bars, S means the upper bar, X means the lower bar, i means the number ofthe bar (i = 1,2, . . . ,42).

This paper uses the method of images to simplify the electromagnetic field calculation of stator endwindings, which can exactly replace the actual medium with a set of image currents [13]. Based on themethod of images, the Biot–Savart Law and superposition theorem, the magnetic flux density B of the pointP produced by all the end windings currents can be expressed as

B =N

∑K=1

µ0

4π

∫LK

idl× rr3 , (2)

where l is the current vector, r is the distance vector between current vector and the point P, Lk is the centerline of the winding.

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Fig. 3. Developed winding diagram of B phase.

According to Ampere’s law, the electromagnetic force density (F) can be expressed as

F = i0×B (N/m), (3)

where i0 is the current of windings.According to Eq. (3), coordinate transformation has been taken. The electromagnetic force density for-

mula in rectangular coordinate system can be expressed asFr(m) = kr1(m)IAIB + kr2(m)IBIB + kr3(m)ICIB

Fθ (m) = kθ1(m)IAIB + kθ2(m)IBIB + kθ3(m)ICIB

FZ(m) = kZ1(m)IAIB + kZ2(m)IBIB + kZ3(m)ICIB

Fh(m) = [F2r (m)+F2

θ(m)+F2

Z (m)]1/2

(4)

where Fr is the radial electromagnetic force density, Fθ is the tangential electromagnetic force density, Fz isthe axial electromagnetic force density, Fh is the resultant electromagnetic force density, and m is indicatedthe number of 28 units of the stator end windings (m = 1,2, . . . ,28), kr1, kr2, kr3, kθ1, kθ2, kθ3, kz1, kz2, kz3are the correlation coefficients.

3. ELECTROMAGNETIC FORCE CHARACTERISTICS

Based on the model and the equations in Section 2, the maximum electromagnetic force densities havebeen calculated, which acted on phase belt boundary of the upper and the lower. Then the regulations ofdensities changes have been analyzed, which acted on the involute part at the moment of the phase currentmaximizing. Finally, the radial and tangential electromagnetic force densities acted on the nasal tip of thestator end windings have been calculated and compared.

3.1. Analysis of Maximum Electromagnetic Force DensityBased on the densities calculation of all windings, the positions of maximum electromagnetic force densitieshave been found. The maximum values exist in the phase belt boundary at the moment of the phase currentmaximizing. So the windings which stay at the phase belt boundary (such as S7, X24, S21, X38, S14, X31)are chosen as an example to analyze the electromagnetic force densities, the maximum electromagneticforce densities curves are indicated in Fig. 4.

According to Fig. 4, a general conclusion of the upper and lower windings whose interval is a pitch (suchas S7 and X24, S21 and X38, S14 and X31) is given as follows:

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Fig. 4. The maximum electromagnetic force densities curve. (a) Resultant electromagnetic force densities, (b) axialelectromagnetic force densities, (c) radial electromagnetic force densities and (d) tangential electromagnetic forcedensities.

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Table 1. Statistical results of maximum electromagnetic force densities.Location Maximum electromagnetic force densities (N/m)

Fh Fr Fθ FzB S7 value 1768.5 –1061.7 1299.7 1186.8

No. 16 27 4 16X24 value 1376.5 –739.5 894.6 –1014.6

No. 13 4 12 14A S21 value 1630.7 –1365.2 –1202.2 1385.1

No. 27 27 25 28X38 value 1299.8 901.2 1089.9 –959.1

No. 26 27 25 24C S14 value 2384.4 –1227.2 1513.2 1758.2

No. 17 27 17 16X31 value 1973.4 –699.5 1244.7 –1487.5

No. 16 4 16 16

1. The variation trends of resultant electromagnetic force densities are the same, and the values have alittle difference.

2. The axial electromagnetic force densities are zero at the exit part of slot and the nasal tip area, thevariation trends are opposite in the involute part, and the saltation happens in the nasal tip area.

3. The variation trends of radial electromagnetic force densities are the same at the exit of slot whileopposite in the involute part and nasal tip area.

4. The variation trends of tangential electromagnetic force densities are opposite in the nasal tip area.Figures 4(b), (c) and (d) show the important locations of the electromagnetic force in which thedirection of electromagnetic force changed. Such locations must be paid more attention to prevent thefatigue wear and other faults.

Through comparison of the maximum electromagnetic force density in different places, the positions andthe values of maximum force densities could be found, statistical results are showed in Table 1.

Table 1 shows that the locations of maximum electromagnetic force densities are different in differentdirections, but the orders of magnitude are the same. According to Table 1 and Fig. 2, the maximumelectromagnetic force densities appeared in four places:

1. the transition section from the exit part of slot to the involute part (No. 4),

2. the middle position of the involute part (Nos. 16–17),

3. the transition section from the involute part to the nasal tip area (No. 25 or 26), and

4. the transition section from the line segment to the connection of the nasal tip (No. 27 or 28).

3.2. Maximum Electromagnetic Force Density in Involute PartsAs is generally known, the shape of the end windings is a complex involute and fixed weakly. The structureof single bar is similar to a simply supported beam. In view of the structure, as in Fig. 1, the end windingscannot be a strong rigid structure. So nowadays it is unprocurable to fix the winding reliably by binding.But the way that the end windings are bound in the design and manufacturing stages at the location in whichthe maximum densities appeared would play an important role in increasing the stiffness of end windings.

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Table 2. The position of the extreme value of electromagnetic forces.Phase Upper bar Lower bar

Location Number Location Numberminimum maximum minimum maximum

B S1 15 23 X18 9 17S2 14 22 X19 10 18S3 13 21 X20 11 19S4 12 20 X21 12 20S5 11 19 X22 14 22S6 9 18 X23 15 23S7 8 16 X24 16 24

C S8 15 24 X25 9 17S9 14 22 X26 10 18

S10 13 21 X27 11 19S11 12 20 X28 13 21S12 11 19 X29 14 22S13 9 18 X30 15 23S14 8 16 X31 16 24

A S15 15 23 X32 9 17S16 14 22 X33 10 18S17 13 21 X34 11 19S18 12 20 X35 13 20S19 11 19 X36 14 21S20 9 18 X37 15 22S21 8 16 X38 16 23

Figure 5 shows the electromagnetic force densities of seven B phase bars, which are located in the samepole pitch and layer. And the characteristics of the maximum electromagnetic force densities are analyzedas follows.

Figure 5 also shows the similar variation trends of resultant, axial, radial and tangential electromagneticforces in the involute parts (Nos. 5–24), and the values of electromagnetic force densities associated with theposition of the winding closely. Generally speaking, the maximum electromagnetic force densities appearedin phase belt boundary such as S7, and the maximum value of density in every bar is associated with thelocation of the bar.

Based on the analysis, the position of extreme value of electromagnetic forces can be given as Table 2.This table shows, in the belt boundary (such as S7, S8, X24, X25, etc.), that the maximum electromag-netic force is located in Nos. 15–16 or the 23–24 pieces of the upper bars. In the same way, the maxi-mum value is located in Nos. 16–17 or the 23–24 pieces of the lower bars in this part. And these placesshould be paid more attention because of the maximum electromagnetic forces. According to Table 2,the positions of extreme value in the windings are not only certain, but also neighboring between adja-cent bars in this part. And this proposed result has referenced value for the binding to the end windings.The extreme point of upper and lower bar in one phase can be derived by the above conclusion. Becauseof the symmetry of the electromagnetic density, the rule of maximum value position can be expanded toall the end windings, and provide beneficial reference for the selection of the colligation and supportinglocation.

3.3. Maximum Electromagnetic Force Density in Nasal Tip AreaBecause of the weak binding, a large portion of accidents happened in the nasal tip area. The installationposition of the sensor is generally located in this part, so the nasal tip area is chosen to analyze the radialand tangential electromagnetic force density.

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Fig. 5. Electromagnetic force densities in one pole pitch. (a) Resultant electromagnetic force, (b) axial electromagneticforce (c) radial electromagnetic force, and (d) tangential electromagnetic force.

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Fig. 6. Average values of tangential and radial electromagnetic force densities. (a) Tangential electromagnetic forcedensities. (b) Radial electromagnetic force densities.

In order to understand the distribution of force densities in the nasal tip area, the calculation and compari-son of average radial and tangential electromagnetic force densities have been taken. For the convenience ofdescription, it is defined as follows: inner circle indicates the upper bar, outer circle indicates the lower bar,the length of the line segment indicates the size of the electromagnetic force density, the inside indicates thenegative direction of the electromagnetic force, the outside indicates the positive.

The average radial and tangential electromagnetic force densities are shown in Fig. 6. This figure showsthe direction of the radial electromagnetic force densities is opposite in the upper and lower bar, the upperinward contraction and the lower outward expansion. In the same way, the directions of the tangentialdensities are also opposite. So the radial electromagnetic forces make the bar far away from each other andthe tangential ones make the friction growing. In conclusion, the position of the maximum repulsive forceand static friction force can provide a reference for the selection of vibration measuring point.

Table 3 shows the size relations of the tangential and radial electromagnetic force densities in peripherallocation within one pole pitch.

As mentioned above, the values of the electromagnetic force densities in the same direction, position anddifferent layers are very close, and the values have a big difference in different directions. The tangentialand radial electromagnetic force densities are different in different place, but they have the same order ofmagnitude.

According to Table 3, we can find that the maximum average value of the tangential electromagneticforce density is 549.6 N/m, located at the slot No.2 in upper bar, and the radial one is 534.4 N/m, locatedat slot No. 17 in upper bar. Based on the previous calculation and comparison, a general conclusion canbe drawn: the maximum average tangential and radial densities in the nasal tip area have the same order ofmagnitude. So the tangential densities should be paid attention as well, when analyzing the electromagneticforce characteristics of generator stator end windings.

The radial and tangential electromagnetic force densities have been compared when putting the upperand lower bar as a whole structure, which acted on the same position in the nasal tip area. By comparison,the size relationship of the radial and tangential electromagnetic force densities can be given as follows:the radial one is larger than the tangential one in eight places (slot Nos. 6, 9, 10, 11, 12, 13, 14 and 21),smaller than in one place (slot No. 15), and they are similar in 12 places (slot Nos. 1, 2, 3, 4, 5, 7, 8,16, 17, 18, 19 and 20). According to the relationship, a preliminary reference for the position of sensor

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Table 3. Values of tangential and radial electromagnetic force densities.

installation in nasal tip area can be given: When testing the vibration of the end windings, not only the radialvibration but also the tangential one should be paid enough attention. In this case, the sensor measuringthe radial vibration should be installed in slot Nos. 6, 9, 10, 11, 12, 13, 14 and 21 positions, the sensormeasuring the tangential one should be installed in the slot No. 15 position and the sensor measuring theradial as well as tangential one should be installed in slot Nos. 1, 2, 3, 4, 5, 7, 8, 16, 17, 18, 19 and20 positions.

4. EXPERIMENTAL VERIFICATION

4.1. Relationship between Experiment and TheoryIn order to verify the theoretical results, this paper set up a corresponding experiment. It is difficult to testthe electromagnetic force directly because of the time-varying characteristics of electromagnetic forces. Inthis paper, the vibration testing is used to replace the forces testing. In the following some statements forthis transition are given.

This paper focuses on the relationship between the radial and the tangential electromagnetic force den-sities, and the vibration measurement aims to explain the relationship only. Hu et al. [4] introduced “themain factor affecting the end winding vibration is the electromagnetic force density as well as the po-sition and fastness level of the winding bind”. In this research, the radial and tangential vibration aremeasured at the same time, in other words, the vibration is measured in the same position and fastnesslevel of the winding bind condition. So, the electromagnetic force affects the vibration only in this cer-tain condition. As above, choosing the vibration testing to verify the theoretical result can meet ourrequirements.

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Slot No. and force direction

Force density（N/m） Slot No. and force direction

Force density（N/m） Upper

barLower

bar Resultant Upper bar

Lower bar Resultant

1 tangential –546.6 473.3 –73.3 12 tangential 126.0 –119.3 6.8 radial –405.9 351.0 –55.0 radial –457.2 372.5 –84.7

2 tangential –549.6 501.3 –48.3 13 tangential 137.0 –124.1 12.9 radial –413.6 348.6 –65.0 radial –449.5 346.8 –102.8

3 tangential –537.1 495.1 –42.0 14 tangential 169.4 –128.9 40.5 radial –406.3 334.4 –71.9 radial –410.4 270.8 –139.7

4 tangential –506.6 470.0 –36.6 15 tangential –287.0 156.8 –130.1 radial –395.9 317.4 –78.6 radial –457.2 439.8 –17.4

5 tangential –455.5 427.3 –28.2 16 tangential –383.5 323.3 –60.2 radial –384.3 296.6 –87.7 radial –520.7 476.8 –44.0

6 tangential –373.5 364.8 –8.7 17 tangential –457.1 408.3 –48.8 radial –369.6 265.5 –104.1 radial –534.4 476.2 –58.2

7 tangential –218.6 294.7 76.0 18 tangential –506.3 461.4 –44.9 radial –326.5 186.2 –140.4 radial –534.0 467.2 –66.8

8 tangential 146.5 –172.8 –26.3 19 tangential –535.9 494.1 –41.8 radial –371.9 353.8 –18.1 radial –526.8 453.3 –73.6

9 tangential 133.5 –136.0 –2.5 20 tangential –545.8 510.4 –35.4 radial –436.6 391.3 –45.4 radial –514.3 433.2 –81.2

10 tangential 125.4 –123.5 2.0 21 tangential –523.6 516.3 –7.3 radial –453.6 392.4 –61.2 radial –489.2 394.8 –94.4

11 tangential 123.1 –119.0 4.1 11 radial –458.4 385.6 –72.8

341

Table 4. Installation locations of sensors.Test No. Location of the sensors Direction1 10 o’clock in exciter end tangential2 10 o’clock in exciter end radial3 12 o’clock position of parallel ring axial4 12 o’clock position of parallel ring radial5 12 o’clock position of turbine end connecting line tangential6 12 o’clock position of turbine end connecting line radial7 1 o’clock in exciter end tangential8 1 o’clock in exciter end radial9 2 o’clock in exciter end tangential10 2 o’clock in exciter end radial11 Half past 2 o’clock position in exciter end tangential12 Half past 2 o’clock position in exciter end radial

Fig. 7. Location of sensors. (a) Measure point in end windings, (b) measuring point in parallel ring, and (c) measuringpoint in steam end connecting line.

4.2. Experimental InvestigationUnder the rated hydrogen pressure, the vibrations of generator in rated speed without exciting current andrated operating conditions have been tested respectively.

The experiment is based on the 600 MW large-capacity turbo-generator. The stator winding is two- layerand pitch-shortening, and the corresponding parameters are:

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(a) (b)

Fig. 8. Test installation. (a) Location of sensors and (b) leads of optical fiber in sealing flange.

Table 5. Vibration values in rated speed without exciting current condition (µm).Test Vibration Value of Value of Test Vibration Value of Value ofNo. value 50 Hz 100 Hz No. value 50 Hz 100 Hz1 9.11 6.12 0.60 7 13.87 8.39 0.512 6.30 3.66 0.80 8 16.07 5.65 0.983 8.09 5.59 1.49 9 23.88 12.87 0.334 9.03 6.48 1.54 10 15.07 8.00 0.685 12.77 8.72 0.62 11 28.96 16.28 0.266 6.85 4.72 0.58 12 10.72 4.61 1.04

Rated capacity: 600 MW Rated exciting current: 4128 ARated power factor: 0.9 Rated exciting voltage: 421.8 VRated current: 19245 A Rated speed: nr=3000r/minRated voltage: 20000 V

The arrangements of vibration measuring points are shown in Table 4 and Fig. 7.FODA-2500 type optical acceleration sensor has been chosen to testing the vibrations, which is not af-

fected by electromagnetic interference completely. Sensor is fixed with epoxy glue and polyester glass ropeat the appropriate place, fiber of sensor is fixed with glass rope in a interval of 100 mm on the outer supportring and come out through the flange. Finally, the seal is fastened to make sure there is no leakage at the airtight test and rated hydrogen pressure working condition (shown in Fig. 8).

The vibration values of the generator works in rated speed without exciting current condition are shownin Table 5.

Table 5 shows that the vibration frequency is mainly composed by the rotors’ frequency (50 Hz), when thegenerator worked in rated speed without exciting current condition. In this condition, the exciting currentand stator current are equal to zero, so the vibration in 100 Hz is very small.

The vibration values of the generator works in rated operation condition are shown in Table 6. Thevibrations of time-domain and frequency-domain of Nos. 9 and 10 are shown in Fig. 9.

Table 6 and Fig. 9 show the periodicity of vibration signal. Because of the excitation of rotor, the vibrationfrequency is mainly composed by 50 and 100 Hz, even 150 and 200 Hz appeared in some test point. Wherethe vibration of the 50 Hz is derived from the rotor vibration, the 100 Hz is derived from the electromagnetic

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Table 6. Vibration values in rated operation condition (µm).Test Vibration Value of Value of Test Vibration Value of Value ofNo. value 50 Hz 100 Hz No. value 50 Hz 100 Hz1 14.05 3.85 6.52 7 31.86 11.14 12.512 23.05 9.89 6.81 8 36.37 10.85 20.273 25.75 10.92 16.67 9 34.39 12.27 14.174 18.93 12.07 10.32 10 38.37 8.23 21.175 24.04 7.08 7.79 11 52.43 16.28 13.076 20.19 8.04 6.64 12 38.65 15.88 21.29

Fig. 9. Vibration of time-domain and frequency-domain of Nos. 9 and 10. (a) Vibrations of point 9 and (b) vibrationsof point 10.

vibration of the end windings. Table 6 shows the radial and tangential vibrations component in 100 Hzhave the same order of magnitude, but the radial vibration amplitude is slightly larger than the tangentialcomponent. According to the date, the radial vibration component plays a major role and the tangentialone cannot be ignored at the same time. And this conclusion is in conformity with the theoretical results.Making a general survey of on-line vibration monitoring system, they are paid less attention when comparedto the tangential vibration. In order to ensure the accuracy of test result, the monitoring of the tangentialvibration should be paid more attention.

5. CONCLUSIONS

This paper deduced the formula of electromagnetic force acted on the stator end windings, analyzed theregular of maximum electromagnetic force densities, and provided the experiment verifications. The con-clusions drawn from the theoretical and experimental investigation can be given as follows:

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1. The maximum values of radial and tangential electromagnetic force densities have the same order ofmagnitude.

2. The maximum electromagnetic force density appears in the phase belt boundary.

3. The axial, radial and tangential electromagnetic force densities have similar variation trends, whichact on the involute part of bar stay in the same layer of one phase belt.

4. The average values of the tangential and radial electromagnetic force density in the nasal tip area havethe same order of magnitude, and this point verified the theoretical results. The tangential and radialvibration should be monitored in an on-line vibration monitoring system of stator end windings.

ACKNOWLEDGEMENTS

This work is supported by the Natural Science Foundation of China (No. 51177046), Natural Science Foun-dation of Hebei province (No. E2011502024) and the Fundamental Research Funds for the Chinese CentralUniversities (No. 12MS101).

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