# Effect of Static Rotor Eccentricity on End Winding Forces and Vibration Wearing

Authors: Hong-Chun Jiang; Gui-Ji Tang; Yu-Ling He; Kai Sun; Wei-Jun Li; Lun Cheng
Journal: International Journal of Rotating Machinery
 (2021)

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## Abstract

In order to study the vibration wearing regularity and the strength failure point of stator end windings before and after static rotor eccentricity, the three-dimensional electromagnetic forces and the subsequent mechanical responses are studied in this paper. The electromagnetic force, stress, and deformation on the end winding of the QFSN-600-2YHG turbo-generator are calculated by the finite element method (FEM) through an electromagnetic-structure coupling. The radial vibration characteristics of the winding are verified by experiments. It shows that the vibration wearing in the same layer is more serious than that between two neighboring layers. For different layers, the interphase coils endure a larger wearing risk than the innerphase coils. Inside the same phase, the last coil along the rotating direction has the highest risk of insulation wearing. The occurrence of static rotor eccentricity will significantly increase the electromagnetic forces and the vibration amplitudes on some coils. The end-phase coil which is close to the minimum air-gap point is the most dangerous one due to the lasting overstresses and the intensified deformations.


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## Body

## 1. Introduction

The stator windings are the key components which implement the energy conversion. As the capacity increases, the stator windings in generators, especially in large turbo-generators, will afford larger electromagnetic forces. Such electromagnetic forces will cause the winding to vibrate [1] and endure stresses [2, 3].Scholars have paid much attention to the electromagnetic forces/stresses on end windings. Chow et al. used the integral-equation method to determine the magnetic field and the electromagnetic forces on conductors [4]. Khan et al. set up the force analysis model based on the image current analysis [5]. The key point of these two methods is to use Biot-Savart’s law to calculate the magnetic field on the end region based on the integral performance. Albanese et al. further confirmed the effectiveness of such a method [6]. The advantage of such a method is that it is able to obtain the magnetic flux density at an arbitrary point. Afterwards, the finite element method (FEM) was fast developed and more and more researchers employed FEM to study the electromagnetic forces. For instance, Kim et al. analyzed the distribution of the electromagnetic force on the end windings based on the Biot-Savart law and 3D FEM [7], while Fang et al. carry out the calculation of the stress distribution on end windings in a large water filling submersible motor during both steady state and transient operations [8, 9]. Also, Salon et al. presented the winding force variation under the transient operation in larger turbine generators [10]. Besides the running conditions (steady state and transient case), it is also found that some other factors such as the speed, the power factor, the saturation of the stator teeth, the magnetic field distribution, the winding connection, and the shape of the salient poles will also affect the force magnitude [11, 12]. Subsequently, to reduce the fluctuation of the resulting electromagnetic forces, Liang et al. designed a new structure of the stator windings according to the electromagnetic force characteristics [13]. Complementally, by analyzing the electromagnetic force of different rods at varied moments, Bao et al. found that the electromagnetic force mutation often occurs in the transition part of the involute [14]. Ohtaguro et al. studied the mechanical strain of the end windings in high-voltage rotating machines and employed the strain gauges for the validation test [15], finding that the strains of the stator end windings due to the electromagnetic forces will be larger as the machine starts. The authors have also carried out the 3D FEM calculation on the end winding forces as well as the mechanical response in 600 MW turbo-generators, getting that the nose part, the root, and the middle part are the most dangerous positions to be damaged [1].In addition to the electromagnetic forces, stresses, and strains, there is still another key parameter, i.e., the winding vibrations. The electromagnetic forces and the vibrations in axial flux permanent magnet synchronous machines (PMSM) with three different winding types were calculated by Lu et al. based on the multiphysics analysis [16]. Patel and Butler employed the optical displacement follower to test the end winding vibrations in a large 2-pole generator [17]. It is found that the nominal vibration amplitude is about 76-102 μm, while under the leading-power factor pattern, the vibration will be increased to 185 μm. Actually, more and more large generators have installed the optical-electrical conversion type sensors such as the fiber optic accelerometers, to monitor the radial vibration of the end windings. Since the electromagnetic force on the end windings has the prominent 2nd harmonic, it is requested for large generators that the natural frequency of the end windings should be away from the 2nd harmonic frequency as much as possible [1]. To find out a more comprehensive result, Mori and Ishikawa clarified the relationship among the radial electromagnetic force, the natural frequencies, and the vibration velocities [18], while Yang and Chen presented the general expression of the frequency and the corresponding mode number of the radial force harmonics for a permanent magnet synchronous machine (PMSM) which had three-phase symmetrical double-layer windings [19]. In addition, Ishibashi et al. calculated the natural frequencies of the steel core and the windings, finding that as the configuration of the ring becomes more complex, the number of the natural frequencies will be increased, while the exact values of the natural frequencies will be decreased [20].By far, most of the investigations on the end windings are focused on the normal conditions/nominal operations, while the faulty cases are paid with much less attention. It is found that the electromagnetic force on the end windings will be significantly increased in the 3-phase short circuit condition [21]. In the meantime, Li et al. have also discovered that the magnetic field in the end region will be greatly changed when synchronizing out of phase occurs, and the large impulse current may damage the winding insulation [22]. Moreover, it is also pointed out that either the short circuit degree or the short circuit position will affect the electromagnetic forces as well as the vibration characteristics of the end windings in synchronous generators [23]. The authors have also carried out some preresearches and found that the odd harmonics of the electromagnetic force will be increased, while the even harmonics will be decreased as the rotor winding interturn short circuit occurs [24]. Besides, the studies on the rotor vibration [25], the magnetic flux density [26], the phase current [27], the electromagnetic torque [28], the rotor vibration properties [29], and the shaft voltage [27, 30] under static air-gap eccentricity have also been presented. However, how will the end winding force change and how the vibration response varies in the static rotor eccentricity cases have still not been studied.As the rotor eccentricity happens, the electromagnetic forces as well as the vibrations of some windings will be aggravated, and the subsequent insulation wearing failure may happen after a long-term performance. Since the rotor eccentricity is pretty common and exists in almost every generator, it is significant to study the winding force properties as well as the vibration responses. In this paper, we study the electromagnetic force properties and also the subsequent mechanical responses of the end windings in a 600 MW turbo-generator. The remainder of this paper is constructed as follows. The qualitative formulas of the electromagnetic force on the end windings under static rotor eccentricity are derived in Section2, while the finite element analysis and the experimental study are carried out in Section 3 and Section 4, respectively. Based on the theoretical, the FEM, and the experimental study, the primary conclusions are finally drawn up in Section 5.


## 2. Analysis on Electromagnetic Force

The rotor winding distribution and the connection relationship are shown in Figures1(a) and 1(b). The winding magnetomotive force (MMF) can be obtained based on the superposition principle. The MMF generated by the ith coil is shown in Figure 1(c), and the MMF at an arbitrary point whose circumferential position angle is α (as shown in Figure 1(d)) can be written as
(1)friα=Irwr1−αi,‐αiπ,αiπ,−Irwrαi,others,with
(2)αi=βi2,i=1to8,1−βi2,i=9to16,Figure 1
Air-gap and MMF: (a) sectional view of rotor winding distribution, (b) connection relationship of rotor windings, (c) MMF of theith rotor coil, (d) eccentric air-gap, (e, f) rotor winding MMF waveform and spectrum, (g) winding factor harmonics, and (h) relationship of rotor winding MMF and stator wingding MMF.
(a)(b)(c)(d)(e)(f)(g)(h)whereIr and wr are the current and the number of turns, respectively, and βi is the ratio of the ith rotor coil pitch to the pole pitch. The rotor winding MMF and its Fourier series can be written as
(3)frα=∑i=116fri=∑i=116A02+∑n=1∞Ancosnα+Bnsinnα,where
(4)A0=12π∫−ππfrdα=0,An=1π∫−ππfrcosnαdα=2Irwrnsinnαiπ,Bn=1π∫−ππfrsinnαdα=0.The factorswr and βi have the same values for the ith and (17-i)th coils, and the rotor is rotating at the angular frequency of ωω=2πf,f=50Hz. The rotor winding MMF can be reduced as
(5)frα,t=∑i=116∑n=1∞2Irwrnπsinnαiπcosnωt−α=∑n=1,3,5⋯Frncosnωt−α,Frn=∑i=184Irwrnπsinβi2nπ.The MMF waveform and the spectrum are exhibited in Figures1(e) and 1(f), respectively. It is shown that the MMF is a stair wave which mainly contains odd harmonic components.The permeance per unit area (hereafter, it is written as unit permeance for short) in the air-gap is in inversed proportion to the radial air-gap length, which can be expanded via power series and written as(6)Λα=μ0δα=μ0δ01−ςcosα≈Λ01+ςcosα,where Λ0=μ0/δ0 is the normal unit permeance, ζ=e/δ0 is the eccentricity rate, and e is the eccentricity value, as indicated in Figure 1(d).The magnetic flux density (MFD) of the air-gap can be obtained through multiplying MMF by the unit permeance, that is,(7)Brα,t=frα,tΛα=Λ01+ςcosα∑n=1,3,5⋯Frncosnωt−α.Thus, the stator winding current can be written as(8)Ist=EtZ=∑n=1,3,5⋯qwskwnBrLvZ=∑n=1,3,5⋯Isncosnωt−α−ψ−π2,Isn=1ZqwskwnFrnΛ01+ςcosαLv,whereE is the induced electromotive force, L is the effective axial length of the winding, v is the velocity that the coil bar cuts the magnetic flux lines, Z is the reactance of the winding, q is the slot number per phase, ws is the turn number per coil, ψ is the internal power angle of the generator, and kwn is the winding factor of the nth harmonic. The winding factor can be described as
(9)kwn=sinn90∘⋅y/τsinnqα1/2qsinnα1/2,n=1,3,5,⋯,where α1 is the angle between the two neighboring slots, y is the stator coil pitch, and τ is the stator pole pitch. The harmonics of the winding factor are shown in Figure 1(g).The stator winding MMF can also be obtained according to the superposition principle. And it can be written as(10)fsα,t=∑n=1,3,5⋯Fsncosnωt−α−ψ−0.5π,Fsn=1.352Isnwskwnq.According to Equations (8) and (10) and Figures 1(f) and 1(g), the stator winding current contains only odd harmonics, with the 1st harmonic as the primary one. Therefore, the MMF produced by the stator windings should also contain odd harmonics only, and the 1st harmonic should be the prominent. The static rotor eccentricity will generally affect the permeance but has little influence on the magnetomotive force (MMF) [14]. For the sake of analysis convenience, we ignore the higher-order harmonics, and the MMF at an arbitrary point which has the circumferential position angle of α can be written as
(11)fα,t=∑n=1,3,5⋯Frncosnωt−α+Fsncosnωt−α−ψ−0.5π=∑n=1,3,5⋯Fcncosnωt−α−λn≈Fc1cosωt−α−λ1,Fc1=Fs12cos2ψ+Fr1−Fs1sinψ2,λ1=arctanFs1cosψFr1−Fs1sinψ,where Fs1 and Fr1 are the 1st harmonics of the stator and the rotor MMF, respectively, Fc1 is the composite fundamental-frequency MMF, and λ1 is the angle between Fr1 and Fc1, as shown in Figure 1(h).For the sake of clarity, the circumferential angle of the upper layer of theIth stator coil is defined as αI. For instance, the position of the 34th coil is defined as α34. Since there are many points on the end part of each coil, we define the circumferential position of the Kth point as αI+αk, as shown in Figure 2(a). Given that the magnetic field in the end region is generally the leakage field which has a significantly smaller magnetic magnitude, we set a coefficient fk for the modification. Then, ignoring the higher-order harmonics of MMF, the electromagnetic force of the end winding can be obtained via
(12)FIk→=fkB→×Is→dl=FIkx→,FIky→,FIkz→dl,FIk=fkBIssinθk=1ZfkqwsLvΛ021+ςcosαI+αk1+ςcosαI∑n=1,3,5⋯Fcncosnωt−αI−αk−λn∑n=1,3,5⋯kwnFrncosnωt−αI−ψ−0.5πsinθk≈1Zfkqwskw1LvFc1Fr1Λ02sinθkcosωt−αI−αk−λ11+ςcosαI+αkcosωt−αI−ψ−0.5π1+ςcosαI=fkqwskw1LvFc1Fr1Λ02sinθkdl2Z1+Δcosαk+λ1−ψ−0.5π+cos2ωt−2αI−αk−λ1−ψ−0.5π,Δ=ς2cosαI+αkcosαI+ςcosαI+αk+ςcosαI,αk∈0,145.7°,where θk is the angle between the magnetic flux density and the stator current at point K.Figure 2
(a) Sectional view of three-phase winding distribution and (b) winding connection relationship of phase A.
(a)(b)Further, the radial, the tangential, and the axial electromagnetic force components of coilI can be obtained through
(13)FIr=∫lendFIkxcosθ+FIkysinθdl,FIt=∫lend−FIkxsinθ+FIkycosθdl,FIa=∫lendFIkzdl,where θ is the vector angle of point K in the cylinder coordinate, and lend is the coil curve of the end winding.According to (12), the electromagnetic force on the end winding has the DC component and the 2nd harmonic. If taking into account the higher-order MMF harmonics, there should be still other even harmonics such as the 4th, 6th, and 8th. The occurrence of the static rotor eccentricity will affect the amplitudes of these harmonics but will not induce new harmonics. The amplitude variation depends on the eccentricity rate ζ and the coil position αI. In this paper, αI+αk∈−90°,90° when αI∈−90°,−55.7, which results in Δ>0. Consequently, the electromagnetic force will be increased as the static rotor eccentricity happens.


## 3. Finite Element Analysis

### 3.1. Model Setup

In this paper, we take the QFSN-600-2YHG type turbo-generator as the study object, and the 3D transient solution type is employed for FEA. During the calculation, we establish the same physical model and external coupling circuit as Ref. [1]. The static rotor eccentricity is set along the X axis, and the eccentricity rate is 0.03.The key parameters of the generator are listed in Table1. The axial length of the stator core is 6300 mm which requests many computing resources in 3D calculation. To balance the calculation accuracy and the resource, one-tenth of the stator, namely, 630 mm, is set for the finite element analysis [1]. The stator core has 42 slots, with the detailed slot distribution illustrated in Figure 2(a). Each slot has two layers of coils, and each coil has an upper line and a lower line inside these slots. For instance, the 34th coil has the upper line in slot 34 and the lower line in slot 9. The windings are of double Y connected, and the winding distribution of phase A is illustrated in Figure 2(b).Table 1
Key parameters of the QFSN-600-2YHG turbo-generator.
ParameterValueParameterValueRated capacity667 MVARated exciting current4128ARated voltage20000 vNumber of stator coils in series per phase7Rated rotating speed3000 rpmNumber of turns in per rotor slot8 (6 for slot 1 and slot 16)Number of pole pairs1Length of rotor6250 mmConnection mode of stator winding2YStack length6300 mmPitch17Inner diameter of stator1316 mmPower factor0.9Outer diameter of stator2674 mmOperating temperature75°CInner diameter of rotor500 mmIndexing slots of rotor50Outer diameter of rotor1130 mmReal slots of rotor32Air-gap length93 mmNumber of stator slots42Stacking factor0.95Rotor slot dimensions60∗40Stator slot dimensions160∗70


### 3.2. Electromagnetic Force

The radial magnetic flux density variation is illustrated in Figure3(a). It shows that the magnetic flux density curves display as a cosine function. As the eccentricity occurs, the MFD value near the 0 degree (X axis) will be increased, while near the 180 degrees, the MFD will be decreased. Such result well accords with Equation (7).Figure 3
MFD variation: (a) circumferential distribution and (b) axial distribution.
(a)(b)The MFD distribution in the axial direction is indicated in Figure3(b), where the coordinate origin is set at the central point of the stator core. Since half of the stator length is 315 mm, the MFD will obtain the larger values in [0, 315], while outside the stator core, the MFD amplitude will be decreased. The occurrence of the static rotor eccentricity will generally increase the MFD inside the stator core, namely, the 0-315 mm region.To better analyze the electromagnetic force properties, the force of the line part and the force on the end part are calculated individually. The force density is firstly calculated. Then, the further integral operation is carried out via the field calculator to obtain the radial, the tangential, and the axial force components, respectively. The positive directions for the radial force and the axial force are set along the radial and the axial axis, respectively, both from inside to outside. For the tangential force, the positive direction is set along the forward rotating orientation. According to the aforementioned theoretical result, the electromagnetic force of the coil within (-90°, -55.7°) will be increased. Specifically, we set the 34th coil (the position is -72.9°) as the particular analysis object.The electromagnetic forces on the end windings are illustrated as Figure4. Normally, the two coils which have the circumferential distance of 180 degrees generally have the same electromagnetic force. For instance, the 13th coil and the 34th coil illustrated in Figures 4(a)–4(d) have the same force amplitudes. This is consistent to Equation (12). However, the occurrence of static rotor eccentricity will make the magnetic flux distribution no longer even, leading to the force increment of the coils near the small air-gap side. For example, the 34th coil has an obvious force increment. As a verification, we feed the detailed angle value αI=−72.9° into Equation (12) and find that Δ>0, which means the force has a positive increase.Figure 4
Force on windings: (a) axial force, (b) radial force, (c) tangential force, and (d) resultant force.
(a)(b)(c)(d)For a better illustration, the force spectra of the 34th coil before and after static rotor eccentricity are shown in Figure5. As indicated in Figure 5, no matter in normal or the static rotor eccentricity condition, the electromagnetic forces include an obvious DC component and the 2nd harmonic. In addition, there are still other even harmonics such as the 4th and the 6th ones, but with much smaller amplitudes. Since the constant force component will not create vibrations (periodic movements/deformations) and the higher-order even harmonics are tiny, the 2nd harmonic force is the primary factor for coil vibration.Figure 5
Force spectrum on coil 34: (a) normal condition and (b) eccentricity condition.
(a)(b)The amplitudes of the 2nd force harmonic of each coil are illustrated in Figure6. It is shown that the radial force component has larger force amplitudes than the tangential and the axial components. Consequently, the coils will have a more intensive vibration in the radial direction. That is why tie lines are applied to restrain the coil’s radial vibration. Since the end windings are organized in two layers and constructed as a basket form, the tangential and the axial vibration will cause insulation wearing between two neighboring layers. Inside the same layer, the radial and the axial vibrations will also cause insulation wearing. Since the radial forces have larger amplitudes, the insulation wearing in the same layer will be more serious than that in different layers.Figure 6
2nd harmonic amplitudes of electromagnetic forces: (a) axial, (b) radial, (c) tangential, and (d) resultant force.
(a)(b)(c)(d)It is shown in Figure6(d) that the phase-end coils (the last coil along the rotating direction of a phase, for instance, the 7th, 14th, 21st, 28th, and 42nd coils) have the max electromagnetic forces and therefore also the more serious vibration and insulation wearing. Moreover, the interphase coils (the coil neighboring to another phase, such as the 7th and 8th coil) have the max tangential force, as shown in Figure 6(c). Therefore, for the different-layer wearing, the interphase coil endures a larger wearing risk than the innerphase coils.As the static rotor eccentricity happens, some coils’ electromagnetic forces as well as the vibration-caused insulation wearing will be increased. Specifically, the radial and the axial force components of the 27th-41st coils will have the increasing tendencies, while the increased tangential forces will appear on the 4th-23rd coils which are about 180 degrees behind the coils that have the increasing radial/axial forces. In general, the 29th-42nd (including the aforementioned 32nd-36th coils in the (-90°, -55.7°) region) coils will have the enlarged resultant forces. Such a phenomenon well accords with the previous theoretical analysis (see Equation (12)).


### 3.3. Mechanical Response

Due to the action of the electromagnetic forces, the coils will endure the time-varying stresses and deformations which are highly potential to cause material fatigue and insulation wearing after a long period. A significant way for antiwearing is to improve the manufacturing technique, for example, spreading a wear-resistant coating on the coil surface of the dangerous region. The key point is to find out the exact dangerous region. In this paper, we carry out the multifield coupling calculation to obtain the mechanical responses such as the deformation and the equivalent stress. The electromagnetic forces are imported from the Maxwell 3D module to the transient structural module. And the detailed results are illustrated in Figure7. It shows that the 35th coil which is at the end of phase C has the max stress and deformation, since it is just next to the minimum air-gap point.Figure 7
Mechanical response: (a, b) stress and (c, d) deformation in normal and eccentricity condition, respectively.
(a)(b)(c)(d)To better illustrate more details about the coil response, the 34th coil is picked as the study object and is divided into 16 segments, as indicated in Figure8(a). The specific max stresses and deformations for each segment are shown in Figures 8(c) and 8(d), respectively. It is shown that the max stress appears on segments O and P which are actually the root part (connecting point between the line part and the end part) of the end winding. Besides, segments G and H which are the middle part of the end winding have the second-largest stresses. For the deformation, the middle part of the end winding has the largest values (segment G). In addition, segment A to F also has an intensified deformation. Since either the overstress or the excessive deformation will lead to the potential damage to the coil, the root part and the top half near the nose are the most dangerous regions overall. These positions need specific reinforcement.Figure 8
Coil 34 involute mechanical response: (a) probe position, (b) simplified model, (c) max stress, (d) max deformation, and (e–g) shear force, bending moment, and deflection, respectively.
(a)(b)(c)(d)(e)(f)(g)In order to explain the max stress and deformation waveform, we simplify the upper involute as a beam, and the force load analysis is shown in Figure8(b), where q′ is the electromagnetic force density whose more details can be found in Ref. [1]. F1 is the support force provided by the stator core, F2 is the towing force brought by the bottom involute, and they can be obtained via
(14)Mnose=F1l−∫0lq′l−zdz=0,F1+F2‐∫0lq′dz=0,where Mnose is the bending moment of the nose part, and l is the beam length in the Z direction.Since the stress is related to the bending moment, we plot the schematic diagrams of the shear and the bending moments, as shown in Figures8(e) and 8(f). It is indicated that the middle parts have the max bending moments. Therefore, the middle part will have the largest stress. This can also be further explained by the bending stress equation (15). However, since the root part is the shape mutating position, it has to face the stress concentration. Consequently, the max stress actually appears on the root part, as shown in Figure 8(c).
(15)σb=MzW,where M is the bending moment, and W is the bending section coefficient.Due to the action of the bending moment, deflections will be produced. The differential equation of the beam deflection can be written as(16)f′′z=−MzEIm,f0=0,f′0=0,where E is the elastic modulus, and Im is the inertia moment.The deflection tendency can be obtained through the twice integration performance on the bending moment, as shown in Figure8(g). For the positions from P to G, the deflection will generally become larger. Such a result is similar to the simulated deformation illustrated in Figure 8(d). But the deformations of G to A in Figure 8(g) are somewhat different from Figure 8(d). This is caused by the difference between the deflection (the deformation in the radial direction, i.e., the partial deformation) and the deformation (the composite one includes not only the radial component but also others).In engineering, the stress of the coil is hard to test due to the space limit and the data transfer difficulty inside a strong electromagnetic field circumstance. However, it is possible to test the vibration (the vibration amplitude is actually the deformation) by using the fiber optic acceleration sensors. In this paper, we carry out the experimental study by fixing two acceleration sensors at two different coils for the vibration signal sampling and verification; more details can be found in the next section.


## 3.1. Model Setup

In this paper, we take the QFSN-600-2YHG type turbo-generator as the study object, and the 3D transient solution type is employed for FEA. During the calculation, we establish the same physical model and external coupling circuit as Ref. [1]. The static rotor eccentricity is set along the X axis, and the eccentricity rate is 0.03.The key parameters of the generator are listed in Table1. The axial length of the stator core is 6300 mm which requests many computing resources in 3D calculation. To balance the calculation accuracy and the resource, one-tenth of the stator, namely, 630 mm, is set for the finite element analysis [1]. The stator core has 42 slots, with the detailed slot distribution illustrated in Figure 2(a). Each slot has two layers of coils, and each coil has an upper line and a lower line inside these slots. For instance, the 34th coil has the upper line in slot 34 and the lower line in slot 9. The windings are of double Y connected, and the winding distribution of phase A is illustrated in Figure 2(b).Table 1
Key parameters of the QFSN-600-2YHG turbo-generator.
ParameterValueParameterValueRated capacity667 MVARated exciting current4128ARated voltage20000 vNumber of stator coils in series per phase7Rated rotating speed3000 rpmNumber of turns in per rotor slot8 (6 for slot 1 and slot 16)Number of pole pairs1Length of rotor6250 mmConnection mode of stator winding2YStack length6300 mmPitch17Inner diameter of stator1316 mmPower factor0.9Outer diameter of stator2674 mmOperating temperature75°CInner diameter of rotor500 mmIndexing slots of rotor50Outer diameter of rotor1130 mmReal slots of rotor32Air-gap length93 mmNumber of stator slots42Stacking factor0.95Rotor slot dimensions60∗40Stator slot dimensions160∗70


## 3.2. Electromagnetic Force

The radial magnetic flux density variation is illustrated in Figure3(a). It shows that the magnetic flux density curves display as a cosine function. As the eccentricity occurs, the MFD value near the 0 degree (X axis) will be increased, while near the 180 degrees, the MFD will be decreased. Such result well accords with Equation (7).Figure 3
MFD variation: (a) circumferential distribution and (b) axial distribution.
(a)(b)The MFD distribution in the axial direction is indicated in Figure3(b), where the coordinate origin is set at the central point of the stator core. Since half of the stator length is 315 mm, the MFD will obtain the larger values in [0, 315], while outside the stator core, the MFD amplitude will be decreased. The occurrence of the static rotor eccentricity will generally increase the MFD inside the stator core, namely, the 0-315 mm region.To better analyze the electromagnetic force properties, the force of the line part and the force on the end part are calculated individually. The force density is firstly calculated. Then, the further integral operation is carried out via the field calculator to obtain the radial, the tangential, and the axial force components, respectively. The positive directions for the radial force and the axial force are set along the radial and the axial axis, respectively, both from inside to outside. For the tangential force, the positive direction is set along the forward rotating orientation. According to the aforementioned theoretical result, the electromagnetic force of the coil within (-90°, -55.7°) will be increased. Specifically, we set the 34th coil (the position is -72.9°) as the particular analysis object.The electromagnetic forces on the end windings are illustrated as Figure4. Normally, the two coils which have the circumferential distance of 180 degrees generally have the same electromagnetic force. For instance, the 13th coil and the 34th coil illustrated in Figures 4(a)–4(d) have the same force amplitudes. This is consistent to Equation (12). However, the occurrence of static rotor eccentricity will make the magnetic flux distribution no longer even, leading to the force increment of the coils near the small air-gap side. For example, the 34th coil has an obvious force increment. As a verification, we feed the detailed angle value αI=−72.9° into Equation (12) and find that Δ>0, which means the force has a positive increase.Figure 4
Force on windings: (a) axial force, (b) radial force, (c) tangential force, and (d) resultant force.
(a)(b)(c)(d)For a better illustration, the force spectra of the 34th coil before and after static rotor eccentricity are shown in Figure5. As indicated in Figure 5, no matter in normal or the static rotor eccentricity condition, the electromagnetic forces include an obvious DC component and the 2nd harmonic. In addition, there are still other even harmonics such as the 4th and the 6th ones, but with much smaller amplitudes. Since the constant force component will not create vibrations (periodic movements/deformations) and the higher-order even harmonics are tiny, the 2nd harmonic force is the primary factor for coil vibration.Figure 5
Force spectrum on coil 34: (a) normal condition and (b) eccentricity condition.
(a)(b)The amplitudes of the 2nd force harmonic of each coil are illustrated in Figure6. It is shown that the radial force component has larger force amplitudes than the tangential and the axial components. Consequently, the coils will have a more intensive vibration in the radial direction. That is why tie lines are applied to restrain the coil’s radial vibration. Since the end windings are organized in two layers and constructed as a basket form, the tangential and the axial vibration will cause insulation wearing between two neighboring layers. Inside the same layer, the radial and the axial vibrations will also cause insulation wearing. Since the radial forces have larger amplitudes, the insulation wearing in the same layer will be more serious than that in different layers.Figure 6
2nd harmonic amplitudes of electromagnetic forces: (a) axial, (b) radial, (c) tangential, and (d) resultant force.
(a)(b)(c)(d)It is shown in Figure6(d) that the phase-end coils (the last coil along the rotating direction of a phase, for instance, the 7th, 14th, 21st, 28th, and 42nd coils) have the max electromagnetic forces and therefore also the more serious vibration and insulation wearing. Moreover, the interphase coils (the coil neighboring to another phase, such as the 7th and 8th coil) have the max tangential force, as shown in Figure 6(c). Therefore, for the different-layer wearing, the interphase coil endures a larger wearing risk than the innerphase coils.As the static rotor eccentricity happens, some coils’ electromagnetic forces as well as the vibration-caused insulation wearing will be increased. Specifically, the radial and the axial force components of the 27th-41st coils will have the increasing tendencies, while the increased tangential forces will appear on the 4th-23rd coils which are about 180 degrees behind the coils that have the increasing radial/axial forces. In general, the 29th-42nd (including the aforementioned 32nd-36th coils in the (-90°, -55.7°) region) coils will have the enlarged resultant forces. Such a phenomenon well accords with the previous theoretical analysis (see Equation (12)).


## 3.3. Mechanical Response

Due to the action of the electromagnetic forces, the coils will endure the time-varying stresses and deformations which are highly potential to cause material fatigue and insulation wearing after a long period. A significant way for antiwearing is to improve the manufacturing technique, for example, spreading a wear-resistant coating on the coil surface of the dangerous region. The key point is to find out the exact dangerous region. In this paper, we carry out the multifield coupling calculation to obtain the mechanical responses such as the deformation and the equivalent stress. The electromagnetic forces are imported from the Maxwell 3D module to the transient structural module. And the detailed results are illustrated in Figure7. It shows that the 35th coil which is at the end of phase C has the max stress and deformation, since it is just next to the minimum air-gap point.Figure 7
Mechanical response: (a, b) stress and (c, d) deformation in normal and eccentricity condition, respectively.
(a)(b)(c)(d)To better illustrate more details about the coil response, the 34th coil is picked as the study object and is divided into 16 segments, as indicated in Figure8(a). The specific max stresses and deformations for each segment are shown in Figures 8(c) and 8(d), respectively. It is shown that the max stress appears on segments O and P which are actually the root part (connecting point between the line part and the end part) of the end winding. Besides, segments G and H which are the middle part of the end winding have the second-largest stresses. For the deformation, the middle part of the end winding has the largest values (segment G). In addition, segment A to F also has an intensified deformation. Since either the overstress or the excessive deformation will lead to the potential damage to the coil, the root part and the top half near the nose are the most dangerous regions overall. These positions need specific reinforcement.Figure 8
Coil 34 involute mechanical response: (a) probe position, (b) simplified model, (c) max stress, (d) max deformation, and (e–g) shear force, bending moment, and deflection, respectively.
(a)(b)(c)(d)(e)(f)(g)In order to explain the max stress and deformation waveform, we simplify the upper involute as a beam, and the force load analysis is shown in Figure8(b), where q′ is the electromagnetic force density whose more details can be found in Ref. [1]. F1 is the support force provided by the stator core, F2 is the towing force brought by the bottom involute, and they can be obtained via
(14)Mnose=F1l−∫0lq′l−zdz=0,F1+F2‐∫0lq′dz=0,where Mnose is the bending moment of the nose part, and l is the beam length in the Z direction.Since the stress is related to the bending moment, we plot the schematic diagrams of the shear and the bending moments, as shown in Figures8(e) and 8(f). It is indicated that the middle parts have the max bending moments. Therefore, the middle part will have the largest stress. This can also be further explained by the bending stress equation (15). However, since the root part is the shape mutating position, it has to face the stress concentration. Consequently, the max stress actually appears on the root part, as shown in Figure 8(c).
(15)σb=MzW,where M is the bending moment, and W is the bending section coefficient.Due to the action of the bending moment, deflections will be produced. The differential equation of the beam deflection can be written as(16)f′′z=−MzEIm,f0=0,f′0=0,where E is the elastic modulus, and Im is the inertia moment.The deflection tendency can be obtained through the twice integration performance on the bending moment, as shown in Figure8(g). For the positions from P to G, the deflection will generally become larger. Such a result is similar to the simulated deformation illustrated in Figure 8(d). But the deformations of G to A in Figure 8(g) are somewhat different from Figure 8(d). This is caused by the difference between the deflection (the deformation in the radial direction, i.e., the partial deformation) and the deformation (the composite one includes not only the radial component but also others).In engineering, the stress of the coil is hard to test due to the space limit and the data transfer difficulty inside a strong electromagnetic field circumstance. However, it is possible to test the vibration (the vibration amplitude is actually the deformation) by using the fiber optic acceleration sensors. In this paper, we carry out the experimental study by fixing two acceleration sensors at two different coils for the vibration signal sampling and verification; more details can be found in the next section.


## 4. Experiment Study

### 4.1. Method

The experiments are taken on the MJF-30-6 prototype generator in the State Key Laboratory of Alternate Electric Power System with Renewable Energy Sources, China, as illustrated in Figure9(a). The primary parameters of the generator are listed in Table 2. The air-gap is tested by a group of high-precision gauges, and the test shows that the minimum air-gap point (static rotor eccentricity position) is 120°, as shown in Figure 10(a). Two accelerometers are fixed to the end winding and marked as A and B, as illustrated in Figure 9(b). Sensor A is next to the minimum air-gap point, while sensor B is on the coil in the neighboring phase. To be more clarified, the section diagram of the generator is displayed in Figure 10(a). Particularly, the aforementioned finite element calculation has the similar probe position as the experiment, as indicated in Figure 10(b). Sensor A is on the 34th coil, while sensor B is on the 41st coil. The 34th coil’s center is on the minimum air-gap position (see Figures 1(d) and 2(a)), and the 41st coil is in the neighboring phase.Figure 9
Fault simulating generator: (a) general outlook and (b) vibration sensor settings.
(a)(b)Table 2
Key parameters of the MJF-30-6 prototype generator.
ParametersRated capacityRated rotationRated exciting currentRated power factorPole pairsPithValues30 kVA1000 r/min1.8A0.838Figure 10
Probe positions of radial acceleration: (a) in experiment and (b) in simulation.
(a)(b)


### 4.2. Results

The vibration result (acceleration signal) is illustrated in Figures11(a)–11(d). Obviously, the 2nd harmonic is the prominent component, showing a good consistency to the previous theoretical result (see Equation (12)). Moreover, the vibration amplitude of sensor A which is closer to the minimum air-gap position is larger than that of sensor B. The simulation data of probes A (on coil 34) and B (on coil 41) shows a similar result. Such a result also matches Figure 6(b) well, and therefore, it is a good verification of the simulation.Figure 11
Time-domain waves and spectra of radial accelerations: (a, b) sensor A in experiment, (c, d) sensor B in experiment, (e, f) probe A in simulation, and (g, h) probe B in simulation.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.1. Method

The experiments are taken on the MJF-30-6 prototype generator in the State Key Laboratory of Alternate Electric Power System with Renewable Energy Sources, China, as illustrated in Figure9(a). The primary parameters of the generator are listed in Table 2. The air-gap is tested by a group of high-precision gauges, and the test shows that the minimum air-gap point (static rotor eccentricity position) is 120°, as shown in Figure 10(a). Two accelerometers are fixed to the end winding and marked as A and B, as illustrated in Figure 9(b). Sensor A is next to the minimum air-gap point, while sensor B is on the coil in the neighboring phase. To be more clarified, the section diagram of the generator is displayed in Figure 10(a). Particularly, the aforementioned finite element calculation has the similar probe position as the experiment, as indicated in Figure 10(b). Sensor A is on the 34th coil, while sensor B is on the 41st coil. The 34th coil’s center is on the minimum air-gap position (see Figures 1(d) and 2(a)), and the 41st coil is in the neighboring phase.Figure 9
Fault simulating generator: (a) general outlook and (b) vibration sensor settings.
(a)(b)Table 2
Key parameters of the MJF-30-6 prototype generator.
ParametersRated capacityRated rotationRated exciting currentRated power factorPole pairsPithValues30 kVA1000 r/min1.8A0.838Figure 10
Probe positions of radial acceleration: (a) in experiment and (b) in simulation.
(a)(b)


## 4.2. Results

The vibration result (acceleration signal) is illustrated in Figures11(a)–11(d). Obviously, the 2nd harmonic is the prominent component, showing a good consistency to the previous theoretical result (see Equation (12)). Moreover, the vibration amplitude of sensor A which is closer to the minimum air-gap position is larger than that of sensor B. The simulation data of probes A (on coil 34) and B (on coil 41) shows a similar result. Such a result also matches Figure 6(b) well, and therefore, it is a good verification of the simulation.Figure 11
Time-domain waves and spectra of radial accelerations: (a, b) sensor A in experiment, (c, d) sensor B in experiment, (e, f) probe A in simulation, and (g, h) probe B in simulation.
(a)(b)(c)(d)(e)(f)(g)(h)


## 5. Conclusion

In this paper, the electromagnetic force and the subsequent mechanical response of the stator end windings in turbo-generators before and after static rotor eccentricity are studied. The primary conclusions based on the theoretical analysis, the FEM calculation, and the experimental study can be drawn up as follows.(1)
The electromagnetic force contains a DC component and the even harmonics. The occurrence of the static rotor eccentricity will cause the force variation. The sufficient condition that the coil force will be increased is that the coil falls within the scopeαI∈−90°,−55.7°(2)
The radial force component is larger than both the axial and the tangential components, leading to the more intensified vibrations in the radial direction. As the static rotor eccentricity happens, the coils next to the minimum air-gap point will be increased(3)
The insulation wearing of the coils inside the same layer is more serious than that between two neighboring layers. For the different-layer wearing, the interphase coil endures a larger wearing risk than the innerphase coils. In normal condition, the vibration on the phase-end coils (the 7th, 14th, 21st, 28th, 35th, and 42nd coils) has the largest amplitude. However, in the static rotor eccentricity case, the amplified vibration will primarily appear in the 29th-42nd coils(4)
Either the overstress or the excessive deformation will lead to the damage to the coil insulation. The phase-end coil which is close to the minimum air-gap point is the most dangerous one. The root part and the middle part of the coil have the largest stress, while the top half near the nose has the largest deformation. These two regions need particular attention and special reinforcement during manufacturingIn the next step, the study of the electromagnetic forces on the rotor windings will be carried out as a supplement to this paper. We will conduct the same analyses for the rotor windings under normal condition and in the rotor eccentricity case, respectively.



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*Source: 5554914-2021-03-30.xml*