Effect of Internal Power-Angle on Turbo-Generator Rotor … · · 2014-04-14EFFECT OF INTERNAL...

EFFECT OF INTERNAL POWER-ANGLE ON TURBO-GENERATOR ROTOR VIBRATIONCHARACTERISTICS UNDER ECCENTRICITY FAULTS

Shuting Wan, Yuling He and Changgeng ZhanDepartment of Mechanical Engineering, North China Electric Power University, Baoding 071003, China

E-mail: [email protected]

Received May 2013, Accepted November 2013No. 13-CSME-124, E.I.C. Accession 3258

ABSTRACTThis paper investigates the effect of the turbo-generator internal power-angle on the rotor radial vibrationcharacteristics under air gap eccentricity faults. Firstly the air gap magnetomotive force, the magnetic per-meance and the magnetic flux density of the eccentricity faults is deduced, and the formula of the magneticpull per unit area is obtained. Then the restrictive factors of the magnetomotive force and the internal power-angle are analyzed. The unbalanced magnetic pull (UMP) that acts on the rotor is further deduced, and therotor vibration characteristics are given. Finally the experiments are taken on a SDF-9 non-salient fault sim-ulating generator to verify the theoretical results. The investigation results of this paper will be beneficial toair gap eccentricity faults diagnosis of turbo-generator.

Keywords: turbo-generator, internal power-angle, eccentricity faults, vibration characteristics, rotor.

EFFET DE L’ANGLE INTERNE DE PUISSANCE SUR LES CARACTÉRISTIQUESVIBRATOIRES DU ROTOR D’UN TURBOGÉNÉRATEUR SOUMIS À DES DÉFAUTS

D’EXCENTRICITÉ

RÉSUMÉCette recherche s’intéresse aux effets de l’angle interne de puissance d’un turbogénérateur sur les carac-téristiques vibratoires radiales du rotor soumis à des défauts d’excentricité de l’entrefer. Premièrement, laforce magnétomotrice de l’entrefer, la perméance et la densité du flux magnétique des défauts d’excentricitésont déduits, on obtient ainsi la formulation de la zone d’attrait magnétique par unité. Ensuite les facteursrestrictifs de la force magnétique et l’angle interne de puissance sont analysés. L’attrait magnétique asymé-trique (UMP) qui agit sur le rotor est déduit à nouveau, et on obtient ainsi les caractéristiques vibratoires.Finalement, les expériences sont poursuivies sur un générateur SDF-9 du type à pôles lisses pour vérifier lesrésultats théoriques. Les résultats de cette recherche seront utiles pour le diagnostic de défauts d’excentricitéde l’entrefer du turbogénérateur.

Mots-clés : turbogénérateur ; angle interne de puissance ; défauts d’excentricité ; caractéristiques vibra-toires ; rotor.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014 63

NOMENCLATURE

f magnetomotive force (A), and frequency (Hz)t time (s)F value of magnetomotive forces (A), and unbalanced magnetic pull (N)U vector of armature phase voltageU value of armature phase voltageI vector of armature phase currentI value of armature phase currentE0 vector of electromotive force caused by the field currentg radial length of the air gap (mm)q magnetic pull per unit area (N/m2)B magnetic flux density (Wb/m2)L axial length of the air gap (m)R radius of the rotor (m), and resistive component of the load (Ω)X inductive and capacitive component of the load (Ω)M number of sampling pointsxs synchronous reactance (Ω)y vibration speed (mm/s)

Greek symbolsα circumferential angle (rad)ω angular frequency (rad/s)ψ internal power-angle of generator (rad)θ power-angle of generator (rad)ϕ power-factor angle of generator (rad)β angle between magnetomotive forces (rad)δ relative air gap eccentricity (mm)Λ magnetic permeance (H)µ0 magnetic permeability of the air (H/m)

Subscriptss denotes static eccentricity, stator or synchronousd denotes dynamic eccentricityr denotes mechanical rotation or rotorm denotes mechanical parametera denotes average parameterc denotes composite parameterO denotes constant parameterX denotes components in X directionY denotes components in Y directioni denotes the order of the parametermax denotes the maximum parameter

1. INTRODUCTION

Air gap eccentricity is one of the main mechanical faults, which can exist in almost every electric generator.This fault can produce unbalanced magnetic pull (UMP) and vibration of the generator, deform the statorcore and damage the winding insulation. Thus, investigation of the influential factors of the UMP and thevibration amplitudes under the presence of this fault is of primary importance.

64 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

The UMP problem has been discussed by many authors, such as Dorrell and Smith [1], Dorrell [2, 3],Frauman et al. [4] and Pennacchi and Frosini [5], and published in many papers about this topic. Most ofthese papers are about motors. Published research about generators is scarce.

The diagnosis, causes, and preventive maintenance for air gap eccentricity of generators were investigatedfirst by Rosenberg [6]. Then, simulation, modeling and several computing methods were presented [7–10].Hamid and Nabil [7] and Tabatabaei et al. [8] used the modified winding function method to model andsimulate the salient-pole synchronous generator, presented the effect of dynamic eccentricity and verifiedthe results by the finite element method. Zhu et al. [9] used a 2D multi-slice model to analyze the effectsof the skewed slots, and illustrated the influence of the eccentricity on the branch induced voltage. Wan andHe [10] analyzed the stator circulating current characteristics in normal and air gap eccentric fault conditionrespectively, and put forward a diagnosis method for the combined faults based on current characteristics.

Meanwhile, UMP and the vibrations of the rotor caused by the air gap eccentricity were studied [11–18].Ohishi et al. [11] studied a new simple analytical method for radial magnetic pull force in salient polemachines with eccentric rotors, and improved the estimation of radial magnetic pull. Burakov and Arkkio[12] investigated two common types of eccentric rotors and found that the parallel stator windings can bemore effective in mitigating the UMP than the rotor cage. Perers et al. [14] reported an investigation ofsaturation effects on the UMP, and gave the conclusion that saturation significantly affects the magnitudeof the UMP. Keller et al. [15] presented a combined analytical and numerical method for computation ofUMP. Wan et al. [16, 17] investigated the stator and the rotor vibration characteristics of turbo-generatorunder the air gap eccentricity fault, gave the conclusion that the static eccentricity causes the rotor vibrationat 2 fr ( fr is the rotor rotating frequency), while the dynamic eccentricity causes the rotor vibration at fr,and this result could provide probably a basis for further research. However, other factors could affect thevibration characteristics, and there is much room for improvement. Besides radial vibration, axial vibrationalso exists [18].

All of these published papers put forward important findings and they are the basis of the on-line and theoff-line detecting systems for this fault. However, none of them studies the effect of the internal power-angleon the UMP or the vibration characteristics. As an improvement, this paper intends to study the UMP on therotor, and investigates the effect of the internal power-angle on the rotor radial vibration characteristics undereccentricity faults. Based on Wan et al. [16, 17], the present paper extends research to the eccentricity fault,by discussing the relationship between the internal power-angle ψ and the angle between magnetomotiveforces β , rearranging the formula of UMP. Conclusions about the influence of the internal power-angle aredrawn.

2. ECCENTRIC AIR GAP MAGNETIC FLUX DENSITY

2.1. Basic Electromagnetic RelationshipNeglecting the magnetic saturation, the electromotive force equation of the non-salient pole synchronousgenerator can be given as follows [19]:

E0 = U + jIxs, (1)

where E0 is the vector of electromotive force caused by the field current, U is the vector of armature phasevoltage, I is the vector of armature phase current, xs is the synchronous reactance. The vector diagramshown in Fig. 1, where the angle between U and I is ϕ (called the power-factor angle), the angle between E0and U is θ (called the power-angle), and the angle between E0 and I is ψ (called the internal power-angle);ψ is denoted by the characters of armature reaction.

Since generators always operate under the condition that the current I lags the phase voltage U and theelectromotive force E0 is in phase, the calculations of the internal power-angle and the power-factor angle


Fig. 1. The vector diagram of the electromagnetic parameter.

can be given as ψ = arctan U sinϕ+Ixs

U cosϕ

ϕ = arccos R√R2+X2

(2)

where U indicates the value of armature phase voltage, I indicates the value of armature phase current, Rindicates the resistive component of the load, X indicates the inductive and the capacitive component of theload.

2.2. Flux Density under the Eccentric FaultThe magnetic flux depends on the magnetomotive force and the magnetic permeance. When eccentricityfaults occur, the magnetic permeance will change while the magnetomotive force remains stable.

The general magnetomotive force can be expressed as [16]

f (αm, t) = Fs cos(

ωt−αm−ψ− π

2

)+Fr cos(ωt−αm) = Fc cos(ωt−αm−β ) (3)

where Fr is the main magnetomotive force, Fs is the magnetomotive force produced by the armature reaction,Fc is the composite magnetomotive force:

Fc = [F2s cos2 ψ +(Fr−Fs sinψ)2]1/2

β = arctg Fs cosψ

Fr−Fs sinψ

(4)

For convenience, the coordinate system shown in Fig. 2 is adopted. The x-axis is defined as the zeroattitude of the angle αm, along which is located the minimum air gap aperture. Dorrell [2], Frauman et al.[4] and Perers et al. [14] considered only the dynamic eccentricity when deducing the air gap formula. Asan improvement, the present paper considers the static eccentricity and the dynamic eccentricity at the sametime, and the air gap length is

g(αm, t) = ga[1−δs cosαm−δd cos(ωrt−αm)] (5)

where ga is the average length of the radial air gap, δs is the relative static eccentricity, δd is the relativedynamic eccentricity, and ωr is the mechanical angular frequency. For turbo-generators, ωr is equal to theelectrical angular frequency ω . In the remainder, angular frequency is written as ω , and fr is written as f .


Fig. 2. The eccentric air gap of turbo-generators.

The magnetic permeance per unit area can be expanded as a power, and the higher harmonic componentscan be ignored because of their small values. Then the magnetic permeance per unit area is [2, 4, 8, 12, 14]

Λ(αm, t) =µ0

g(αm, t)= Λ0[1+δs cosαm +δd cos(ωt−αm)] = Λ0 +Λs cosαm +Λd cos(ωt−αm) (6)

where Λ0 is the constant component of the magnetic permeance, Λs = Λ0δs is the component caused by thestatic eccentricity, and Λd = Λ0δd is the component caused by the dynamic eccentricity.

The magnetic flux density is

B(αm, t) = f (αm, t)Λ(αm, t) = Fc cos(ωt−αm−β )[Λ0 +Λs cosαm +Λd cos(ωt−αm)] (7)

3. MAGNETIC PULL PER UNIT AREA

Referring to the method used by Ohishi et al. [11] and Pennacchi et al. [5], when the eccentricity faultsoccur, the magnetic pull per unit area is given by Eq. (8), while without considering the internal power-angle,the magnetic pull per unit area is given by Eq. (9) [16]:

q(αm, t) =B(αm, t)]2

2µ0=

[ f (αm, t)Λ(αm, t)]2

2µ0

=F2

c

8µ0

(2Λ

20 +Λ

2s +Λ

2d)+(4Λ0Λs cosαm)+(Λ2

s cos2αm)+

(Λ2

d2

cos2β

)+(2ΛsΛd cosωt)

+ [4Λ0Λd cos(ωt−αm)]+ [2ΛsΛd cos(ωt−2αm)]+ [ΛsΛd cos(ωt−2β )]+ [2Λ0Λd(ωt−αm−2β )]

+ [ΛsΛd cos(ωt−2α−2β )]+ [Λ2d cos(2ωt−2αm)]+

[Λ2

s

2cos(2ωt−2β )

]+ [2Λ0Λs cos(2ωt−αm−2β )]+ [2Λ0Λs cos(2ωt−3αm−2β )]+

[Λ2

s

2cos(2ωt−4αm−2β )

]+ [(2Λ

20 +Λ

2s +Λ

2d)cos(2ωt−2αm−2β )]+ [ΛsΛd cos(3ωt−2αm−2β )]

+ [2Λ0Λd cos(3ωt−3αm−2β )]+ [ΛsΛd cos(3ωt−4αm−2β )]+

[Λ2

d2

cos(4ωt−4αm−2β )

](8)


q(αm, t) =F2

c

4µ0

[Λ

20 +

(Λ2

s

2+

3Λ2d

4

)+

ΛsΛd

2cosαm−2Λ0Λs cosαm +

Λ2s

2cos2αm

]+

[3ΛsΛd

4cos(ωt)−3Λ0Λd cos(ωt−αm)+ΛsΛd cos(ωt−2αm)

]+

[ΛsΛd

2cos(2ωt−αm)+(Λ2

0 +Λ2d +Λ

2s cos(αm)−2Λ0Λs cosαm)cos(2ωt−2αm)

]−Λ0Λd cos(3ωt−3αm)+

ΛsΛd

2cos(3ωt−4αm)+

Λ2d

4cos(4ωt−4αm) (9)

It is clear that Eqs. (8) and (9) have the same form, but Eq. (9) does not have the parameter β , whichdepends on ψ according to Eq. (4). As indicated in Eqs. (4) and (8), the internal power-angle ψ affectsthe magnetic pull via β . Since ψ depends on the generator’s load nature, it will have different valuesbecause of the electric network’s changing load. Therefore, analyzing the magnetic pull and the vibrationcharacteristics without considering β is not precise. Moreover„ the effect of the internal-power angle on themagnetic pull per unit cannot be indicated in Eq. (9). The purpose of this paper is to analyze the influenceof the internal-power angle on the vibration characteristics, for which Eq. (8) is a better choice.

4. LIMITING CONDITIONS

4.1. Limiting Condition of β and ψ

Since load of an electric network is generally inductive, the range of the internal power-angle ψ should bein the range 0−π/2, based on the calculation of ψ . Moreover, to transform the mechanical energy to theelectric energy, armature reaction of quadrature axis is necessary. Besides, Fr > Fs, so that 0 < β < π/2,based on Eq. (4).

4.2. Relation among FC,β and ψ

As indicated in Eq. (4), when 0 < β < π/2, FC will be reduced as ψ increases. According to the extremaltheorem and the monotonicity judging principle, β will be a maximum when ψβmax = arcsin(Fs/Fr). In theinterval (0,ψβmax],β will decrease as ψ increases. However, in the case the range [ψβmax,π/2), the result isopposite. Figure 3 shows the relationship between β and ψ when Fr/Fs = 2. As the value of Fr/Fs increases,the curve gradient will be reduced, and the maximum point will shift to the left. However, if the value ofFr/Fs reduces, the result will be reversed.

5. EFFECT OF INTERNAL POWER-ANGLE ON ROTOR VIBRATION CHARACTERISTICS

The mechanical model of the rotor is a solid cylinder. The unit magnetic pull is not enough to cause rotorvibration due to its rigidity. The essentially influential factor for the rotor vibration characteristics is thepulsating peculiarity of the unbalanced magnetic pull (UMP). And the UMP can be calculated as follows [3,11, 14, 15]:

FX = LR∫ 2π

0q(αm, t)cosαmdαm

FY = LR∫ 2π

0q(αm, t)sinαmdαm

F = (F2X +F2

Y )1/2

(10)

where F is the composite UMP and FX and FY are its components in x-axis and y-axis.


Fig. 3. The relation between β and ψ .

The UMP on rotor under the eccentricity fault is given by

FX =LRF2

C π

4µ0[2Λ0Λs +2Λ0Λd cosωt +Λ0Λd cos(ωt−2β )+Λ0Λs cos(2ωt−2β )]

FY =LRF2

C π

4µ0[2Λ0Λd sinωt +Λ0Λd sin(ωt−2β )+Λ0Λs sin(2ωt−2β )]

F =LRF2

C π

4µ0

[5Λ2

s +5Λ2d +4Λ2

d cos2β +10ΛdΛs cosωt

+8ΛdΛs cos(ωt−2β )+4Λ2s cos(2ωt−2β )

]1/2

(11)

There are three types of air gap eccentricities, i.e. static eccentricity, dynamic eccentricity, and mixedeccentricity that is composed of both. Based on Eqs. (8) and (11), further analysis of the rotor vibrationcharacteristics is given as below.

5.1. Static Air Gap EccentricityIn this case, Λd = 0,Λs 6= 0, then Eq. (11) can be simplified as

FX = LRF2C π[2Λ0Λs +Λ0Λs cos(2ωt−2β )]/4µ0

FY = LRF2C πΛ0Λs sin(2ωt−2β )/4µ0

F = LRF2C πΛ0Λs[5+4cos(2ωt−2β )1/2/4µ0

(12)

As indicated in Eq. (12), the forced rotor vibration of frequency 2 f exists. The value of the compositemagnetic pull changes when the rotor is rotating, but the direction of the composite UMP remains stable be-cause of static eccentricity. The internal power-angle ψ affects FC and β so that the UMP will be influenced,both at the beginning of the excitation and during the running process.

Setting Fr/Fs = 2 and ignoring the constant factor LRπΛ0Λs/4µ0, the relationship between F and ψ under25% static eccentricity is shown in Fig. 4. It is clear that the value of the composite UMP is reduced as ψ

increases, which means that the rotor vibration is also reduced as the internal power-angle increases.


Fig. 4. The relation between F and ψ under static eccentricity.

Fig. 5. The relation between F and ψ under dynamic eccentricity.

5.2. Dynamic Air Gap EccentricityIn this case, Λd 6= 0,Λs = 0 and hence Eq. (11) can be simplified as

FX = LRF2C π[2Λ0Λd cosωt +Λ0Λd cos(ωt−2β )]/4µ0

FY = LRF2C π[2Λ0Λd sinωt +Λ0Λd sin(ωt−2β )]/4µ0

F = LRF2C πΛ0Λd [5+4cos2β ]1/2/4µ0

(13)

As indicated in Eq. (13), the values of FX and FY will change as the rotor rotates while the value of thecomposite UMP will stay the same. However, since it is a dynamic eccentricity, the eccentric location of theair gap and the composite UMP’s direction will change as the rotor rotates. Thus, the dynamic eccentricityimposes on the rotor a forced vibration of frequency f . The internal power-angle ψ affects FC and β , so thatthe UMP will be influenced, both at the beginning of the excitation and during the whole running process.

Setting Fr/Fs = 2 and ignoring the constant factor LRπΛ0Λd/4µ0, the calculated relation between F andψ under 25% dynamic eccentricity is shown in Fig. 5. Since the composite UMP is unrelated to the time,


Fig. 6. The relation between F and ψ under mixed eccentricity.

as indicated in Eq. (13), the figure is two-dimensional. In fact, the figure is generally the same as the curvein Fig. 4, for the composite UMP has the similar amplitude formula under static eccentricity and dynamiceccentricity. It is clear that the composite UMP is reduced as ψ increases. Thus, the rotor vibration is alsoreduced as the internal power-angle increases.

5.3. Mixed Air Gap EccentricityIn this case, Λs 6= 0,Λd 6= 0, then Eq. (11) is the UMP expression.

As indicated in Eq. (11), the forced rotor vibration of frequencies f and 2 f exist. The amplitude and thedirection of the composite UMP changes as the rotor rotates. Obviously, ψ affects FC and β . Thus, bothat the beginning of the excitation and during the whole running process, the UMP will be influenced by theinternal power-angle.

Setting Fr/Fs = 2 and ignoring the constant factor LRπΛ0/4µ0, the relation between F and ψ under 25%static eccentricity and 25% dynamic eccentricity is shown in Fig. 6. It is clear that the UMP is reduced as isψ increased, which means that the rotor vibration is also reduced as the internal power-angle increases.

6. EXPERIMENTAL VERIFICATION

6.1. Experimental Equipment and MethodThe experiment is based on a SDF-9 non-salient pole simulating generator, as shown in Fig. 7(a). The statorwinding is two-layer and pitch-shortening, and the corresponding parameters are:

• Rated capacity: 7.5 kVA.

• Rated exciting current: 1.5 A.

• Rated speed: nr = 3000 r/min.

• Number of pole-pairs: p = 1.

• Radial length of air gap: 0.8 mm.


Fig. 7. SDF-9 non-salient fault simulating generator. (a) The general layout of the simulating generator set, (b) thepictorial diagram of simulation platform, (c) the method to set static eccentricity on the generator in front, and (d) theback of the set.

• Pitch-shortening value: 0.966.

• Number of exciting turns per pole: 480.

• Rated voltage: 400 V.

• Synchronous reactance: xs = 2.802 Ω.

The generator stator can be moved along the horizontally radial direction while the rotor remains stable.The movement can be controlled by two dial indicators so that the static eccentricity can be simulated, asshown in Figs. 7(c) and (d). A CD-21S speed sensor and a CD-21C speed sensor (both made by Beijingvibration instrument factory, with sensitivity 30 mv/mm/s) are respectively set on the bearing block in thehorizontal direction and the stator core in the vertical direction, as shown in Fig. 8.

A U60116C collector made by Beijing Bopu Co. Ltd is used. The red one is used to measure the vibrationof stator, the blue one is used to measure the vibration of rotor through the bearing seat. The samplingfrequency is 10 kHz. During the experiment, the generator is operated under the rated condition (linevoltage is 400 V, phase voltage is 230 V and phase current is 4.9 A) and is not connected to the electricnetwork. A 25% static eccentricity is firstly set. Then different values as shown in Table 1 are respectivelyset.

To verify the effect of ψ on the vibration characteristics, a G03-30/16 induction voltage regulator made byBeijing Changping electric motor factory and three slider rheostats are connected to the generator to simulatethe symmetrically triphase load, as shown in Fig. 9. When the slider rheostats change their resistance values


Fig. 8. The method to set speed sensors.

Fig. 9. The method to set internal power-angle: (a) the slider rheostat, (b) the induction voltage regulator, (c) theequivalent experimental circuit model.

at a same level, the generator load nature will be also changed. By this means, the change of can besimulated.

6.2. Experimental Results and DiscussionAs has been verified and previously analyzed [16], the static air gap eccentricity fault causes the rotorto vibrate at frequency 2 f . Thus, the vibration component of 2 f is the essentially effective component.


Table 1. The Experimental data set of the internal power-angle.Order Resistance (R) Inductance (X) ϕ ψ

1 950Ω j210Ω 12.46 15.75

2 570Ω j210Ω 23.85 26.90

3 190Ω j210Ω 47.86 50.06

4 0Ω j210Ω 90 90

Fig. 10. Spectra of rotor vibration under static eccentricity: (a) ψ = 15.75, (b) ψ = 26.90, (c) ψ = 50.06, and(d) ψ = 90.

Figure 10 shows the vibration speed spectra, and Table 2 offers the detailed data. As indicated in Table 2,the effective vibration component of frequency 2 f decreases as the internal power-angle increases.

To further verify the influence of the internal power-angle on the rotor vibration characteristics in thewhole bandwidth when the eccentricity faults happen, the vibration intensity method in time domain isadopted. The vibration intensity stands for the average vibration level. The experimental time domainvibration conditions of the rotor are shown in Fig. 11. The vibration intensity is given by Eq. (14) [20],where yi is the vibration speed, m is the total number of sampling points and y is the vibration intensity. Theresults are shown in Table 3.

y =

(m

∑i=1

y2i /m

)1/2

(14)


Table 2. Rotor vibration speeds of each frequency under static air gap eccentricity fault (mm/s).Internal power Frequency (Hz)

angle ψ f 2 f 3 f15.75 4.1 11.6 326.90 4 11.0 3.1550.06 4 10.0 3.15

90 1.95 8.75 1/1

Table 3. Rotor vibration intensities under static air gap eccentricity fault (mm/s).Value of ψ Rotor vibration intensity15.75 714.879326.90 689.456850.06 675.652590 577.1885

Fig. 11. Time domain diagrams of rotor vibration under static air gap eccentricity fault. (a) ψ = 15.75, (b) ψ = 26.90,(c) ψ = 50.06, and (d) ψ = 90.

As is seen in Table 3, the radial vibration intensity is reduced as the internal power-angle increases underthe static air gap eccentricity. In other words, the general vibration of the rotor decreases as the internalpower-angle increases.

To compare the previously analytical results with the experimental ones, the theoretical and the experi-mental vibration developing trends under static air gap eccentricity fault are shown in Fig. 12. It is clear thatas the internal power-angle increases, the theoretically magnetic pull amplitude, the experimentally radial vi-


Fig. 12. Developing trend comparison between theoretical analysis and experiment results in static eccentricity condi-tion. (a) Theoretical magnetic pull amplitude. (b) Tested vibration speed.

bration speed of the effective component, and the experiment vibration intensity decrease. The experimentalresults are in agreement with the theoretical analysis results.

7. CONCLUSIONS

This paper studies a particular problem that has been discussed by very few researchers previously. Inthis work, the effect of the internal power-angle of turbo-generator on the rotor vibration characteristicsunder eccentricity faults is investigated, which may have significant value for the air gap eccentricity faultdiagnosis and offers a reference for the further research in this topic. The presented research is based on thededuction of theoretical formula and the experimental verification. The conclusions drawn from the researchcan be given as follows:

1. This paper obtained the relationship of the internal power-angle ψ and the angle β : the internal power-angle ψ of generators and the angle β between the main magnetomotive force and the compositemagnetomotive force have a limited range of 0 to π/2.

2. This paper analyzed the UMP characteristic and the relation between UMP and internal power-angletheoretically: eccentricity faults cause the UMP acted on the rotor, and the UMP will decrease as theinternal power-angle increases no matter which one of the three eccentricity faults occurs.

3. This paper analyzed the influence of vibration caused by internal power-angle theoretically and ex-perimentally: the internal power-angle will affect the rotor vibration characteristic under eccentricityfaults. No matter which one of the three eccentricity faults occurs, the radial vibration in each fre-quency component and the vibration intensity of the rotor will decrease as the internal power-angleincreases.

4. This research is useful for the qualitative vibration analysis of rotating machines. As we know, vi-bration intensity becomes severe when the eccentricity occurs. In order to decrease the vibration andkeep the machine working, the operator could increase the internal power-angle by reducing the pro-portion of the inductance artificially. In this case, the method can weaken the eccentric vibration in anemergency situation.


The experimental results under static eccentricity match well the theoretical ones. Based on research resultsin this paper, a device or a system might be developed to monitor rotor vibration. And this result makesit possible to monitor the vibration of generator under the eccentricity faults comprehensively. However,experiments of dynamic air gap eccentricity fault and mixed air gap eccentricity fault are still needed.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (Nos. 51177046 and51307058), Natural Science Foundation of Hebei province (No.E2011502024) and the Fundamental Re-search Funds for the Chinese Central Universities (Nos. 12MS101 and 13QN49).

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Journal of Academy of Armored Force Engineering, Vol. 22, No. 4, pp. 46–49, 2008

APPENDIX

According to Eqs. (6) and (7), the formula of the magnetic pull per unit area (Eq. 8) can be given as follows,considering the influence of the internal power-angle:

q(αm, t) =[B(αm, t)]2

2µ0=

[ f (αm, t)Λ(αm, t)]2

2µ0

=Fc cos(ωt−αm−2β )[Λ0 +Λs cosαm +Λd cos(ωt−αm)]2

2µ0

=F2

c

8µ0

(2Λ

20 +Λ

2s +Λ


s cos2αm)+

(Λ2

d2

cos2β

)+(2ΛsΛd cosωt)

+ [4Λ0Λd cos(ωt−αm)]+ [2ΛsΛd cos(ωt−2αm)]+ [ΛsΛd cos(ωt−2β )]

+ [2Λ0Λd(ωt−αm−2β )]+ [ΛsΛd cos(ωt−2αm−2β )]+ [Λ2d cos(2ωt−2αm)]

+

[Λ2

s

2cos(2ωt−2β )

]+[2Λ0Λs cos(2ωt−αm−2β )]+ [(2Λ

20 +Λ

2s +Λ

2d)cos(2ωt−2αm−2β )]

+ [2Λ0Λs cos(2ωt−3αm−2β )]+

[Λ2

s

2cos(2ωt−4αm−2β )

]+[ΛsΛd cos(3ωt−2αm−2β )]

+ [2Λ0Λd cos(3ωt−3αm−2β )]+ [ΛsΛd cos(3ωt−4αm−2β )]+

[Λ2

d2

cos(4ωt−4αm−2β )

]Wan et al. [16] neglected the influence of the internal power-angle, and the derivation of Eq. (9) can be

given as follows:

q(αm, t) =[B(αm, t)]2

2µ0=

[ f (αm, t)Λ(αm, t)]2

2µ0

=Fc cos(ωt−αm)[Λ0 +Λs cosαm +Λd cos(ωt−αm)]

2µ0

=F2

c

8µ0

(2Λ

20 +Λ

2s +Λ


s cos2αm)+Λ2

d2

+(2ΛsΛd cosωt)

+ [4Λ0Λd cos(ωt−αm)]+ [2ΛsΛd cos(ωt−2αm)]+ [ΛsΛd cosωt]+ [2Λ0Λd(ωt−αm)]

+ [ΛsΛd cos(ωt−2αm)]+ [Λ2d cos(2ωt−2αm)]+

[Λ2

s

2cos2ωt

]+[2Λ0Λs cos(2ωt−αm)]+ [(2Λ

20 +Λ

2s +Λ

2d)cos(2ωt−2αm)]+ [2Λ0Λs cos(2ωt−3αm)]


+

[Λ2

s

2cos(2ωt−4αm)

]+[ΛsΛd cos(3ωt−2αm)]+ [2Λ0Λd cos(3ωt−3αm)]

+ [ΛsΛd cos(3ωt−4αm)]+

[Λ2

d2

cos(4ωt−4αm)

]=

F2c

4µ0

[(Λ

20 +

Λ2s

2+

3Λ2d

4

)+

ΛsΛd

2cosαm−2Λ0Λs cosαm +

Λ2s

2cos2αm

]+

[3ΛsΛd

4cos(ωt)−3Λ0Λd cos(ωt−αm)+ΛsΛd cos(ωt−2αm)

]+

[ΛsΛd

2cos(2ωt−αm)+(Λ2

0 +Λ2d +Λ

2s cos(αm)−2Λ0Λs cosαm)cos(2ωt−2αm)

]−Λ0Λd cos(3ωt−3αm)+

ΛsΛd

2cos(3ωt−4αm)+

Λ2d

4cos(4ωt−4αm)


Effect of Internal Power-Angle on Turbo-Generator Rotor … · · 2014-04-14EFFECT OF INTERNAL...

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