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An improved analytical model for salient pole synchronous machines under general eccentricity fault.

1. INTRODUCTION

To model electrical machines under different types of faults, winding function approach (WFA) [1] and finite element method (FEM) [2] have been introduced as appropriate modeling methods. Although FEM provides precise modeling, its application is complicated and time consuming especially for the analysis of electrical machines with asymmetry in the motor body such as eccentricity [2, 3].

WFA is based on the basic geometry and winding layout of machine [1]. By this approach, it is possible to analyze the behavior of any machine with any winding distribution and air gap length. Hence this method has gained a broad application in the analysis of faulty electrical machines, such as broken rotor bars [4] and fault condition in stator windings [5].

Analysis of electrical machines with non-uniform air gap using WFA was not accurate until MWFA was proposed [6]. This theory has been applied to analyze static, dynamic and mixed eccentricity in induction and synchronous machines [7-9]. In [10], MWFA was extended for radial and axial non-uniform air gap in SPSMs.

An essential step of this approach is the calculation of machine inductances. An accurate inductances calculation is necessary to improve the accuracy of the analysis of electrical machines. In [12], an analytical equation of the air gap for an eccentric SPSM was presented. The inductances have been calculated by expressing the turn functions and inverse air-gap function in their Fourier series expansions. A numerical integration has been used to determine the values of inductances.

In the previous works, based on winding function theory, the calculation of SPSM inductances is carried out using numerical integration or inexact analytical equations based on approximated Fourier series expansions of the inverse air gap function [11-16], except [17], in which, the air gap function has been estimated by a step function.

In the present paper, more realistic inverse air-gap function of an eccentric SPSM is used to derive pure analytical expressions for inductances without any development in Fourier series. This will ensure that all space harmonics ignored by the Fourier series expansions of the inverse air gap function are included in the model. The influence of the flux fringing effect on the air gap function has been taken into account by using an effective air gap function. This results an increase in the accuracy of calculating machine inductances.

To the best of our knowledge, this is the first study for developing analytical method for calculation of SPSM inductances under different kinds of eccentricities, considering flux fringing effect, without any development in Fourier series. It is clear that the proposed technique decreases the time and computation process and leads to more accurate results.

To develop analytical expressions for inductances, the effective inverse air gap function of the eccentric SPSM has been defined, and its indefinite integral has then been determined. Developed analytical expressions make it possible to calculate inductances of SPSM with any eccentricity type and degree in the frame of a single program. By using the presented method, inductances of a SPSM with eccentric rotor are calculated. The effects of several rotor asymmetries on the inductances are shown and the inductances are evaluated. Theoretical fundamentals and experimental results that validate the proposed technique are presented. Simulation results show that the result with proposed method is closer to the experimental result than that of available analytical methods.

The contributions of the present paper includes: 1) Defining the inverse air gap function of the eccentric SPSM and determining its indefinite integral, taking into account the flux fringing effect and 2) Developing a new closed form analytical expression for calculation of SPSM inductances under different types of eccentricities, using more realistic air gap function.

2. ECCENTRICITY MODELING

When eccentricity fault occurs in an electrical machine, three geometric centers; stator ([o.sub.s]), rotor ([o.sub.r]) and rotor rotation ([o.sub.c]) don't coincide with each other.

There are three types of air-gap eccentricity: static, dynamic and mixed eccentricity [8]. Static eccentricity (SE) occurs when the rotor rotates about its own centerline, but this centerline does not coincide with that of the stator bore. Dynamic eccentricity (DE) occurs when the rotor geometric center is not at the center of rotation, producing an air-gap periodic variation as a rotor position function. In mixed eccentricity (ME) both rotor geometric center and the center of rotation are displaced individually in respect to the stator geometric center. Figure 1 shows an elementary SPSM for three types of eccentricities.

The air gap length variation under ME for an induction machine can be described by air gap function as follows [18]:

g([phi], [[phi].sub.m], [delta]) = [g.sub.0] (1 - [delta] cos([phi] - [[phi].sub.m])) (1)

where

[delta] = [[[[delta].sup.2.sub.s] + [[delta].sup.2.sub.d] + 2[[delta].sub.s][[delta].sub.d] cos ([theta] - [[phi].sub.s])].sup.1/2] (2)

[[phi].sub.m] = [[phi].sub.s] + [tan.sup.-1] ([[delta].sub.d] sin ([theta] - [[phi].sub.s)/ [[[delta].sub.s] + [[delta].sub.d]cos ([theta] - [[phi].sub.s])] (3)

where, [g.sub.0] is air gap length for healthy machine, [theta] the rotor position angle in stator reference frame, [phi] the angle in stator reference frame, [[phi].sub.s] the angle at which rotation and stator centers are separated, and [[delta].sub.s] and [[delta].sub.d] are static and dynamic eccentricity levels, respectively.

These relations are used in the following section for developing analytical expressions of inductances of eccentric SPSM.

3. ANALYTICAL EQUATIONS OF INDUCTANCES FOR GENERAL ECCENTRICITY

According to MWFA, mutual inductance between two windings a and b in electrical machines, can be calculated by the following equation [6]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[[mu].sub.0] is the free space permeability, l the axial stack length of the machine, [g.sup.-1] the inverse air gap function, [n.sub.a] the turns function of winding a, and [M.sub.b] the modified winding function of winding b.

Available analytical methods for calculation of inductances of eccentric SPSM are based on the winding function theory. Although available analytical methods provide good knowledge about performance of eccentric SPSM, they are based on simplification and geometrical approximation of unsymmetrical models of the machine under different eccentricities. In [12], the values of inductances of an eccentric SPSM have been determined using Fourier series expansions of the inverse air gap function and numerical integration. In the recent studies [17, 20], analytical expressions have been presented assuming step function (Figure 2) for air gap distribution. In the present paper, effective air gap function which considers flux fringing effect is used to develop analytical expressions without any development in Fourier series.

To derive analytical expressions for inductances, the varying air gap length, due to the pole saliency, eccentricity and slots should first be modeled.

The influence of the pole saliency, can be taken into account by using the effective pole opening distance. The effective opening between the poles is slightly larger than the actual opening due to flux fringing. The air-gap function is computed by modeling the flux paths through the air-gap regions using straight lines and circular arc segments. The details of air gap function modeling in a salient pole synchronous machine considering flux fringing effect have been given in [19].

Knowing the actual opening and air gap length, the effective opening is calculated as [19]:

O' = [[gamma]/[beta]]g (6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where, o is the actual opening length. The effective air gap length within the effective pole opening, for a healthy SPSM can be expressed as

[g.sub.eff] ([empty set]) = g/[1-[beta] (1 - cos 2[pi]/o'[phi])] (8)

The eccentricity can be taken into account according to (1). Therefore, the effective air gap length of an eccentric SPSM, in interpolar space, can be described by air gap function as follows

[g.sub.q] ([phi], [[phi].sub.m], [delta]) = [g.sub.0](1- [delta]cos([phi] - [[phi].sub.m]))/ [1-[beta] (1 - cos 2[pi]/o'[phi])] (9)

Elsewhere, in polar space, the air gap length is

[g.sub.d]([phi], [[phi].sub.m], [delta]) = [g.sub.0] (1 - [delta] cos ([phi] - [[phi].sub.m])) (10)

The subscripts d and q refer to the polar and interpolar regions, respectively.

In the case of static eccentricity, [delta] = [[delta].sub.s] and [[phi].sub.m] = [[phi].sub.s] and in dynamic eccentricity [delta] = [[delta].sub.d] and [[phi].sub.m] = [[phi].sub.d]. In mixed eccentricity, both [delta] and [[phi].sub.m] are rotor position dependent as (2) and (3). Therefore we can define air gap function for SPSM with a general eccentricity fault, including static, dynamic and mixed eccentricities.

Figure 3 shows the effective air gap length for a four pole SPSM under 30% SE condition. This effective function is used in the present study for developing analytical equations.

Finally, the stator slots effect is taken into account by Carters' factor referred to the slots [19].

3.1. Inductances of Stator Phases

To derive the analytical equations for inductances of stator phases, it is assumed that the functions yd and yq are indefinite integrals as follows

[Y.sub.d] ([phi], [[phi].sub.m], [delta]) = [integral] [g.sup.-1.sub.d] ([phi], [[phi].sub.m], [delta])[d.sub.[phi]] (11)

[Y.sub.d] ([phi], [[phi].sub.m], [delta]) = [integral] [g.sup.-1.sub.q] ([phi], [[phi].sub.m], [delta])[d.sub.[phi]] (12)

where, [g.sup.-1.sub.d] and [g.sup.-1.sub.q] are inverse air gap functions in polar and interpolar regions, respectively. Expression (11) is elaborated to yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Function [y.sub.q] is obtained as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where, o" = 2[pi]/o'. [y.sub.q] can be defined as follows

[y.sub.q] ([phi], [[phi].sub.m], [delta]) = [[1 - [beta]]/[g.sub.0]][y.sub.d] + [[beta]/[g.sub.0]] [y.sub.q1] (15)

where, [y.sub.d] is derived earlier in the paper, and [y.sub.q1] can be obtained by some mathematical manipulations. This function is given in Appendix A.

It is assumed that functions [y.sub.1] and [y.sub.2] are the definite integrals as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Separating the associated terms of polar and interpolar spaces, (16) results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where, [[theta].sub.q] = o' (effective pole opening distance) and [[delta].sub.d] = 2[pi]/p - [[theta].sub.q] in radian. Using the definitions given in (11) and (12), [y.sub.1] is obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where [y.sub.d] and [y.sub.q] are calculated from (13) and (15), respectively. In a similar way, [y.sub.2] is calculated in the whole range of [phi]. Function y is defined as follows:

y([[phi].sub.m], [theta], [delta]) = [y.sub.2]([[phi].sub.m], [theta], [delta]/[y.sub.1]([[phi].sub.m], [theta], [delta] (20)

Considering (5), (16), (17) and (20), MWF of winding b, [M.sub.b] is derived as

[M.sub.b]([phi], [[phi].sub.m], [theta], [delta]) = [n.sub.b]([phi]) - y([[phi].sub.m], [theta], [delta]) (21)

To drive an expression for stator phases inductances, (21) is replaced in (4), then the associated terms of polar and interpolar regions are separated. Finally, [L.sub.ab] is obtained as (22). The details of evaluation of [L.sub.ab] have been given in Appendix B.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

In (22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the stator slots being in polar region [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the stator slots being in interpolar region ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

3.2. Inductances of Rotor Windings

The procedure presented in the previous section for stator phases inductances, can be extended to rotor ones. To begin, (4) and (5) are rewritten in the rotor reference frame.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](23)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

[beta] is angle in rotor reference frame. It is assumed that the functions [y.sub.rd] and [y.sub.rq] to be the indefinite integral of the functions [g.sup.-1.sub.d] and [g.sup.-1.sub.q] in the rotor reference frame, respectively. Using some integration rules, [y.sub.rd] and [y.sub.rq] will be obtained. [y.sub.rd] and [y.sub.rq] are the same as [y.sub.d] and [y.sub.q], but [phi] and [[phi].sub.m] are replaced with [beta] and [[beta].sub.m] respectively.

Equation (16) is rewritten in rotor reference frame, as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [y.sub.rd] and [y.sub.rq] are derived earlier in the paper. In the same way, [y.sub.r2] can be derived.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

where, [[beta].sub.di] = 2[pi](i - 1)/p and [[beta].sub.di] = 2[pi](i - 1)/p + [pi]/4. [M.sub.b] in rotor reference frame is defined as follows

[M.sub.b]([beta], [[beta].sub.m], [delta]) = [n.sub.b]([beta]) - [y.sub.r2]([[beta].sub.m], [delta]/[y.sub.r1]([[beta].sub.m], [delta]) = [n.sub.b])[beta]) - [y.sub.r]([[beta].sub.m], [delta] (27)

where [y.sub.r1] and [y.sub.r2] are calculated from (25) and (26).

Replacing (27) in (23) and separating the associated terms of polar and interpolar regions, the expression for rotor winding inductance is derived as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

where [y.sub.r] and [y.sub.rd] and [y.sub.rq] are derived earlier in the paper.

3.3. Mutual Inductances between Stator and Rotor

For mutual inductances between rotor windings and stator phases, the stator reference frame is used. An analytical expression for mutual inductances is obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

where a and b are accounted for the rotor windings and stator phases, respectively. Functions y, [y.sub.d] and [y.sub.q] are defined earlier in the paper.

Therefore, the inductances of SPSM under different kinds of eccentricities can be calculated in a unified technique by means of (22), (28) and (29). In the presented analytical expressions, unlike previous proposals, effective form of pole shoes considering flux fringing effect is taken into account.

4. COMPUTATION OF INDUCTANCES

In this section, by means of developed analytical expressions, inductances of a SPSM for healthy and different eccentricity conditions are calculated in a unified technique, where the eccentricity type and degree are selectable. A typical SPSM, with specifications given in Appendix C, is considered in this paper. It should be noted that, magnetic saturation and leakage flux are not included.

4.1. Magnetizing and Mutual Inductances of Stator Phases

Magnetizing inductance of stator phase a and mutual inductance between phases a and b of the stator have been computed under different eccentricity conditions and are shown in Figures 4 and 5.

Comparison of plots in Figures 4 and 5 for the three types of eccentricity shows that while ME causes asymmetrical self and mutual inductances, SE and DE cause symmetrical inductances. By increasing the eccentricity severity, the magnitude of these inductances increases.

4.2. Magnetizing Inductance of Rotor Winding

The plots of magnetizing inductance of rotor winding for different eccentricities are shown in Figure 6. Comparison of plots in Figure 6 shows how SE, DE and ME affect the profile of the magnetizing inductance of rotor winding. ME causes a pulsating magnetizing inductance of rotor winding, whereas, DE and SE cause a symmetrical inductance with a larger magnitude compared to the non-eccentric condition. By increasing the eccentricity level, the magnitude of this inductance increases.

4.3. Mutual Inductances between Stator Phases and Rotor Winding

Figure 7 shows the mutual inductance between stator phase a and rotor winding for different eccentricity cases.

Comparison of the inductance profiles in Figure 7 indicates that SE and DE increase the magnitude of mutual inductance whereas there is no asymmetry in the inductance distribution and ME creates asymmetrical inductance distribution.

5. EXPERIMENTAL RESULTS

To validate the theoretical and simulated results, experiments were performed on a machine identical to the one simulated. A technique was used to make the machine temporarily eccentric with different types and degrees of eccentricities. The stator and the rotor bearings at the two ends of the motor were separately mounted on the test bed. Therefore, it was possible to displace any of bearings separately and create different static eccentricities. To be able to create dynamic eccentricity, bearings were replaced by other bearings with different diameter. Special screws were used to fix the bearings. The position of bearings was accurately checked by a clock. The clock and special screw are shown in Figure 8.

To obtain the mutual inductance between stator phase a and stator phase b, a sinusoidal voltage was applied in stator phase a and the voltage induced in the stator phase b was measured as a function of the rotor position from 0[degrees] to 360[degrees] taking 3[degrees] steps. The mutual inductance was calculated as follows:

[L.sub.ab] ([theta]) = [V.sub.b]([theta])/[V.sub.a]([theta]) ([theta]) (30)

where [V.sub.a]([theta]) and [V.sub.b]([theta]) are induced voltages in stator phase a and stator phase b, respectively and [L.sub.aa]([theta]) is self inductance of stator phase a, previously calculated by means of phase a voltage and current measurements.

The induced voltage in phase A was derived as follows:

[V.sub.A] = [square root of ([V.sup.2.sub.s] - [([r.sub.A][i.sub.A]).sup.2])] (31)

where [V.sub.s] is the applied voltage, [i.sub.A] is the current following in stator phase A and [r.sub.A] is stator phase A resistance (1.36 [OMEGA]). To avoid the magnetic saturation, a low ac voltage was applied. The phase A voltage drop was negligible in the experiments.

The mentioned process was repeated to obtain the mutual inductance between stator phase a and rotor winding.

Figures 9 and 10 show the profiles of mutual inductances under eccentricity conditions for the experiment and simulation. It is apparent that the inductance profiles coming from the simulation and experiment demonstrate a satisfactory mach. The reasons of this agreement are as follows: 1) a more realistic distribution of the air gap in the presence of eccentricity considering flux fringing effect is used, 2) development in Fourier series of the inverse air gap function has not been used, but closed form analytical equations are employed for inductances calculation and 3) real spatial distribution of stator phases and rotor winding has also been taken into account.

Figure 11 shows the mutual inductance between stator phase a and phase b for machine under 50% static eccentricity from simulations and experiment. The inductance was calculated using proposed analytical method and available analytical method in which, the air gap function is estimated by a step function. The comparison between inductance profiles indicates that the maximum value of the inductance from the proposed method is slightly lower than the available method result. The main reason is that in the proposed method the flux fringing effect is considered, while in previous analytical method, based on winding function theory, this effect is excluded.

As shown, the result with proposed method is closer to the experimental result than that of available method. Probably, the difference between results from the proposed method and the experiment is due to neglecting the slots flux leakages. This claim needs more detailed study.

6. CONCLUSIONS

In this study, a new analytical model for SPSM under general eccentricity fault has been developed, capable of accounting for the machine geometry, effect of flux fringing, slots effect and spatial distribution of machine windings. While in previous analytical methods, air gap function has been estimated and therefore geometrical model has been simplified, in this paper, effective air gap function which considers flux fringing effect is taken into account.

By using the effective air gap function of the eccentric SPSM and determining its indefinite integral, precise analytical equations for inductances have been obtained. Derived comprehensive equations allow calculating time varying inductances of SPSM under any eccentricity type and degree, with a general simulation technique. Analytical equations prevent imprecision due to numerical differentiations. Moreover, the computational time is decreased thanks to an exact and analytical definition of the distribution functions, in contrast to methods based on only an approximation with Fourier series.

Simulation results show that the result with the proposed method is closer to the experimental result than that of available analytical methods. The main reason is that in the proposed method the flux fringing effect is considered, while in previous analytical method, based on winding function theory, this effect is excluded and the air gap distribution is estimated by step function. Since the calculation of inductances is an essential step for simulation and analysis of fault in electrical machines, the proposed method will improve the online diagnosis of the fault. The author plans to apply the evaluated inductances in a coupled electromagnetic model of SPSM in the future for fault analysis.

APPENDIX A.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX B.

Replacing (21) in (4) and separating the associated terms of polar and interpolar regions, Lab is obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B1)

where a and b are accounted for the stator phases. It is assumed that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the stator slots being in polar region [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the stator slots being in interpolar region ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B2)

where [[phi].sub.i] is angle of stator slot i center, [[phi].sub.ti] is angular position between stator slot i and stator slot i + 1 (angle of stator tooth i). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be easily determined at every time step of simulation. Using the definitions given in (11) and (12), [L.sub.ab] is obtained as (22).

APPENDIX C.

Table C1. Specifications of simulated machine.

Pole arc (degrees)             57
Inner radius of stator (mm)    80
Air gap length (mm)           0.6
Active axial length (mm)      130
Number of turns per slot       20
Number of stator slots         30
Stator slot pitch (degrees)    12
Rotor outer radius (mm)       79.4


ACKNOWLEDGMENT

This work was supported by the Office of Vice Chancellor for Research, Yazd Branch, Islamic Azad University, Yazd, Iran. The author would like to thank Dr. H. Meshgin Kelk for providing information regarding the installed 9 KVA synchronous machine in the electrical machines research Lab, Tafresh University, Iran and Prof. J. Milimonfared from Amirkabir University of technology, Tehran, Iran.

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Hamidreza Akbari *

Department of Electrical Engineering, Yazd Branch, Islamic Azad

University, Yazd, Iran

Received 1 January 2013, Accepted 5 March 2013, Scheduled 7 March 2013

* Corresponding author: Hamidreza Akbari (hamid_r_akbari@yahoo.com).
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