Zeszyty Problemowe – Maszyny Elektryczne Nr 93/2011 1

Jan Rusek AGH University of Science and Technology, Chair of Electrical Machines

INDUCTION MACHINE DIAGNOSTICS BASED ON FFT ANALYSIS OF CURRENT SPACE VECTOR

Abstract: FFT analysis of one of the stator currents has become a standard procedure for diagnosing the state of an induction machine. In this procedure the harmonics are all as if positive. However, if the stator currents are aggregated to a space vector, a direction of rotation of complex harmonics is determinable. The sign of a harmonic conveys additional information, important for diagnostic purposses. The paper presents analyses of calculated currents, based on an FFT algorithm for two currents. As, typically, a machine is fed through three cables, two currents are sufficient to establish a space vector of stator currents. The space vector is a complex quantity which can be put into an FFT analysis. A developed software is fitted with graphical interface. Unlike traditional display, separate amplitudes of harmonics are displayed above and below a horizontal frequency axis. The harmonics rotating in positive direction are displayed above, and the others below the frequency axis. The software automatically indicates zones where the slot harmonics are expected. The algorithm for this is based on a specific harmonic balance model. Based on the frequency of a slot harmonic, the rotor speed is determined. Preliminary analyses are uncluded. 1. Introduction

The so called Current Signature Analysis [1], based on analysis of one current, has now become a standard in motor diagnostics. However, the current harmonics are deprived of one of a very important attribute: they are sign agnostic, i.e. they are always as if positive. But if all stator currents are considered, and if they are aggregated to a space vector, being a complex quantity, the direction of rotation of a complex harmonic can be determined. That could be of some importance for proper diagnosis in more complex cases, met in real industrial diagnoses. The paper presents a software accomplishing an FFT algorithm for two currents. As, typically, a machine is fed through three cables, without neutral, two currents allow establishing a space vector of stator currents. This complex quantity undergoes FFT analysis, always carried out in a complex domain. The software was developed in one of the contemporary programming environments, in a C# language. The developed software Sp2ph is fitted with a graphical interface. Unlike traditional display, separate amplitudes of the harmonics are displayed above and below the horizontal frequency axis. The harmonics rotating in positive direction are displayed above, and the others below the x axis. The software automatically indicates zones where slot harmonics are to be looked for. This was accomplished via making use of a specific form of the harmonic balance model. The paper contains preliminary examples of applications of the developed software.

2. Space vector of stator currents

If instant currents, flowing through stator phas-es, are i1, i2, i3, then the space vector is

(1)

where a = exp(j2π/3).

For a typical three-phase supply, without neu-tral conductor, the sum of all three currents vanish. Hence

(2)

Recognizing (2) in (1) the real and imaginary parts of a space vector are:

, (3)

, 2 /√3 (4)

As input currents i1 and i2 are sequences of samples, taken at the same time instants, also the real and imaginary parts (3) and (4) are sequences of, say, N samples. In accordance with Shanon’s theorem, the sampling apparatus must include an anti-aliasing filter rejecting harmonics of orders higher or equal to N/2. The FFT algorithm requires that the number of samples N is a power of two. It can be taken e.g. from [2]. As, generally, N samples do not cover exactly an integer number of periods of currents, current samples must be multiplied by e.g. Hann window function. The FFT algorithm transforms N samples (3) and (4) into N com-plex numbers. In the case of classical Current Signature Analysis, based on a single current, the components resulting from the FFT algo-

Zeszyty Problemowe – Maszyny Elektryczne Nr 93/2011 2

rithm, the order of which is higher than N/2, are complex conjugate to appropriate harmonics of orders lower than N/2. Hence, they do not con-vey any new information. This, however, is no longer the case if the input signal is complex, as is a space vector in (1). In fact a harmonic of the order of N-µ (after FFT analysis) rotates with exactly the same speed as does the har-monic of the order µ, except that it rotates in opposite direction. If the sequence of input cur-rents i1 and i2 is such that i1 leads i2, then the fundamental harmonic rotates forward. If it is not the case, current i2 should be considered as i1 and vice versa. Such obvious change is in-cluded in the developed software, used in the analyses further down.

3. Harmonic balance model

Application of a harmonic balance model [3] for direct calculation of amplitudes of separate harmonics, though possible [4], is not practical, mainly due to the fact that strict constancy of rotor speed leads to overestimation of the am-plitudes of calculated harmonics. It, however, can constitute a starting point for deriving a specific harmonic balance model [1,5]. The assumptions for the latter one are:

the three-phase stator winding is sym-metrical, without parallel branches,

the air-gap is uniform: there is neither static nor dynamic eccentricity,

the slot dents are of no meaning, the cage is symmetric, the iron is of infinitely high permeance, the shaft flux vanish.

Despite that none of the machines fulfills these rather strong assumptions, the model proved to be very fruitful in diagnosing even big power machines run in the industry [6]. It allows to foresee the value of the so called main slot harmonic index mShi, also referred to as a parameter h, indicating a zone where the main slot harmonic is to be looked for. The specific harmonic balance model also allows establishing of the so called sequence index Si, also referred to as a category, indi-cating the sequence of symmetrical compo-nents of higher harmonics. Both, the mShi and Si indexes result from one, rather simple, equation (5), derived in [1] and [5], with p meaning the number of pole-pairs and Nr the number of rotor slots.

102

3 (5)

Equation (5) is to be understood in a following manner. One looks for the smallest natural number h fulfilling (5) for either 1, or 0, or 2 from the curly brackets. This value of h coin-cides with the value of the mShi index. If the above fulfillment takes place for 1, from the curly brackets, the sequence index Si = 012, if for 0 the Si = 111, and if for 2 the Si = 210.

For Si = 012 the sequence of symmetrical com-ponents of stator current higher harmonics, in equations of the specific harmonic balance model, is:

,

,

,

(6)

For Si= 111 this sequence is:

,

,

,

(7)

For Si= 210 this sequence is:

,

,

,

(8)

In (6) to (8), out of an infinite number of sym-metrical components of higher harmonics, only the three central ones are explicitly exposed. The superscripts are indexes of symmetrical components. The subscripts determine the speed of rotation of separate symmetrical com-ponents, of higher harmonics. For example, the subscript 1,hNr in (6) determines the angular speed ωh of rotation of a complex higher har-monic:

1 (9)

where

ωe = 2πfL, fL = 50 Hz, is a power supply fre-quency,

Ωm – is a mechanical speed, in rad/s.

Formulas (6) to (8) determine the layout of a specific harmonic balance model. However, for further analysis, of importance are the fol-lowing notes:

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