X International PhD Workshop OWD 23, 9 22 October 23 Simulink models of faulty induction machines for drivers applications Fernández Gómez, Alejandro J., Energy Smartops - Politechnika Krakowska (6.2.23, prof. zw. dr hab. inż. prof. PK Tadeusz Sobczyk, Politechnika Krakowska - Instytut Elektromechanicznych) Abstract Induction machines are the most common element in industry due to their characteristic and became the heart of many industrial processes. The conversion of electric energy into mechanical energy and physical phenomena inside the machine can be easily explained with mathematical equations. The use of mathematical models of faulty induction machine has proved a useful tool for prediction of effects due to faults. Their performance under faulty conditions has been studied for years but still today there is no % reliable methods allowing us to diagnose a fault and its severity. Therefore, research in that field is still needed. The purpose of this study is to give an answer to the following question: is it possible to create dynamic models of faults following the classical model of Induction Machine (IM)? The answer is: yes. Under certain assumptions the initial set of equations can be reduced to only four electrical equations and two mechanical equations. Differences in the final model of Induction Machine regarding the assumptions made will be discussed as well. Based on this approach Matlab/Simulink models can be created for fault diagnosis purposes. Index Terms-- Induction Machine, Power Electronics, Fault Diagnosis, Motor Current Signature Analysis (MCSA), Fourier Analysis, Matlab/Simulink.. Introduction Nowadays it is almost impossible to find a simple factory where the use of electrical machines as process motion system is not necessary. Their use has increased due to their properties, such as reliability, robustness, simplicity, lower price. Since these machines are often cruitial in industry, their reliability is very important issue. In the last 3 years, the introduction of power This work has the financial support from the Marie Curie FP7-ITN project "Energy savings from smart operation of electrical, process and technical equipment ENERG-SMARTOPS", Contract No: PITN-GA-2-26494 is gratefully acknowledged. electronic drives with motors has led to new design opportunities. Indeed, the increased integration of drives and machines in recent years has created a quantum leap in productivity, efficiency and system performance. Moreover the integration of power electronic drives as control brain of the electrical machine has made more difficult the analysis of faults due to the additional harmonics introduced by drives []. Drives of electrical machines have a time constrain. A fast response under different load conditions of the drive or the calculation of the machine parameters in each operation mode are some of the challenges for manufacturers. In order to adapt the control of faulty electrical machines simple models are needed. This paper attempts to describe in a simple way the procedure of mathematical fault representation of induction motors following the classical model of IM [2] as well as the consequences of the assumptions taken, for drives applications. It is well known that each fault has a specific frequency component in the spectra of stator phase currents [3]. With the aim of model validation an alternative method for Fourier analysis has been used [4]. This method allows us to calculate directly those frequencies saving time on calculations. Summarizing the aim of the paper is to present the process of modeling induction machines from the mathematical point of view for their implementation in electrical drives, improving the performance of the control techniques in case of faulty machines. Matlab/Simulink has been chosen to test the models developed under fault condition. 24 2. Mathematical modeling of electric machines 2. BACKGROUND An induction or asynchronous motor is an AC electric motor in which the electric current in the rotor needed to produce torque is induced by electromagnetic induction from the magnetic field of the stator winding. An induction motor, therefore, does not require mechanical commutation, separate-
excitation or self-excitation for all or part of the energy transferred from stator to rotor, as in universal, DC and synchronous motors. The most common configuration of the motor is the squirrel cage induction motor. The cage contains conductive bars of aluminum or copper connected both ends by shorting rings inside of a magnetic iron core. Fig. Inducction Machine Electrical signals such as voltage or currents are the sum of infinite sum of sinusoidal waves called frequency harmonics. In AC power systems frequency harmonics are multiples of the fundamental frequency of the system working and whose amplitude is decreasing with increasing multiple. For power systems with net frequency 5 Hz, harmonics can appear, 5, 2 Hz. It is well known in literature the classical model of IM represents mathematically the performance of a healthy motor. () where U and I are the vector of voltages and currents, R matrix of resistance, L matrix of inductances and W matrix of speed. 2.2 OBJECTIES The benefits of the mathematical models respect to new approaches such as Finite Element Methods software is the simplicity of the problem statement. Another advantage of mathematical modeling is that the complexity of the model can be selected by the engineer regarding the analysis objectives. Moreover, the information required for mathematical modeling can be obtained directly from the motor applying the well-known non load and blocked rotor test. The assumptions taken will approximate as much as possible the results of the model to the real machine signals. Time is an important constraint in online controlling of electrical machines by drives, hence, the complexity of mathematical models should be relatively simple. Therefore, the creation of simple mathematical models which represent the main effects of fault condition seems to be the best approach for drives applications. Only effects of main harmonics have been applied. In addition the classical model of IM has been used as reference. Dynamic models of faults can be represented by the same set of equations. 2.3 APPROACH Magnetic fields are the fundamental mechanism by which energy is converted from one form to another in electrical machines. Linked flux created by the magnetic field interaction between rotor and stator through the air-gap causes the emergence of a rotational torque, hence, the movement of the rotor. All those electro-mechanical phenomena inside the motor can be represented by mathematical formulation. For calculating the magnetic field is need to know the magneto-motive force function MMF usually represented by the Fourier series due the periodicity of the function. In case of distributed coils in stator the general form used for the MMF function is well known in literature,,, 4, cos #$ (3) 2 where i(t) is the current flowing through the coil i",, is the winding factor for -harmonic, is number of turn of the coil, the order of the harmonic and α is the angle between coils. Similar formulation is used for rotor frame. %,, %, 4 cos #$ (4) 2 As it has been mentioned the conversion of electrical energy into mechanical energy is thanks to the linked flux between the stator and rotor produced by MMF. Therefore, an important step of the formulation is the definition of the air-gap length, the magneto-motive force lines. Permeance function λ describes changes in the length of the air-gap around the inner circumference of the stator. Permeance function it also periodic and can be represented by Fourier series as well. In case of healthy machine λ is constant. & /) (5) where δ is the length of the air-gap. Knowing the MMF function and the permeance function is possible calculate the magnetic field B and linked flux Ψ, hence, the self and mutual inductances *, * %% and * %. Magnetic field B depends on currents. For a sinusoidally distributed windings and considering infinite permeability of the stator and rotor iron core, the total field of the machine is situated in the air-gap. Thus the air gap B-field is sinusoidally distributed in space and is also rotating. It can be calculated by following equation Bx, -. λ,, # 2 4567 λ,, 4 4567 2 λ, 4 8 (6) 25
In literature exists different methods to calculate the value of linked flux: a) Direct method b) Use of the constrain method c) Use of the equivalent sinusoidal windings In current analysis the equivalent sinusoidal winding method was chosen because is based on MMF function. Amper-turns function w(x,t) is needed to calculate the Linked flux density which is based on MMF function as well wx,, :# 2 ; sin> # $? So final expression for linked flux of the winding n due to magnetic field forced by the winding k is G H,I Ψ A,B C* DE :# 2 ; A A, sin> #$? G H,I J K I F G M,N E B B L,,L8O G I,N (7) (8) where order of harmonic R and L are the radio and length of the machine When distribution of the windings of stator and rotor is known, self and mutual inductances can be computed using formula: * A,A Ψ A,A A P N Q H N. * A,B Ψ A,B B P N Q M N. (9) () 2.4 TRANSFORMATION OF CONERTER EQUATIONS Some special transformations applied to voltages and currents are well known in electrical engineering [5]. They are applied in order to eliminate dependences of inductances matrix on the angle φ (angle between the reference axis of the stator and rotor). As a result, those equations are simplified and the calculations made by computer are reduced. Symmetrical components: reduces equations of a three phase system to three independent circuits, which reduces the number of operations made by the computer while solving the equations. Alpha-beta or Clarke s transformation: is a space vector transformation of time-domain signals (e.g. voltage, current, flux) from a natural three-phase coordinate system (ABC) into a stationary two-phase reference frame (α\β\γ). Fig.3 Alfa/Beta and dq tranformation 3. Modeling faults 3. CAGE ASMMETR Cage asymmetry means that one of the elements of the cage is partially damage, incipient fault, or totally broken. Generally the breakage process is progressive; therefore early detection is important in order to avoid future serious problems especially in high-power machines that suffer greater stress. Those elements are the bars and the end ring segments. Typical approach for modeling purposes is to increase the value of the element s resistance at least 2 times. The resistance matrix size of the motor, R((n+)x(n+)), depends on the total number n of rotor bars plus one extra term of cage mesh. That matrix can be reduced to C RS (2x2) rotor equivalence matrix. Changes in the equivalent matrix due to faults are modeled by asymmetry and symmetry factors, T and respectively [6]. U W C C C % Z [ C % Z \ C Z % \ C Z % ]^ [ T COS2d % \ T ef2d % ] # T COS2d % () where, C, C % Z stator and equivalent rotor resistances. Fig.2 Symmetrical component transformation Park s transformation (Direct quadrature zero or zero direct quadrature transformation): rotates the reference frame of three-phase systems and reduces the three AC quantities to two DC quantities. 3.2 MECHANICAL FAULTS Electrical machines are usually the fundamental part of complex systems connected to a wide number of mechanical devices and they are also built using mechanical components such as bearings. Their functionality can be affected by a fault in those mechanical elements. For example, an unbalanced rotation shaft due to a fault in a gearbox or a broken 26
ball in a bearing creates an additional torque and in some cases an asymmetry in the air-gap. The reaction of the induction motor due to this type of faults can be studied through the set of two mechanical equations adapting the load torque component to the mechanical fault studied. g d % 6 d # i d % d Ƭ RkRl # Ƭ kmtn (2) d % d o % (3) where J is the moment of inertia of the rotor and the load, D the dumping term, Ƭ RkRl the electromechanical torque in the rotor created by the stator currents, Ƭ kmtn the resistive torque in shaft, % is the geometrical angle of the rotor and o % the mechanical rotor speed. Usually the dumping term can be neglected. For example, an oscillating periodic torque, time dependent, can be expressed by Ƭ kmtn Ƭ kmtn Ƭ qr sin o qr (4) where Ƭ qr is the amplitude of the torque oscillations and o qr 2 s m, or if the nature of the fault is dependent on the angle, e.g. when a tooth of gearbox is broken, Ƭ kmtn Ƭ kmtn Ƭ F sin t 2 (5) where Ƭ F is a percentage of the load torque. 3.3 ECCENTRICIT ibrations, internal faults, mechanical faults, produce a certain level of eccentricity in electrical machines. When an eccentricity occurs, the geometrical axis of the rotor is displaced with respect the geometrical axis of the rotor. Therefore the length between the outer circumference of the rotor and inner circumference of the stator, air-gap, is no longer constant. In a healthy induction machine the size of the air-gap is approximately constant. A certain degree of static eccentricity is an inherent part of every induction machine. Eccentricities are classified in three types: Static eccentricity: the geometrical and rotation axis of the rotor is displaced respect to the geometrical axis of stator. The length of the minimal air-gap is constant and fix in space. Dynamic eccentricity: the geometrical axis of the rotor is still displaced respect to the geometrical axis of stator but the rotational axis is different. The position of the minimal air-gap is constant but rotates with angle ϕ. Mixed eccentricity: none of the axes coincide. The length of the minimal air-gap is not constant and depends on the position of the rotor. In order to model the physical phenomena the permeance function is used. Changes of the magnetic field force lines in the air-gap can be represented through the permeance function λ. The unbalance air-gap causes asymmetry of mutual and self-inductances of both stator and rotor because the magnetic field distribution on the air-gap is not uniform [7]. Tab.. Permeance function Healthy Machine Static Eccentricity λ δ R λ Ʌ % w x%4 Dynamic eccentricity λφ Ʌ w xz J %J Mixed Eccentricity λ, φ Ʌ %, w x%4 w xz %J J where Ʌ %, Ʌ and Ʌ %, are the Fourier coefficients, x angle between stator winding and stator reference axis. Those Fourier coefficients have a particular distribution in the 2-D spectra like it is show in the following figure. Fig.4. Fourier coefficients Ʌ {, for mixed eccentricity Summarizing, the permeance function describes the changes of the minimal air-gap, hence, magnetomotive force lines, regarding the type of eccentricity. 4. Fourier Analysis Nowadays there are different non-invasive techniques for fault diagnosis of electrical machines such as Motor Current Signature Analysis (MCSA) or vibration analysis. This paper focuses on MCSA applied to the study of the stator phase currents on frequency domain. Each fault in the IM has associated certain additional frequencies used for diagnosis [3]. Fast Fourier Transform or Discrete Fourier Transform, Wavelet transform, Hilbert transform are the typical approaches to localize those frequencies in the spectra. Instead of those techniques, an iterative algorithm for determining Fourier spectra of stator currents of AC machine for steady-state analysis has been applied. The algorithm can calculate the Fourier Spectra of a dynamic system described by a set of non-linear differential equations. The solving method allows more precise prediction of the Fourier spectra of stator phase currents. 27
a a d Usd Isd Isd Ia b b Usq c c Isq Isq Ib Supply System alpha stator q ws Ird alpha s Ic Scope Reference frame Block wr Reference frame Block 2 phi r Irq Equation Block Angle Reference Stator speed ws Stator angle Rotor speed wr Rotor speed calculated Rotor angle Isd Isq Ird Irq slip Load Set up Rotor Speed Slip Fig.4. General view of Simulink system block of induction motor Currents are considered quasi-periodic or periodic time functions; hence, they can be represented by Fourier Series. For an infinite set of points S, the Fourier coefficient } q of the time domain current signal, q, can be written in the form } q ~ J q (6) where q q,, q, q., q J, q J (7) } q } q,,} q,} q.,} q J, } q J (8) and W [ [ [. [ J [ J [ [ [. [ J [ J ~ [. [. [. [. [. [ w xg (9) [ J [ J [. [ [ U[ J [ J [. [ [ ^ ~ J 2 ] f ~ˆ (2) The matrix T fulfills the condition Š Œ Ž (2) where I is the identity matrix. 5. Mathematical Models As an example some of the models describes in the previous sections will be shown in this chapter. 5. CAGE ASMMETR W W # W W { # { { { { U { ^ U { { ^ U { ^ U { ^ (22) 5.2 ASMMETR SELF-INDUCTANCE OF STATOR DUE TO MIXED ECCENTRICIT * 6Z,F *, * *.,F W # 2 # 2 # # 2 2 U # 2 # 2 ^ W ] e2 γ ] e:2 γ# 2 3 ; ] e:2 γ#4 3 ; ] e:2 γ# 2 3 ; ] e:2 γ# 4 3 ; ] e2 γ U ] e:2 γ# 4 3 ; ] e2 γ ] e:2 γ#2 3 ;^ (27) W # 2 # W ] e2 γ ] e:2 γ# 2 2 #* Z,F # # 3 ; ] e:2 γ#4 3 ; #* Z,F ] e:2 γ# 2 2 2 U # 3 ; ] e:2 γ#4 3 ; ] e2 γ 2 # 2 ^ U ] e:2 γ# 4 3 ; ] e2 γ ] e:2 γ#2 3 ;^ (28) (29) 6. Simulink Simulink is a software package for modeling, simulating and analyzing dynamical systems. It provides graphical user interface for building models as block diagrams. It has available libraries with predefined blocks but it also allows to create and to customize our own blocks. The equations presented in the previous chapters are also suitable for implementation in Simulink. In the figure above the general view of the block diagram for an induction machine is shown in which it is possible to add or to remove blocks to represent different types of faults. In order to validate the model, following figures show the spectra of simulated currents and real currents, for an induction motor available in our laboratory. RMS values coincide. W { š š œ žžÿ { U { š ŒŽŸ { { š ŒŽŸ { { š š œ žžÿ { ^ (23) W { U { ^ { W { { U { ^ (24) g d % 6 d #id % d Ƭ RkRl # Ƭ kmtn (25) d % d o % (26) 28 Fig.5. Phase A current in Simulink; rms value = 3 (A)
consequences due to failures can be represented by simple or complex mathematical models. Only the main effects derived from faults will be reflected. This way we can avoid a possible confusion with the noise existing in real signals. Fig.6. Real Phase currents; rms phase A = 2.9766 (A); rms phase B = 2.9263 (A); rms phase C = 2.967 (A) As an example, Fig. 7 shows the impact on the current spectrum due to the presence of a broken bar. Fig.7. Phase A current in Simulink with broken bar Finally, frequency spectra of phase current A in symmetrical components in which the additional components associated with the fault can be observed. Fig.8. Frequency spectra phase current A in symmetrical components for broken bar 7. Conclusions In this paper the mathematical model procedure for faulty induction motors for drives applications has been presented under assumptions of main magneto-motive harmonics. Equations presented in the paper are suitable to be implemented in Matlab/Simulink for diagnosis purposes. Simplifications applied to the models reduce the time calculation. An alternative method of Fourier Analysis of stator phase currents has been explained. The electromechanical energy conversion, the physical interactions in the machine and 8. Bibliography [] A. J. Marques Cardoso, Nuno M. A. Freire, Fault-Tolerant Converter for AC Drives using ector- Based Hysteresis Current Control, IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics & Drives (SDEMPED), pp 329-336, Aug. 23. [2] A. Muñoz, R. Lipo, A. Thomas, Complex vector model of the squirrel-cage induction machine including instantaneous rotor bar currents. Industry Applications, IEEE Transactions on, vol. 35, no 6, p.p 332-34, 999. [3] S. Nandi, H. A. Toliyat, X. Li, Condition monitoring and fault diagnosis of electrical motors-a review, Energy Conversion, IEEE Transactions on, vol. 2, no 4, pp. 79-729, Dec. 25. [4] T.J. Sobczyk, Direct determination of two periodic solutions for non-linear dynamic systems, COMPEL International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 3, No. 3, 994, pp.59 529. [5] A. Bellini, F. Filippetti, F. Franceschini, T.J. Sobczyk, C. Tassoni, Diagnosis of induction machines by d-q and i.s.c rotor models, in Proc. 5th IEEE Symp. on Diagnostics of Electric Machines, Power Electronics and Drives, the SDEMPED 25, pp.4-46, Sep. 25. [6] T.J. Sobczyk, W. Maciołek, Influence of pole-pair number and rotor slot number on effects caused by cage faults, IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics & Drives (SDEMPED), pp. 99-24, Sep 2. [7] T. J. Sobczyk, P. as, C. Tassoni, Models for Induction motors with air-gap asymmetry for diagnosis purposes, ICEM Proceedings International Conference on Electrical Machines, vol. II, pp. 79-78, Sep. 996. Author: MSc. Alejandro Fdez Gómez Cracow University of Technology, Institute on Electromechancial Energy Conversion, Early Stage Researcher within Energy-Smartops Project ul. Warszawska 24 3-55 Kraków (Poland) Phone: +48 88 37 2 email: afernandezpk@gmail.com (http://www.energy-smartops.eu/) 29