SENSORLESS SPEED ESTIMATION IN THREE PHASE INDUCTION MOTORS

Transcription

1 SENSORLESS SPEED ESTIMATION IN THREE PHASE INDUCTION MOTORS by Matthew Govindsamy NHD: Electrical Engineering A research dissertation submitted in compliance with the requirements for the degree Magister Technologiae: Electrical Engineering in the Faculty of Engineering Port Elizabeth Technikon Promoter: Dr H. A. van der Linde Phd:Elctrical Engineering

2 DECLARATION This dissertation has not been submitted previously for qualification purposes but has been created by the author during 2001/2002. The references are utilized to establish the background. 20 January M. Govindsamy Date i

3 ABSTRACT This thesis proposes a technique to determine and improve the performance of a sensorless speed estimator for an induction motor based on Motor Current Signature Analysis (MCSA). The theoretical concepts underlying the parameter based observer are developed first and then the model of the observer is built using Simulink. The observer is developed based on Model Reference Adaptive System (MRAS). The dynamic performance of the observer and its behavior due to variation of machine parameters is studied. The error in speed estimated using this observer is shown and the ability of MCSA to retune the rotor speed from the stator current spectrum. The spectrum estimation technique has been implemented using a software routine in Matlab. Both the observer and MCSA techniques were implemented practically on an induction motor. The performance of the combined sensorless speed estimation system was tested and verified. ii

4 ACKNOWLEDGEMENTS The following persons are acknowledged for their valued participation that contributed to the successful completion of this research project: * Dr. A vd Linde for his continued academic guidance, motivation and dedication during the course of my study. My family who provided me the mental support and motivation, to keep up my spirit and carry out my work successfully. Mrs N Sam for her meticulous administrative assistance. Port Elizabeth Technikon for their financial commitment. iii

5 TABLE OF CONTENTS DECLARATION ABSTRACT ACKNOWLEDGEMENTS CONTENTS LIST OF TABLES LIST OF ABBREVIATIONS AND TERMS i ii iii iv ix x CHAPTER 1: INTRODUCTION BACKGROUND PROBLEM STATEMENT OBJECTIVES METHODOLOGY SCOPE OF THE DISSERTATION SIGNIFICANCE OF THE RESEARCH HYPOTHESIS GENERAL 5 iv

6 1.8.1 Speed Estimation using Induction Motor Models Stator field orientation based Estimation Back emf based Estimation Speed Estimation Independent 7 Of Secondary Resistance Speed Estimation using the Extended Kalman Filter Approach Model Reference Adaptive System Speed Estimation using Motor Current Signature Analysis Fine Tuning for Better Speed Estimation STRUCTURE OF THE DISSERTATION 14 v

7 2 MOTOR CURRENT SIGNATURE ANALYSIS 2.1 INTRODUCTION MATHEMATICAL ANALYSIS OF MCSA REVIEW OF SENSORLESS SPEED ESTIMATION USING MCSA LABVIEW IMPLEMENTATION OF MCSA REAL TIME IMPLEMENTATION USING DSP DISCUSSION ON RELATED WORK IN SENSORLESS SPEED ESTIMATION 22 3 OBSERVER BASED SPEED ESTIMATION 3.1 INTRODUCTION INDUCTION MACHINE MODEL OPEN LOOP OBSERVER CLOSED LOOP OBSERVER Model Reference Adaptive System (MRAS) 37 vi

8 3.4.2 MRAS In Speed Estimation Design And Synthesis Of Observer Analysis Of Dynamics Of The Observer System Performance Analysis Of The Observer REAL TIME IMPLEMENTATION OF THE SPEED OBSERVER IMPLEMENTATION OF SENSORLESS SPEED ESTIMATION 4.1 INTRODUCTION EXPERIMENTAL SETUP FOR SPEED ESTIMATION Current And Voltage Transducers Analog Interface Induction Motor And Load SPEED ESTIMATION AND FINE TUNING Speed Estimation Using Observer Effect Of Parameter Variation 67 vii

9 4.3.3 Speed Estimation Using MCSA Fine Tuning Of The Observer Speed Estimate SUMMARY AND CONCLUSION 5.1 SUMMARY SCOPE FOR FUTURE WORK REFERENCES APPENDIX A A1 8. APPENDIX B B1 9. APPENDIX C C1 viii

10 LIST OF TABLES 3.1 Effect of parameter variation on speed estimate Simulation results Effect of parameter variation on speed estimate Experimental results 69 ix

11 CHAPTER 1 INTRODUCTION 1.1 BACKGROUND Electric motors for variable speed drives have been widely used in many industrial applications. In the early years dc motors were widely used for adjustable speed drives because of their ease of control. However, due to advances in both digital technology and power semiconductor devices, ac drives have become more economical and popular. For accurate torque control and precise operating speeds, more sophisticated techniques are necessary in the control of ac motors. These techniques employ high speed Digital Signal Processors and control techniques based on estimation or identification of speed and other machine states. Speed estimation is an issue of particular interest with respect to induction motor drives as the rotor speed is generally different from the speed of the revolving magnetic field. 1

12 The measurement of speed in adjustable speed drives is done using opto-electronic or electromagnetic speed transducers. The opto-electronic transducers experience errors in speed detection as a result of mounting, vibration and the ingress of contaminant; in addition they are usually the least reliable drive component. Therefore sensorless speed detection is highly desirable. 1.2 PROBLEM STATEMENT Commercially available speed measurement devices require direct contact with the shaft of the motor and are often inaccurate and unreliable after prolonged use. 1.3 OBJECTIVES Investigate speed estimation using techniques that are dependant and those that are independant on machine parameters Correction of one technique using the other for greater accuracy. 2

13 1.4 METHODOLOGY A literary review is undertaken in order to establish the required background, new trends in industry as well as the relevancy, and application of the research. The implementation of sensorless speed estimation is carried out experimentally. The method and results are dealt with in chapter SCOPE OF THE DISSERTATION This research dissertation only considers: ac induction motors three phase supply The application of Motor Current Signature Analysis is limited to speed estimation only. 1.6 SIGNIFICANCE OF THE RESEARCH Speed measurement is normally accomplished with a tachometer. Some tachometers require direct contact with the shaft of the motor, whilst others such as photo tachometer and stroboscope tachometer require close proximity to the shaft. 3

14 Many motors are located in inaccessible locations or are operated in hazardous environments e.g. motor operated valves in a nuclear plant. In such instances personal safety may often preclude the monitoring of these motors, even when it otherwise would be desirable. Many motors, even when accessible, do not provide an exposed shaft due to their mounting configurations. For example, many compressors used in air conditioning and refrigeration equipment are coupled to the motors inside a sealed compartment, thus preventing motor speed measurement by all commercially available tachometers. All these problems can be overcome by means of sensorless speed estimation. Sensorless speed estimation permits the speed sensing to be done remotely, even some distance from the motor. All that is needed is access to the motor electric cables. This could even be at the control centre situated remotely. As the proposed technique of sensorless speed estimation is non intrusive, it is a very safe method. 4

15 1.7 HYPOTHESIS The combination of the machine parameter dependent and machine parameter independent techniques will provide accurate and reliable speed estimation in three phase induction motors that does not require contact with the rotating shaft. 1.8 General A brief introduction to observer based speed detection and current based methods is now given Speed Estimation using Induction Motor Models Many control and estimation strategies for induction motor (IM) drives are based on electrical equivalent circuit models of the motor. In many cases, the model is a steady-state equivalent circuit model, but for high performance drives, a transient model of the motor is required. Many schemes based on simplified motor models have been devised to sense the speed of the IM from measured terminal quantities. A few of the techniques based on machine parameters available in the literature are discussed here with their relative merits and demerits. 5

16 Stator Field Orientation based Estimation Some of the earlier work on sensorless speed estimation was based on the method of field orientation, relative to the rotor flux linkages or its time derivative. In [1], the stator flux vector is estimated from measured machine terminal quantities to provide the field transformation angle δ. An estimate of the rotor frequency is obtained from the condition for field orientation. These two can be used to estimate the angular mechanical velocity. At low stator frequencies, stator flux estimation is sensitive to an inaccurate stator resistance value in the estimation model. It has also been shown that the accuracy of the speed estimate is poor under load due to the amplitude error of the stator flux Back-emf based Estimation Another method of speed estimation [2] uses the backemf vector. This is based on the fact that the backemf vector leads the rotor flux vector by 90º, provided the rotor flux magnitudes changes slowly. Here the estimate of rotor speed is based on the stator input voltage and the synchronous speed. This method has moderate dynamic performance at lower speeds. 6

17 Some work has been done based on the stator current and the phase angle of the stator voltage reference vector [2]. Speed estimation here depends on the stator frequency signal and the active stator current, which is proportional to the rotor frequency. The speed estimation techniques discussed so far, are based on stator current or the rotor flux vector and are essentially open-loop types of estimation. More accurate speed can be obtained when compared to the above techniques. A few of these techniques are now presented Speed Estimation Independant of Secondary Resistance In the work done in [3], speed estimation is done without prior knowledge of the rotor resistance. The machine characteristic equations are derived without involving the rotor resistance and the estimate is based on the rotor current and flux vector. Here, the characteristic equations of the induction motor are used to express the rotor current and flux linkages in terms of the stator voltages and currents. 7

18 The speed of the motor is estimated making use of the outer product and inner product of the flux linkages and currents. This method has a disadvantage of division by zero when the machine is supplied from sinusoidal mains. Means to avoid this has been shown, but involves estimation of the rotor resistance. This method is also influenced by parameter variations, especially the errors due to stator resistance, stator and rotor leakage inductances Speed Estimation using the Extended Kalman Filter Approach A different approach to speed estimation is based on the Extended Kalman Filter (EKF) algorithm. The estimation technique [5] is based on a closed-loop observer that incorporates mathematical models of the electrical, mechanical and thermal processes occurring within the induction motor. However, this work addresses only the thermal effects by incorporating a thermal model of the motor in the estimation process. Here, a two twin axis stator reference frame is used to model the motor s electrical behaviour. 8

19 The thermal model is derived by considering the power dissipation, heat transfer and the rate of temperature rise in the stator and rotor. The well known linear relationship between resistance and temperature are also taken into account in the model. These yield a non-linear model, which is linearized for the EKF estimator. The EKF estimator for speed and temperature is a predictor-corrector estimator. It has been shown that the speed estimation correlates with the measured speed in both the transient and steady state conditions. Though this method of speed estimation is independent of the drive s operating mode, closed loop estimation is possible only if the stator current is nonzero Model Reference Adaptive System The Model Reference Adaptive System (MRAS) is one of the more recent techniques in speed estimation based on the machine model [4]. Here the induction motor is used as the reference and a vector-controlled induction motor model is used as the adjustable model. 9

20 This model is adjusted to drive the error in speed between the two models to zero. The method described here uses the synchronous reference frame in the model. In order to obtain an accurate dynamic representation of the motor speed, it is necessary to base the calculation on the coupled circuit equations of the motor. This technique is used in [6]. In this technique of speed estimation, the IM is modelled based on a state-space model of the machine using two axis variables. This may be done in the stationary or synchronous frames, both having been used widely. Since the motor voltages and currents are measured in the stationary reference frame, it is convenient if the motor equations are also in the stationary reference frame. With complete knowledge of the motor parameters and variables like the resistance, inductance, poles, electrical angular velocity, stator voltages and current, the instantaneous speed of the rotor can be estimated on a closed-loop basis from the equations of the machine. This technique will be dealt in chapter 3. 10

21 This method of speed detection has disadvantages because of its dependence on machine parameter. The frequency dependence of the rotor electrical circuit parameters, non-linearity of the magnetic circuit and temperature dependence of the stator and rotor electrical circuits all have an impact on the accuracy of the observer and hence the speed estimation. At high frequencies and no-load conditions these errors are usually quite negligible. However, the speed accuracy is generally sensitive to model parameter mismatch if the machine is loaded, especially in the field-weakening region and in the low-speed range. The parameter contributing to this variation are [1][6]: Rotor resistance variation with temperature Stator resistance variation with temperature Stator inductance variation due to saturation of the stator teeth A parameter independent technique is discussed next. 11

22 1.8.2 Speed Estimation using Motor Current Signature Analysis Motor current signature analysis was developed as a powerful monitoring tool by Oak Ridge National Laboratories for motors and motor driven equipment. It can provide signatures or information regarding the condition of the machine like bearings, windings and speed of the rotor. These signatures arise as a result of the variation in permeance of the air-gap field, which are due to the rotor slotting and eccentricity. Further, this signature is available in the stator current dawn by the machine from the power supply. This avoids the use of a separate cable being used for speed estimation using conventional transducers. The stator current can be sensed using a current transducer and then can be sampled to convert it into a discrete time signal. This is used to analyse the spectrum of the current in the frequency domain using digital techniques by means of a DSP and PC. Frequency domain analyses give a better representation of the contents of the stator current and bring out the harmonics related to speed. 12

23 The transformation from the time domain to the frequency domain is achieved using the Fast Fourier Transform technique. The improvement that can be obtained using other spectral estimation techniques other than the FFT has also been studied. The FFT technique of speed estimation has a disadvantage of poor dynamic performance. A conceptual understanding of the MCSA and the related mathematics is given in the next chapter. It also gives a comparison of the various techniques being followed and their relative merits and demerits Fine-tuning for better Speed Estimation The current harmonics based method of speed estimation, MCSA, has a disadvantage at low speeds and accurate estimation can be made only at the cost of longer response time. On the other hand, observer based techniques used are affected by variations in machine parameters. Hence, it is proposed in this thesis to use the current harmonic method to fine tune observer based estimation technique already presented in the literature. 13

24 1.9 STRUCTURE OF THE DISSERTATION The remainder of this thesis is organised into 4 chapters. Chapter 2 has a detailed discussion of the various techniques for sensorless speed estimation using current harmonics. In the 3 rd chapter the various steps involved in developing an observer based speed estimator and the effects of parameter variations are presented. Then, methods of fine-tuning the observer based estimation with the motor current signature analysis based techniques is presented in the 4 th chapter. The 5 th chapter concludes the thesis and makes recommendations on further work that can be done in sensorless speed estimation. 14

25 CHAPTER 2 MOTOR CURRENT SIGNATURE ANALYSIS 2.1 Introduction In this chapter the theory underlying sensorless speed estimation using the stator current spectrum, namely motor current signature analysis (MCSA) is discussed. Different techniques that have been employed are explained and an indication of how this thesis follows the previous works by Schauder.C, Zibai.X [7,8], is presented. 2.2 Mathematical Analysis of MCSA In an induction motor, speed associated harmonics arise in the stator current due to variations in airgap permeance interacting with the air-gap MMF, which produces an air-gap flux density. 15

26 B ag ( ϕ θ ) MMF ( ϕ, θ ). P ( ϕ, θ ) s, = (2.1) rm ag s rm ag s rm Where, ϕ s is the stator angular position, θ rm is the mechanical rotor position, MMF ag is the air-gap mmf resulting from the applied stator current, P ag is the air-gap permeance and B ag is the air gap flux density. The variations in air-gap permeance are caused by rotor slotting and rotor eccentricity. The frequencies of the harmonics in the air-gap field due to the rotor slotting and eccentricity are given [9] by f sh s = f ( n. R ± ne). ± nws } (2.2) p f 1 s = f1. ( n. R ± ne ± nor. p). ± nwr s } (2.3) p sh. where f sh slot harmonic frequency f 1 - supply frequency s - per unit slip p - pole-pairs R - number of rotor slots n - 0,1,2,3 16

27 n e - order of rotor eccentricity static n e = 0 dynamic ne = 1,2,3 n order of stator mmf time harmonic = 1.2.3,.. ws n wr - order of rotor mmf time harmonic = 1,2,3,.. n 0 r - order of rotor space harmonic = 0,1,2,3,.. These harmonic fluxes move relative to the stationary stator and therefore induce corresponding voltage harmonics and hence current harmonics in the stator winding. The lowest frequency current harmonics and largest magnitude components in the phase current, are due to dynamic eccentricity and are given by f sh 1 s 1. f1 p = ± (2.4) Substituting n=0, n e = n ws = 1 in (2.2) the above equation is obtained. This equation is independent of the number of rotor bars and hence the rotor speed can be determined with the number of poles known. Using the knowledge of the presence of harmonics related to speed in the spectrum, different techniques have been employed in extracting the speed related 17

28 information. The various techniques differ in the particular harmonic to be detected, dependence on machine parameters, the use of frequency or time domain analyses and methods of implementation. Some of these methods analyse the voltage mmf while most of them analyse the current spectrum, as this is more reliable even at low speeds. 2.3 Review of Sensorless Speed Estimation using Motor Current Signature Labview Implementation of MCSA The sensorless speed estimation technique used in this work has been derived from the initial work done in [7]. Here the speed estimation was carried out in Labview and the possibility of implementing this using a DSP was also discussed. The stator current drawn by an induction motor can be expressed as: i ( π ) cos( 2πf t) ( t) k + k ( 2 f t) + k cos( 2πf t) +... k cos( 2πf t) = o 1 cos i i o (2.5) Where, k i are constants, f i are the frequencies, which depend on the mechanical and electrical systems. F o is the fundamental or supply frequency. 18

29 The above equation can be rewritten as i ( t) m( t) cos( 2πf t) = (2.6) o Where, m ( t) is the amplitude of the stator current. Compared to (2.5), m ( t) k + k ( 2πf t) + k cos( 2πf t) +... k cos( 2πf t) = o 1 cos i i (2.7) The frequency of each term in (2.7) is lower than f o. To extract m(t) from the stator current signals several techniques have been employed. One method is to square the stator current signal given in (2.6) to yield 1 1 = πfot m t (2.8) 2 2 ( ) ( ) 2 ( ) ( ) 2 i t 2 m t + cos 2x 2 Since the frequency of each component in m(t) is lower than f o, a low-pass filter can be used to filter the second term of (2.8). m(t) is extracted after i(t) 2 goes through a low-pass filter and then a square rooted operator. The high cut-off frequency of the low-pass filter must be lower than 2f o in order to filter components with frequency of 2f o. An IIR filter has been implemented to achieve this. Converting this Fourier transforms back from the frequency domain to the time domain gives m(t) 2. 19

30 Finally, amplitude information is separated from the current signal after m(t) 2 is square rooted. After extracting the amplitude information from the current signals the speed spectrum must be searched in the range of (f k to 2f o /p), where p is the number of poles and 2f o /p is the synchronous mechanical speed and f ( ( ) ) o Rr f k = 2 1 0, 5 (2.9) p 2 2 Rs + X ls + X lr The component with the maximum amplitude in the range corresponds to the rotor speed. The complete procedure has been implemented in Labview and proved to perform a good speed estimate. However, this technique requires a higher number of samples in order to get an accurate speed estimate than some of the techniques that are to be discussed soon. Also the transformation from the time to frequency domain, back to the time domain, involve the FFT and the inverse FFT and hence increases the computation process time Real-time Implementation using a DSP This implementation in [8] followed the work done in [7] and has been implemented suing a TMS320C30 DSP. 20

31 Here eqn. (2.4), for the frequency of eccentricity harmonics, has been used to estimate the slip and hence the speed of the rotor. This technique uses the current spectrum and analyses it using the FFT. This is applicable to motors fed from mains or inverters, as the fundamental frequency is tracked at the beginning of the search process. Speed estimation can be done independent of the machine design with knowledge of the number of poles alone. A notable feature of this technique is, it makes use of the Interrupt Service Routine feature of the DSP in making the data acquisition process interrupt drive. An interrupt service routine transfers acquired data from the A/D to an array and at the same time previously acquired data is processed by the speed estimation algorithm. This method improves the time taken for speed estimation by avoiding the time the algorithm has to wait and for data to be acquired. The possibility of using windowing techniques, interpolation and decimation to improve the overall performance and time has also been discussed in this work. 21

32 Further, the speed estimation loop has been closed and the effects of variation of load on the speed estimate was studied. The above method performs satisfactorily at high speeds, but not so at speeds and load that are 50% less than rated values. So, a technique to use this speed estimate in fine tuning an observer based model has been developed and is discussed in our later chapters Discussion on Related Work in Sensorless Speed Estimation The work by Jiang et.al[1] separates the induced rotor slot harmonic voltage and other triplen components from the much larger fundamental emf by summing the three phase voltages in a Wye-connected winding arrangement. Of this, the rotor slot harmonic components exhibit the dominating frequency. ω N ω + ω ( N + 1)ω (2.10) hsl = r s r Where,ω is the angular velocity of the rotor, N r is the number of rotor slots. 22

33 This frequency is extracted using an adaptive band pass filter, which is tuned to the rotor slot harmonic frequency. The filtered signal is digitised and then a software counter is used in computing the digitised rotor position angle, which, on differentiation yields the rotor speed as from an incremental encoder. This scheme yields a poor estimation during transient conditions and at low speeds. Also, since this uses analogue techniques it is not possible to get an accurate speed estimate. The method of speed estimation by Williams B. et.al[9] is based on identifying the rotor eccentricity harmonics whose frequency is as given in (2.4). The frequency spectrum of the stator current is analysed and the slip frequency is calculated from the frequency of eccentricity harmonics. A method to estimate speed of an induction motor fed by an inverter is also discussed here. This has been achieved using the dc-link current, in which the dynamic eccentricity harmonics appear at six times the frequency in a phase current, and represented in the following modified relation 23

34 1 s 6 f1 p = (2.11) f sh 1. 1±. In reality, it is not possible to extract speed at all values of slip using this method, so a method to reconstruct the phase current from the dc-link current has been given. This is used in the speed estimation as explained earlier. The drawback of this method is its use of analogue techniques as well as difficulties in implementing the required filters with acceptable error. The brief review so far represents some of the research in sensorless speed estimation, that were based on analogue techniques for the main part of speed estimation. With the advent of Digital Signal Processors and enhanced digital techniques more work based on rotor slot harmonics for speed estimation has been done. The work by Ferrah et.al[10] is one of the earlier works in this field. Here speed estimation for induction motors supplied from a 3ph mains and a nonsinusoidal source, i.e., using an inverter for varying the supply frequencies has been implemented. 24

35 A tuning mechanism to adjust to the fundamental frequency has been incorporated. A FFT-based spectral estimation technique is used to detect the fundamental frequency and the speed dependant rotor slot harmonic frequency. In the process a Hanning window in the time domain is used to reduce the spectral leakage. The power spectrum density of the windowed signal is obtained by squaring the magnitude of the FFT output coefficients. Following this, a search algorithm is implemented to identify the slot harmonic component with maximum magnitude at a frequency given by (2.1) and this is used to estimate the speed of the rotor. This method of speed estimation has been shown to have a better steady state performance and speed estimation than the previous techniques. However, this algorithm requires knowledge of the number of rotor slots and number of poles in the machine and hence is machine dependent. The same authors have extended their work [11] and have used this technique in tuning a Model Reference Adaptive System (MRAS), which is sensitive to parameter variations. 25

36 A slightly different approach, using FFT for spectrum estimation of current harmonics has been done in [12][13]. This is better than the earlier approach in the sense that they do not require any knowledge of the machine parameters other than the number of poles. They make use of the frequency of the eccentricity harmonics given in eqn.2.4 for an initial estimate of slip. This permits parameter independent speed measurements, but they provide much lower slip resolution than the slot harmonics for a given sampling time. Hence, the obtained slip information is used in an online initialisation algorithm to determine the values of R, n e, n wr and n ws, corresponding to the most significant slot harmonics. Then a speed detection algorithm detects the speed using (2.1), for frequency of the rotor slot harmonics, without any user input. However, this method cannot be used in a fieldoriented control as it has discrete speed updates. To make it continuous, the speed estimated is used to fine-tune a rotor speed observer based on the model of the machine and load. 26

37 The FFT based spectral estimation techniques discussed require a longer sampling time, particularly at low speed. So, parametric spectrum estimation techniques are being used in the work by Hurst K.D et.al [14]. It has been shown that better estimates can be obtained when the amount of data acquired is small. However, if sampling time is not a criterion, both FFT based and parametric estimation techniques perform in a similar manner. Having discussed the intricacies inherent in the different methods, the next chapter examines its applicability to observer based speed estimation techniques. 27

38 28 CHAPTER 3 OBSERVER BASED SPEED ESTIMATION 3.1 Introduction In this chapter, the method adopted in the design and development of the speed estimator is presented. The simulations were done in SIMULINK using a model based on the stationary and synchronous reference frame equations of the induction motor. The derivation of the observer equations is based on the coupled circuit dq equations of the motor. It is convenient to express the machine equations in the stationary reference frame, as real-time measurements of motor voltages as input and the rotor flux as state variables in the calculation of the output. The equations are modified and expressed in the form required for the observer, as (3.1) (for stator) + = q d r m q d r r r r q d i i T L T T p λ λ ω ω λ λ 1/ 1 /. (3.2) (for rotor) + + = q d s s s q d r q d i i p L R p L R v v M L p σ σ λ λ

39 -Where, L s, L r stator, rotor self-inductance; L m mutual inductance; R s -stator resistance; T r rotor time constant; λ- rotor flux; i stator current; v- stator voltage; ω r stator rotor electrical angular velocity; σ - motor leakage coefficient; p. denotes d/dt; and d,q denotes dq-axis components. With the above parameters available for a machine, equation (3.1) and (3.2) can be used in estimating the speed. The estimation can be done in open loop or closed loop. Both open loop and closed-loop observers were simulated. The closed loop observer was selected, as it required a lower number of parameters with improved stability. 3.2 Induction Machine Model The speed observer requires voltage and current as its input variables. These quantities can be acquired and measured in real-time. However, in order to perform simulations of the observer, a model of the induction machine, both in the stationary and synchronous frames of reference have been developed using the dq equations of induction motors. 29

40 The model, in the stationary reference frame, is then used to simulate the current drawn by the induction motor with the same voltage being applied to both the machine model and the observer. The induction motor currents are then fed into the observer and the parameters of the observer timed for optimal performance. The block diagram for the simulation is shown in Fig.3.1. ids iqs Va Va Vd Vb Vb Vq lamda_dr Vc Vc Vo Induction motor dq model lamda_qr Wr V0 Vds Te Vqs Speed Observer West Figure 3.1 : Block diagram showing Simulation setup for Speed Estimation 30

41 where, v ds and v qs are applied stator voltages in the dq axis, i ds and i qs are the stator currents, λdr and λqr are the developed rotor flux components,ω est is a function of state error. The motor model is developed based on the dq equations of the induction motor. The model solves the motor dynamic state-variable expression, shown in (3.3.a-e). di dt ds L L dλ r m dr = Riids qr vds iqs L m LsL + ωλ ω r Lr dt + 2 (3.3-a) di qs dt L L dλ r m qr = Rsids dr vqs ids L 2 m LsL + ωλ + ω r Lr dt (3.3-b) dλ dt dr LmRr Rr = ( ω ωr ) λqr + ids λdr (3.3-c) L L r r dλ dt qr LmRr Rr = ( ωr ω) λdr + iqs λqr (3.3-d) L L r r T P L ( λ i λ i ) m e =. dr qs qr ds (3.3-e) Lr where, v ds and v qs are applied stator voltages in the dq axis, i ds and i qs are the stator currents, λ dr and λ qr are the developed rotor flux components, 31

42 ω is the speed at which the q axis rotates relative to the d axis and ωr is the speed of the rotor in electrical rad/sec. Here ω = 0 for operation in the stationary reference frame and ω = 2Πf for operation in the synchronous reference frame, f being the supply frequency in Hz. The simulink block diagram of the machine model in the stationary reference frame is shown in Fig 3.2. The model was verified for proper operation both in the transient and the steady state operating conditions. Two sets of parameters were used in the simulation of the observer, one is that of the machine used in the experiments and the other is from [16]. The speed-time and speed torque responses with applied load were also verified before being used in simulation. The 3ph voltages were transformed to the dq frame using a simulink as shown in the block diagram in Fig

43 Vqs K- ^ ^ K K K P P K- -K K K 2 K- 1 Vds 1 -K 8 1 ids 3 lamda_dr ^ ^ 4 2 ^ K- + - External load lamda_qr iqs 1 Js+C Mechanical Equivalence 6 Te 5 Wr Figure 3.2 Simulink Model of Induction Motor where, v ds and v qs are applied stator voltages in the dq axis, i ds and i qs are the stator currents, λ dr and λ qr are the developed rotor flux components. A similar model of the induction motor in the synchronous frame was also developed to determine the flux λ 0 developed at specific operating points. This estimated flux is used in determining the gain terms in the observer. 33

44 3.3 Open-Loop Observer The open-loop observer requires knowledge of four constants, which depend on the machine parameters and the applied stator voltage and current to estimate the speed of the machine. This can be done by expressing the rotor flux vector angle(f), and its derivative as =tan 1 λ λ q d (3.3) λ p. = d ( p. λ ) λ ( p. λ ) q q 2 2 ( λ + λ ) d q d (3.4) Substituting for p.λ d and p.λ q in (3.4) from (3.2), we obtain L i p. = qλd idλ m q ω r T ( + ) (3.5) r λd λq Using (3.4) in (3.5), the expression for rotor speed is ω r 2 ( λ + λ ) L m ( λ ( p. λ ) λ ( p λ )) ( i λ i λ ) 1 =. d q q d q d d q T (3.6) 2 d q r In this method, equation (3.1) acts as a flux observer and the estimated flux is used in equation (3.2) to estimate the speed of the machine. The process is indicated as a block diagram in Fig.3.3, and it s Simulink model is shown in Fig

45 Vds Vqs Ids Iqs Flux Observer Eqn. 3.1 Speed Estimator Eqn. 3.2 West Figure 3.3 : Block diagram of Open Loop Observer The observer s parameters depend directly on the motor s parameters. So, its performance is sensitive to the parameter variations in the machine. 1 Vqs -Kdu/dt K- 1 - s ^ ^ 1e -0 u I u I Mux 2 Ids 1 Vqs 2 Ids -Kdu/dt -K- -K K- 1 - s ^ + - u K- + - u[2]/u[1] K- 1 West ^ Figure 3.4 : Simulink Model of Open Loop Observer 35

46 where, v ds and v qs are applied stator voltages in the dq axis, ids and iqs are the stator currents, λdr and λqr are the developed rotor flux components,ω est is a function of state error. A plot of the estimated speed is shown in Fig.3.5. The oscillation of the estimated speed about the steady state is large, both during the steady state and the transient. Performance can be improved by using a closed-loop observer. The underlying concepts and development of the adaptive closed loop observer are discussed in the remaining sections of the chapter Speed. rad/s Time in sec. s Original Speed Estimated Speed Figure 3.5 : Plot of estimated speed using Open Loop Observer 36

47 3.4 Closed-Loop Observer Model Reference Adaptive System (MRAS) The closed loop technique used in the observer is a popular technique in observer based speed estimation and control and is known as Model Reference Adaptive System (MRAS). The basic scheme of a MRAS [15] is shown in Fig.3.6. The scheme comprises a Reference model, Adjustable system and Adaptation mechanism. The reference model specifies, in terms of input and model states, a given index of performance. A comparison between the given and measured Index of Performance (IP) is obtained directly by comparing the outputs of the adjustable system and the reference model using a typical feedback comparator. The difference between the outputs of the reference model and those of the adjustable system is then used by the adaptation mechanism; either to modify the parameters of the adjustable system or to generate an auxiliary input signal. This is to minimize the difference between the two Indicies of Performance, expressed as a function of the difference between the outputs or the states of the adjustable system and those of the model. 37

48 U Reference Model Y + Error Adjustable System Y' Signal Synthesis Adaption Parameter Adaption Adaptive Mechanism Figure 3.6 : Basic Scheme of a Model Reference Adaptive System (MRAS) The observer used here is based on the parameter adaptation technique. A more detailed analysis of MRAS and its application can be found in literature [15]. We now analyse the method, which is used in designing the MRAS based observer MRAS in Speed Estimation In this technique of speed estimation, two independent observers are used to estimate the components of the rotor flux vector, one based on (3.1) and the other on 38

49 (3.2). Of this, the first does not involve the rotor speed information ωr, while the second equation does. Making use of this fact, the observer based on (3.1) can be regarded as the reference model and the other as the adjustable system. The error between the flux estimated by the two model is used to drive a suitable adaptation mechanism, which generates the estimate ωest, for the adjustable model. The block diagram for this method is shown in Fig.3.7. For the adaptation mechanism of a MRAS, it is important that the system is stable and the estimated quantity converges to the desired value. Synthesis techniques for MRAS structures based on hyper-stability are dealt in [15]. A suitable adaptation law has to be incorporated based on the requirements of stability and the system dynamics. 39

50 Ids Iqs Actual Motor Speed ωr Flux λd, λq Vds Vqs Stator Equations Eqn. 3.1 λd λq Rotor equations Eqn. 3.2 λdr λqr Eq Ed West Adaptive Mechanism Figure 3.7 : Structure of MRAS for Speed Estimation where, v ds and v qs are applied stator voltages in the dq axis, i ds and i qs are the stator currents, λ dr and λ qr are the developed rotor flux components,ω est is a function of state error Design and Synthesis of Observer In general, ω r is time varying, resulting in a time varying model. However, in order to derive the 40

51 adaptation law and study system dynamics, it is valid to treat ω r as a constant parameter of the model. To derive the adaptation law, the error in speed has to be expressed in terms of the controllable parameters of the model. This is obtained by subtracting (3.2) from (3.1). We obtain the following error equations p. d q = 1/ T ωr r ω r 1/ T r d q λd + λq ( ω ω ) r est (3.7) where, λ ' andλ d q ' are estimated flux, d = λ λ ', = λ λ ', ω speed estimated by the observer. d d q q q est The above equation can be written in a simplified state form as p[ ] = [ A] [ ] [ W ] Here, ω est is a function of the state error and hence the system can be represented as a non-linear feedback system as shown in Fig

52 + + I/S [E] [W] - - Linear block [A] I/S φ1([e]) West - [ λq ] [ λd ] + Ws Non Linear block φ2([e]) Figure 3.8 : MRAS representation as a non-linear System It has been established that for the system to be stable, the linear time-invariant forward path matrix be Strictly Positive Real (SPR), and the non-linear feedback path satisfies Popov s hyper-stability criterion. Accordingly, it can be shown that the linear block in Fig.3.8 is SPR. For the non-linear system to be stable, the following adatation mechanism has been adopted in the observer. ω ω est = φ ([ ε ]) φ ([ ε ]) dτ Let = est t (3.8) 0 Where, φ and φ 2 are functions of error ε 1. 42

53 For stability, Popov s criterion requires that t o T [ ] [ ] ε W dt. > -γ 2 for all t 1 >0 (3.9) o where, 2 γ 0 is a positive constant. Substituting for [ ε ] and [W] in this inequality, using the definition of ωest, Popov's criterion for the present system becomes t1 [ ] ( [ ] ) t ( [ ] dλq εqλd ' ωr φ ε φ1 ε ) dτ. ε 2 dt. > -γ 2 0 (3.10) 0 0 A solution to this inequality can be found through the following relation: t k ( p. f ( t ) f ( t) dt. > - k. f ( 0), k 2 > 0 (3.11) Using this expression, it can be shown that Popov s inequity is satisfied by the following functions: φ ( ε λ ε λ ') = k ( λ λ ' λ λ ') 1 K2 q d ' d q 2 q d d q = (3.12) φ ( ε λ ε λ ') = k ( λ λ ' λ λ ') 2 k1 q d ' d q 1 q d d q = (3.13) Where, K1 and K2 are adaptation gain constants in the observer. Based on the design technique discussed so far, the observer is realised using Simulink and the block diagram shown in Fig

54 3.4.4 Analysis of Dynamics of the Observer System The observer was designed for two machines of different ratings to verify consistency of its performance. The design of the observer involves the development of an adaptation mechanism, which controls both stability and dynamic performance. This requires the non-linear equations representing the observer to be linearized about a stable operating point. 1 Vds -Kdu/dt /s lambda_d 2 -K- Iqs 1 Vds 2 Ids Kdu/dt -K- 1/s - k lambda_q 1/s Product 1/s lamda_dest 1/s Product err /s 1/s 1/s lamda_qest K1.e +K2 e TransferFon 1/P 1 West -k- Figure 3.9 : Simulink Block diagram of MRAS based Speed Observer where, v ds and v qs are applied stator voltages in the dq axis, i ds and i qs are the stator currents, 44

55 λ dr and λ qr are the developed rotor flux components,ω est is a function of state error. In general, the quantity ω r and ω est are time-variant and each may be regarded as an input to the system described by (3.2). To linearize these equations, they are transformed to a reference frame, rotating synchronously with the stator current vector. Then we obtain the following equations: λ p. λ de qe 1/ Tr = ( ω ( ω o ( 1 Tr ) r0 ω0) / ω ro ) λ λ de qe + L T m r i i de qe λ + λ de0 qe0 ω r (3.14a) λ p. λ ' de ' qe 1/ Tr = ( ω r0 ω 0) ( ω 0 ω 1/ Tr r0 ) λ λ de qe L + T m r i i de qe λ λ de0 qe0 ω est (3.14b) Where ω 0 is the synchronous speed, ω r 0 is the speed about which the equations are linearised, λ de and λqe flux at the point of linearisation, λde and λqe are are small deviations in flux. The subscript e, is to indicate that the equations are in the synchronous frame. The error function ε, has the form vector inner product which is independent of the reference frame in which the vectors are expressed. It may thus be represented by the following linearised expression: ( λ λ λ λ ) ( λ λ λ λ ) ε = qe0 de de0 qe qe0 de de0 qe (3.15) 45

56 It can be shown from eqn 3.9 and 3.10 that the transfer function for the open loop relation between speed and error is: ε ω r = ε ω r = 1 s +. λ T r 2 1 s T r 2 0 ( ω ω ) r0 = G ( s). λ 2 0 (3.16) where λ 0 ( λ de0 + λ qe0) = and it is assumed that λ = ' 0 and 0 λ ω ro = ω ro. The linearised flux λ 0 is determined from the synchronous frame model of the induction motor. The speed at this point corresponds to ω ro. The observer designed can be represented as a system comprising a plant and a controller. Making use of this representation the dynamics of the observer as a closed loop system can be studied. This is shown in Fig

57 The characteristic equation of the closed loop system is given as 1 + K.G(S) = 0 (3.17) 2 K2 where, G 1 (S)= G(S) λ o K1 + (3.18) S and K is the gain of the system. Then the breakaway point is determined by setting the derivative of gain K to zero, i.e., dk ds = 0 (3.19) Solving the above equation yields the breakaway point, based on the location of the zeros and poles [17]. The suitable operating point is determined by choosing the value of gain for which the peak overshoot and the rise time are acceptable. The root locus of the closed loop system is plotted and the breakaway point is verified with the calculated result. This is shown in Fig along with the step response of the system. With this as the starting point, different operating ranges with increased and decreased gains were analysed. The results from various points of operation indicate that, at the breakaway point both transient overshoot and rise time were better compared to other points of 47

58 operation. The values for K 1 and K 2 are calculated, based on the value of gain at the breakaway point, and are used in the adaptive control mechanism. Using these values of gains the observer was found to track the speed of the induction motor accurately. The effect of variation in gain, on the performance of the system can be seen from the plot shown in Fig Close loop root locus Imag Axis Real Axis -15 Figure Root Locus of the closed loop observer 48

59 Amplitude Time (sec) Step response Time (sec) K2/K1 = 15 K2/K1 = 13 K2/K1 = 17 Figure 12 : Root locus of closed loop observer and Step Response for different values of gain Performance Analysis of the Observer The observer was implemented in a Simulink model as shown in Fig.3.9. The simulation requirements for the model are discussed in section (3.5). In the simulations, the induction machine was represented as a model in the stationary frame of reference, to simulate real-time environment. The plot in Fig.3.13 shows the speed observed by the closed loop observer during simulations. 49

60 Estimated speed using Closed-loop observer Speed. rad/s Time in sec. s Estimated Speed Original Speed Figure 3.13 : Plot of Speed estimated using Closedloop Observer Parameter variation is an inherent disadvantage in the observer based speed estimation techniques. In order to study the effects of parameter variation, simulations were done for changes in R s and R r, as they are sensitive to temperature variations during the operation of the machine. 50

61 To study this, the parameters of the observer are to be maintained at a constant value and those of the machine are to be changed. This is to simulate realtime, as with increase in operating temperature, the resistance of stator and rotor vary. The variations in resistance is given by the relation R trise = R original t ta (3.20) where, R trise is the resistance at a temperature t centigrade, t a is the ambient temperature and R original is the initial resistance at t a centigrade. Since the observer parameters are based on ambient conditions of the machine, these parameter variations are not incorporated. The rotor resistance increases with increase in temperature causing the speed of the machine to drop. In order to maintain the torque constant at a particular load condition, the following relation has to be satisfied. It can be shown, that for this condition to be satisfied the speed has to drop with increase in temperature. 51

62 I ' 2 2 R 2 R = ( sω /( 1 s) ) r const. (3.21) r o Rr ω = ω ki (3.22) where, ω r is the speed of the machine, ω 0 is the synchronous speed, s is slip, I 2 is the rotor current and K is a constant. It has been observed that with increase in temperature, the variation in parameters and hence the error in estimated speed increase. The variation of the load applied to the machine also is a factor in determining the accuracy with which the speed can be estimated. The results obtained are shown in Table 3.1 and in Fig The reason for this is that, the machine operates at a different flux level from that calculated by the observer. Actual Temperature Estimated Error in speed(rpm) increase( C) speed(rpm) Estimate(RPM) Ambient temperature Table 3.1: Effect of parameter variations on speed estimate Simulation results 52

63 Fig 3.14 Effect of parameter variations on speed estimate The speed estimated using the observer had some oscillations even in the steady state. This is due to the integration and differentiation of quantities involved in the speed estimation. In order to reduce these oscillations, a low pass filter was included in 53

64 the model. The block diagram with the filters incorporated is shown in Fig The placement of the low pass filters in the path of the adjustable model can be either on the input or output of the adjustable block. Placing the filter on the input is found to give better results. However, in the reference path the filter cannot be placed in the input side of the model as it alters the input to the system. λd Id Reference Model λq S/(s +1Tr) Iq S/(s +1Tr) I'd I'q Adjustable model λdr λqr + - Eq Ed West Adaptive Mechanism Figure 3.15 : Block diagram of Closed Loop Observer with Low Pass filter The effect of including the low pass filter in speed estimation can be seen on the plots shown in Fig Though the low pass filter reduces the oscillations in speed estimation, the overall system stability is affected and makes the system unstable at certain load conditions. 54

65 Estimated Speed Original Speed Estimated Speed Original Speed Figure 3.16 : Effect of including a Low Pass filter in the Observer 3.5 Real-time Implementation of the Speed Observer Simulations were carried out using Matlab 5.0. The integration parameters used were ODE45 of Dormand prince, variable step size (adaptive step size), relative tolerance of 10e-4, absolute tolerance of 10e-6 and a refine factor of 5. 55

66 These parameters were chosen after testing different types of integration and for different operating conditions of the machine and the observer. Originally, the simulations were done using continuous time models. In order to verify performance of the observer in real-time, the simulations were done using fixed step solver ODE45 Runge Kutta with a step size of 10e-4, which corresponds to sampling frequency of 1kHz. The results obtained were satisfactory and showed that the observer can be realized in real-time. The realization in real-time involves representing the observer as a set of differential equations. The equations 3.1, 3.2 and the closed loop observer model were used in arriving at the set of differential equations representing the machine. They are as shown dλ dt dids = v i. R σ (3.23a) d ds ds s. ( Ls ) dt dλ dt q diqs = v i. R σ. (3.23b) qs qs s ( Ls ) dt d λ dt 1 ' d ' ' = ids λ d ωrλq (3.23c) Tr 56

67 d λ dt ' q 1 ' ' = iqs λ q + ωrλ d (3.23d) Tr dω dt r = K1 λ K q ' ' dλ. dλ. d ' q dλ q ' dλ + λd λd λq dt dt dt dt ' ' [ λ λd λ q ] 2 q dλ d + (3.23e) These differential equations obtained were solved using Matlab s ODE solvers. The input currents and voltages were generated using the machine model. They were solved using both variable steps and fixed steps. The results agreed with the results from the simulink model of the observer. Having verified the possible methods of integrating, the equations were then solved using a C program. Initial results using the Euler s integration routine did not yield satisfactory results. After applying different integration and differentiation routines, the Runge-Kutta method performed satisfactorily, in terms of both time taken and rates of convergence. Both 2 nd and 4 th order routines were tested and they performed in a similar manner except over the region where a change in speed occurs. 57

68 A Central-difference method of differentiation is used in solving the equations, except during the initial and final conditions. This is chosen to reduce time in real-time speed estimation as a lower number of points can be used in the speed estimation. The Matlab routines were time consuming as they also involved interpolating values in the differentiation routines. The C routines were also tested with the same input arrays as given to Matlab. They coincide with the results obtained using Simulink and Matlab. The simulation results showed the performance of the observer to be satisfactory and they can be used in real-time to estimate speed. The simulated observer is realized under experimental conditions and fine-tuned using the Motor Current Signature Analysis discussed in Chapter4. 58

69 CHAPTER 4 IMPLEMENTATION OF SENSORLESS SPEED ESTIMATION 4.1 Introduction The implementation of speed estimation using the speed observer and fine-tuning it using MCSA is dealt with in this chapter. This includes the experimental set-up and results. The experiments were carried out on a 250 W, 4pole, 3ph induction motor. The parameters of the machine were determined from the no-load, blocked rotor and dc resistance tests, which are given in Appendix (1). The process of speed estimation and tuning is as shown in Fig

70 Where A/D is an analog to digital converter 4.2 Experimental Set-up for Speed Estimation The experimental set-up for the speed estimation is shown in Fig 4.2. The quantities measured are used as inputs to the observer and the MCSA algorithm for speed estimation. As shown in the block diagram in Fig 4.1, the speed estimates were compared and the induction motor parameters of the observer were tuned to follow the speed estimated using MCSA. 60

71 4.2.1 Current and Voltage Transducers The observer required as its input, the applied stator voltage and the current drawn by the machine. These quantities were used in the observer developed in simulink and in the MCSA algorithm for speed estimation. Hence their measurement had to be done with extreme care. This had been achieved using LEM make current (LA 20-NP) and voltage (LV 25-P) transducers. They provided the necessary isolation between the primary power circuit and the secondary side electronic circuit i.e. the PC hardware for A/D and the PC itself. The current and voltage transducers 61

72 chosen had a good range of linear operation with linearity better than 0,2%. Their response was fast enough to acquire transient currents and voltages. The current transducer required a resistor on its secondary side; the voltage drop across this was applied as an input to the A/D board with respect to a common ground. The voltage transducers required resistors on both the primary and secondary side, so that on the primary side a current proportional to the measured voltage was applied. Both the transducers and resistors were selected to give a voltage output in the range of +2.5V to 2.5V. The current transducer was connected to have a nominal current of 5A and a maximum of 7A in its primary. The voltage transducers can have a primary nominal voltage in the range of 10V 700V. The transducers were tested for their linearity, as errors in their measurement would have affected the performance of both the observer and the MCSA algorithm for speed measurement. The outputs from the transducers were 62

73 given to the A/D board for use in the Simulink model and Matlab routines Analog Interface The analog output from the transducers was converted to a digital signal using an ADC board, DAS1600. This is a 16 channel, 12bit A/D and D/A board. The sampling frequency ranges from 2 to 10kHz. A maximum of 5000 samples at a sampling frequency of 1kHz can be acquired. This was the best possible, as 6 input channels are used to acquire 3 voltages and 3 currents. For a supply frequency of 50Hz, a sampling frequency of 1kHz is sufficient to get an accurate reproduction of the input voltages and currents. The A/D board works in conjunction with a set of C programs using functions in a library called NLIB. This is a library containing functions to determine numerical solutions. A C program was used to control sampling rate, number of samples, number of channels required and was also used to send the output from the A/D to a Matlab M-file. This file stores the 6 input quantities along with the time stamp. 63

74 Another mode of data acquisition was also used to get better and more accurate results using the MCSA method of speed estimation, as it required still higher sampling rates and is largely affected by variation in frequency content of the input current spectrum. This was carried out using a 200MHz Oscilloscope. Using the 4 input channels of the oscilloscope samples of two-phase voltages and currents are acquired. A sampling frequency of 5kHz was used in data acquisition. This gave better results both with the MCSA method and the observer based speed estimation. The two-phase values were used to find the third phase quantity. These three phase quantities were converted to the dq axis voltages and currents for use in the observer. The speed measurements were also verified with an optical sensor Induction Motor and Load The experiments were carried out on a 3Ph, 250W, 200V, 1.7A, 1725 rpm induction motor. The parameters of the 64

75 machine were determined by conducting the no-load, blocked rotor and DC resistance test. The inertia of the machine was determined from the rotor details given by the manufacturer. This was very low and the observer took a larger time to converge with the lower inertia. So, in order to have a better response from the observer, the inertia had to be increased. This was achieved using a DC motor and brake coupled to the machine. This helped in improving the performance of the observer. 4.3 Speed Estimation and Fine-tuning Speed Estimation using Observer The input quantities for the observer were obtained from the acquired data as Matlab m-files. Data from both the Oscilloscope and the DAS1600 board were used in the observer. The observer was developed as described in the previous chapter and was implemented as a Simulink model. The dependence of the observer on the machine parameters is a critical factor and hence the parameters were determined with extreme care. The Simulink block diagram shown in Fig 4.3 has as its inputs, data files containing d, q axis voltages and currents. 65

76 The simulations used fixed-step solvers at a sampling frequency of 1kHz or 5kHz depending on the method of acquisition. The 5 th order Dormand-Prince ODE solver was used as it gave better results than the 4 th order Runge-Kutta methods. The plot in Fig 4.4 shows the response of the observer for the data acquired when the machine was running at 1644 rpm and on no-load. 66

77 The observer had low-pass filters in both the reference path and adaptive paths. This filter was required to filter out the high frequency components present in the supply voltage Effect of Parameter Variation The designed observer is dependent on the parameters of the machine. So, in order to study the effects of parameter variations, the parameters of the observer were varied and the corresponding change in speed estimate was determined. In actual operation the 67

78 machine conditions varied while the observer parameters remained the same. It was valid to study the effect of variation in the observer parameters, as the machine parameters cannot be changed. The plot in Fig 4.5 shows the decrease in estimated speed with an increase in temperature. The results obtained while the machine was running at 1644 RPM and no-load are shown in Table

79 Actual Temperature Estimated Error in Speed(RPM) Increase( 0 C) Speed(RPM) Estimate RPM) 1644 Amb.temp Table 4.1: Effect of parameter variation for speed estimate-experimental results. An increase in temperature caused an increase in the rotor and stator resistance resulting in an error in the speed estimation. In real-time, other parameters of the machine such as the stator and rotor flux linkages also changed and thereby affected the speed estimates. The frequency dependence of the rotor electrical circuit and non-linearity of the magnetic circuits also led to parameter variations. However, the effects of variations of these parameters are not included in this work. The effects of parameter variations due to temperature variations can be studied 69

80 by incorporating a thermal model of the machine in the estimation process [5]. The difference between the estimated speed and the actual speed had to be reduced using a parameter independent method of speed estimation. This was achieved by comparing the two speed estimates and the difference between them was used to tune the parameter of the machine in the observer. This is discussed in section Speed Estimation using MCSA The retuning of the observer was achieved using Motor Current Signature Analysis, the mathematics of which was dealt with in Chapter 2. The speed estimation was implemented in Matlab and can be exported for use in real-time [8]. The estimation in this method as discussed in Chapter 2 was based on identifying the harmonics with the largest magnitude in a specified range, present in the current spectrum due to the rotor slots and rotor eccentricity. 70

81 From (2.4), which has been reproduced here for convenience, the frequency corresponding to this harmonic is given by 1 s f sh1 1. f 1 p = ± Hz where, f sh1 is the slot harmonic frequency, f 1 is the fundamental frequency or the supply frequency, s is percentage slip and p is pole pairs. The process of identifying the rotor eccentricity can be done in the frequency domain using the FFT of the current spectrum or by analysing the Power spectrum density (PSD) of the current signals. The PSD, which is the square of the absolute value of FFT at a particular sample, was found to perform better when compared to the FFT alone. The PSD brings out distinct peaks in the log-magnitude plot, which might not be prominent in the FFT of the spectrum as shown in Fig 4.4. The figure shows both spectra for the current acquired when the machine was running at 1763 rpm with 71

82 no-load applied. Both use samples and the PSD uses a Hamming window of the same size. In order to identify the rotor eccentricity harmonic, the fundamental frequency or the supply frequency is to be identified first. In most practical applications the slip falls in the range of 0% to 10%. This provides a range within which the slot harmonic can be searched for. Making use of this characteristic of induction motors, the range can be determined from (4.1), which are as follows 72

83 1 f shl = 1. f 1 Hz, when s = 0% (4.2) p 0.9 f shu = 1. f 1 Hz, when s = 10% (4.3) p where, f sh1 is the lowest possible value for the rotor eccentricity harmonic and f shu is the highest possible value for a slip of 10%. The harmonic with the highest magnitude in the above frequency range corresponds to the required harmonic frequency. The process can be illustrated by considering a specific operating speed of the motor. The PSD of the current signal acquired while the machine was running at 1644 RPM is shown in Fig 4.7. This shows a prominent peak at 50Hz corresponding to the supply frequency i.e., f 1 is 50 Hz. The fundamental frequency limited the range in which the rotor eccentricity harmonic was to be searched. The lowest frequency in the range was 1 f shl = 1. f Hz p = and highest frequency is 73

84 1 f shu = 1. f1 = Hz p From the spectrum in Fig 4.7 it can be seen that the peak lies at 32.60Hz. i.e., 1 s f sh 1 = 1 ±. f1 = Hz p Using this value for rotor eccentricity frequency in (4.1) and for p = 2 gave a slip 8.67% which represents a speed of 1644 RPM. The performance of the MCSA method was tested for various speeds and load conditions and results were found to be satisfactory. They coincide with the results in [8]. The increased sampling frequency achievable through the use of the oscilloscope for data acquisition was useful in the MCSA method of speed estimation. The sampling frequency and the number of samples acquired played an important role in the real time implementation of speed estimation. They determined both the time taken for the estimation and the accuracy of speed estimate. A detailed study on their effects is shown in [8]. The results from the MCSA method of speed estimation strengthened the choice 74

85 of their use in improving the performance of the parameter based speed estimation using the observer Fine-tuning of the Observer Speed Estimate As the speed estimated by the observer varied with the parameter variations during the operation of the machine, it introduced an error in the speed estimate. From the discussion and experimental results it had been verified that the MCSA method gave an accurate speed estimate independent of the parameters. Hence this method was used to tune the observer to track the speed of the machine. Comparing the speed estimates from both the methods for a specific operating speed, the observer parameters were tuned. Initially the speed estimate with the original parameters in the observer ω est and that from the MCSA algorithm ω sh were verified to be the same, while the machine was running at a specified speed ω r. ω = ω = ω (4.4) r sh est 75

86 Then the rotor and stator resistance in the observer was de-tuned to a different value assuming a particular rise. The relation between the temperature rise and the value of resistance given in (3.20) was used to find the de-tuned resistance. Using these parameters in the observer, the same input current and voltages as that applied to the model with the original parameters were applied to the de-tuned model and speed was estimated. The estimated speed ω er est as expected was different from the original speed of the machine. The difference ω in the two speeds ω sh and ω er est was used to tune the machine parameters in the observer, so that it tracked the original speed ω r.. The tuning of the machine parameters can be done either on-line or off-line. Since the observer and MCSA methods were both off-line, the tuning was also done off-line. The plot in Fig 4.8 shows that the speed can be tracked with considerable accuracy making use of the speed observer. 76

87 The difference in speed estimated ω was multiplied by a gain K corresponding to the ratio between the original and de-tuned parameters of the observer. The gain can be measured in real-time using a thermal model of the machine. However in this thesis, the ratio was found off-line as the de-tuning and tuning back were done off-line. The results using the experimental set-up proved that the observer could be used to obtain accurate speed estimate by tuning it using the non- parameter based, MCSA method of speed estimation. 77