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

i i

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

ii ii

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.

iii iii

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.

iv iv

TABLE OF CONTENTS

DECLARATION i

ABSTRACT ii

ACKNOWLEDGEMENTS iii

CONTENTS iv

LIST OF TABLES ix

LIST OF ABBREVIATIONS AND TERMS x

CHAPTER 1: INTRODUCTION 1

1.1 BACKGROUND 1

1.2 PROBLEM STATEMENT 2

1.3 OBJECTIVES 2

1.4 METHODOLOGY 3

1.5 SCOPE OF THE DISSERTATION 3

1.6 SIGNIFICANCE OF THE RESEARCH 3

1.7 HYPOTHESIS 5

1.8 GENERAL 5

v v

1.8.1 Speed Estimation using Induction

Motor Models 5

1.8.1.1 Stator field orientation based

Estimation 6

1.8.1.2 Back emf based Estimation 6

1.8.1.3 Speed Estimation Independent 7

Of Secondary Resistance

1.8.1.4 Speed Estimation using the

Extended Kalman Filter

Approach 8

1.8.1.5 Model Reference Adaptive

System 9

1.8.2 Speed Estimation using Motor Current

Signature Analysis 12

1.8.3 Fine Tuning for Better Speed

Estimation 13

1.9 STRUCTURE OF THE DISSERTATION 14

vi vi

2 MOTOR CURRENT SIGNATURE ANALYSIS

2.1 INTRODUCTION 15

2.2 MATHEMATICAL ANALYSIS OF MCSA 15

2.3 REVIEW OF SENSORLESS SPEED ESTIMATION

USING MCSA 18

2.3.1 LABVIEW IMPLEMENTATION OF

MCSA 18

2.3.2 REAL TIME IMPLEMENTATION

USING DSP 20

2.3.3 DISCUSSION ON RELATED WORK

IN SENSORLESS SPEED

ESTIMATION 22

3 OBSERVER BASED SPEED ESTIMATION

3.1 INTRODUCTION 28

3.2 INDUCTION MACHINE MODEL 29

3.3 OPEN LOOP OBSERVER 34

3.4 CLOSED LOOP OBSERVER 37

3.4.1 Model Reference Adaptive System

(MRAS) 37

vii vii

3.4.2 MRAS In Speed Estimation 38

3.4.3 Design And Synthesis Of Observer 40

3.4.4 Analysis Of Dynamics Of The Observer

System 44

3.4.5 Performance Analysis Of The Observer 49

3.5 REAL TIME IMPLEMENTATION OF THE SPEED

OBSERVER 55

4. IMPLEMENTATION OF SENSORLESS

SPEED ESTIMATION

4.1 INTRODUCTION 59

4.2 EXPERIMENTAL SETUP FOR SPEED

ESTIMATION 59

4.2.1 Current And Voltage Transducers 61

4.2.2 Analog Interface 63

4.2.3 Induction Motor And Load 64

4.3 SPEED ESTIMATION AND FINE TUNING 65

4.3.1 Speed Estimation Using Observer 65

4.3.2 Effect Of Parameter Variation 67

viii viii

4.3.3 Speed Estimation Using MCSA 70

4.3.3 Fine Tuning Of The Observer

Speed Estimate 75

5. SUMMARY AND CONCLUSION

5.1 SUMMARY 78

5.2 SCOPE FOR FUTURE WORK 80

6. REFERENCES 82

7. APPENDIX A A1

8. APPENDIX B B1

9. APPENDIX C C1

ix ix

LIST OF TABLES

3.1 Effect of parameter variation on speed estimate –

Simulation results 52

4.1 Effect of parameter variation on speed estimate –

Experimental results 69

1 1

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.

2 2

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.

3 3

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 4.

1.5 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.

4 4

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.

5 5

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.

1.8.1 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.

6 6

1.8.1.1 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.

1.8.1.2 Back-emf based Estimation

Another method of speed estimation [2] uses the back-

emf vector. This is based on the fact that the back-

emf 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.

7 7

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.

1.8.1.3 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.

8 8

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.

1.8.1.4 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.

9 9

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.

1.8.1.5 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.

10 10

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.

11 11

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.

12 12

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.

13 13

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.

1.8.3 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.

14 14

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 3rd 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

4th chapter. The 5th chapter concludes the thesis and

makes recommendations on further work that can be done

in sensorless speed estimation.

15 15

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 air-

gap permeance interacting with the air-gap MMF, which

produces an air-gap flux density.

16 16

( ) ( ) ( )rmsagrmsagrmsag PMMFB θϕθϕθϕ ,.,, = (2.1)

Where, sϕ is the stator angular position, θrm is the

mechanical rotor position, MMFag is the air-gap mmf

resulting from the applied stator current, Pag is the

air-gap permeance and Bag 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

( )

±

−±= wsesh np

snRnff 1...11 (2.2)

( )

±

−±±= snp

spnnRnff wroresh .1....1 (2.3)

where

fsh – slot harmonic frequency

f1 - supply frequency

s - per unit slip

p - pole-pairs

R - number of rotor slots

n - 0,1,2,3…

17 17

ne - order of rotor eccentricity

static ne = 0

dynamic ne = 1,2,3…

−wsn order of stator mmf time harmonic = 1.2.3,..

wrn - order of rotor mmf time harmonic = 1,2,3,..

rn0 - 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

1.1

1 fp

sf sh

−±= (2.4)

Substituting n=0, ne = nws = 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

18 18

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

2.3.1 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:

( ) ( ) ( ) ( )( ) ( )tftfktfktfkkti oiio ππππ 2cos2cos...2cos2cos 211 2++++= (2.5)

Where, ki are constants, fi are the frequencies, which

depend on the mechanical and electrical systems. Fo is

the fundamental or supply frequency.

19 19

The above equation can be rewritten as

( ) ( ) ( )tftmti oπ2cos= (2.6)

Where, ( )tm is the amplitude of the stator current.

Compared to (2.5),

( ) ( ) ( ) ( )tfktfktfkktm iio πππ 2cos.....2cos2cos 2211 ++++= (2.7)

The frequency of each term in (2.7) is lower than fo.

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

( ) ( ) ( ) ( )222 22cos21

21

tmtfxtmti oπ+= (2.8)

Since the frequency of each component in m(t) is lower

than fo, 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 2fo in order to filter components with

frequency of 2fo. 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.

20 20

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 (fk to 2fo/p), where p is the number of poles

and 2fo/p is the synchronous mechanical speed and

( )( )

++−= 5,022

12lrlss

rok

XXR

Rpf

f (2.9)

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.

2.3.2 Real-time Implementation using a DSP

This implementation in [8] followed the work done in

[7] and has been implemented suing a TMS320C30 DSP.

21 21

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.

22 22

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.

2.3.3 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.

( )ωωωω 1+≈+= rsrhsl NN (2.10)

Where,ω is the angular velocity of the rotor, Nr is

the number of rotor slots.

23 23

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

24 24

11 .11.6 fp

sfsh

−±= (2.11)

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 non-

sinusoidal source, i.e., using an inverter for varying

the supply frequencies has been implemented.

25 25

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.

26 26

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 on-

line initialisation algorithm to determine the values

of R, ne, nwr and nws, 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 field-

oriented 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.

27 27

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.

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

rr

rr

q

d

ii

TL

TT

pλλ

ωω

λλ

/1/1

. (3.2) (for rotor)

+

+−

=

q

d

s

ss

q

dr

q

d

ii

pLRpLR

vv

MLp

.00

.1 σ

σλλ

29

-Where, Ls, Lr – stator, rotor self-inductance;

Lm – mutual inductance; Rs-stator resistance; Tr – rotor

time constant; λ- rotor flux; i –stator current; v-

stator voltage; ωr – 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.

30

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.

Induction motordq model

Va Vd

Vb Vq

Vc Vo

Va

Vb

Vc

V0

lamda_dr

Speed Observer

lamda_qr

Wr

Te

West

iqs

ids

Vds

Vqs

Figure 3.1 : Block diagram showing Simulation setup for Speed Estimation

31

where, vds and vqs are applied stator voltages in the dq

– axis, ids and iqs are the stator currents, λdr and λqr

are the developed rotor flux components,ωest 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).

qsdsqrdr

r

mdsi

rsm

rds ivdt

dLL

iRLLL

Ldt

diωωλ

λ+

−

−+

−= 2 (3.3-a)

dsqsdrqr

r

mdss

rsm

rqs ivdt

dLLiR

LLLL

dtdi

ωωλλ

+

−

−+

−=

2 (3.3-b)

( ) drr

rds

r

rmqrr

dr

LR

iLRL

dtd

λλωωλ

−+−= (3.3-c)

( ) qrr

rqs

r

rmdrr

qr

LR

iLRL

dt

dλλωω

λ−+−= (3.3-d)

( )dsqrqsdrr

me ii

LLP

T λλ −=. (3.3-e)

where, vds and vqs are applied stator voltages in the dq

– axis, ids and iqs are the stator currents,

λdr and λqr are the developed rotor flux components,

32

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.3.1.

33

+-+

K-

-K

1

8

1

ids

-K+--

-K1

8

K-

^

^

1

8

K-

P

P

-++ -K

-K

-K

K-+-+

2 1

8

2

4

1

3

^

^^ K- +

- 1Js+C

6

5

Te

Wr

Externalload

lamda_qr

iqsVds

Vqs

lamda_dr

MechanicalEquivalence

Figure 3.2 Simulink Model of Induction Motor

where, vds and vqs are applied stator voltages in the dq

– axis, ids and iqs are the stator currents, λdr and λqr

are the developed rotor flux components.

A similar model of the induction motor in the

synchronous frame was also developed to determine the

flux λ0 developed at specific operating points. This

estimated flux is used in determining the gain terms in

the observer.

34

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−

d

q

λ

λ (3.3)

p.∅ =( ) ( )

( )22

..

qd

dqqd pp

λλ

λλλλ

+

− (3.4)

Substituting for p.λd and p.λq in (3.4) from (3.2), we

obtain

p.∅ = ( )

+

−+ 22

qd

qddq

r

mr

ii

TL

λλ

λλω (3.5)

Using (3.4) in (3.5), the expression for rotor speed is

( ) ( ) ( )( ) ( )qddqr

dqqdqd

r iiT

Lpp m λλλλλλ

λλω −

−−

+= ..

122 (3.6)

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.3.4.

35

Flux ObserverEqn. 3.1

SpeedEstimatorEqn. 3.2

VdsVqsIdsIqs

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.

Vqs

+--

-K-du/dt

-K-

1K-

^

^

2

1-s

Ids

Vqs

+--

-K-du/dt

-K-

1K-

2

1-s

Ids

^

+-

^

u ∧ 2

+-

+++

u∧2

1e -0

I u I

Mux

u[2]/u[1]

K- 1

West

+-

-K-

Figure 3.4 : Simulink Model of Open Loop Observer

36

where, vds and vqs are applied stator voltages in the dq

– axis, ids and iqs are the stator currents, λdr and λqr

are the developed rotor flux components,ωest 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.

-50

0

50

100

150

200

250

0 1 2 3 4 5Time in sec. s

Spe

ed. r

ad/s

Original Speed Estimated Speed

Figure 3.5 : Plot of estimated speed using Open Loop

Observer

37

3.4 Closed-Loop Observer

3.4.1 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.

38

ReferenceModel

AdjustableSystem

+ +

+

-

AdaptiveMechanism

U

Y

Y'

Error

Parameter AdaptionSignalSynthesisAdaption

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.

3.4.2 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

39

(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.

40

AdaptiveMechanism

Rotor equationsEqn. 3.2

West

Ed

Eq

λqr

λdr

λq

λdStatorEquationsEqn. 3.1

Actual MotorSpeed ωr

Flux λd, λq

Vds

Vqs

Ids

Iqs

Figure 3.7 : Structure of MRAS for Speed Estimation

where, vds and vqs are applied stator voltages in the dq

– axis, ids and iqs are the stator currents, λdr and λqr

are the developed rotor flux components,ωest is a

function of state error.

3.4.3 Design and Synthesis of Observer

In general, ω r is time varying, resulting in a time

varying model. However, in order to derive the

41

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.( )estr

q

d

q

d

rr

rr

q

d

T

Tωω

λ

λ

ω

ω−

−

−+

∈

∈

−−

−−=

∈

∈

/1

/1

(3.7)

where, '' qd andλλ are estimated flux,

−−=∈−=∈ tesqqqddd ωλλλλ ,,' ' speed estimated by the observer.

The above equation can be written in a simplified state

form as

p[ ] [ ] [ ] [ ]WA −∈=∈

Here, estω is a function of the state error and hence

the system can be represented as a non-linear feedback

system as shown in Fig.3.5.

42

I/S

[A]

I/S φ1([E])

φ2([E])

[W] -

Linear block

Non Linear block

+ +

-

[E]

[ λq ][ λd ]

+ Ws

West-

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.

Let =estω ωest = [ ]( ) [ ]( ) τεφεφ dt

∫+0

12 (3.8)

Where, εφφ erroroffunctionsareand 21 .

43

For stability, Popov’s criterion requires that

[ ] [ ]∫t

o

T dtW .ε > - 2oγ for all 1t >0 (3.9)

where, 20γ is a positive constant. Substituting for [ ]ε

and [W] in this inequality, using the definition of

,estω Popov's criterion for the present system becomes

[ ] [ ]( ) [ ]( )∫ ∫

−−−

1

0 012 ..'

t t

rdqqd dtdτεφεφωλελε > - 20γ (3.10)

A solution to this inequality can be found through the

following relation:

( )( ) ( )∫1

0

..t

dttftfpk > - ( ) kfk ,0.21 2 > 0 (3.11)

Using this expression, it can be shown that Popov’s

inequity is satisfied by the following functions:

( ) ( )'''' 221 qddqqddq kK λλλλλελεφ −=−= (3.12)

( ) ( )'''' 112 qddqqddq kk λλλλλελεφ −=−= (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.3.9

44

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.

Vds

+---K-

du/dt

-K-

1

2

Iqs

Vds

+--

-K-du/dt

-K-

1

2

Ids

-+-

++-

- k-

-k-

1/s

1/s

1/s

1/s

1/s

1/s

1/s

1/s+-

K1.e +K2e

1/P 1

2

err

West

lamda_qest

lambda_q

lambda_d

lamda_dest

Product

Product

TransferFon

Figure 3.9 : Simulink Block diagram of MRAS based Speed

Observer

where, vds and vqs are applied stator voltages in the dq

– axis, ids and iqs are the stator currents,

45

λdr and λqr are the developed rotor flux components,ωest is a function of state

error.

In general, the quantity ω r and estω 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:

( ) rde

qe

qe

de

r

m

qe

de

r

roo

qe

de

ii

TL

Tr

Trp ω

λ

λλλ

ωω

ωωλλ

∆

−+

∆∆

+

∆∆

−−

−−=

∆∆

0

0

00 /1)(

)(/1.

(3.14a)

estde

qe

qe

de

r

m

qe

de

r

r

qe

de

i

i

TL

Tr

Trp ω

λ

λ

λ

λ

ωω

ωω

λ

λ∆

−

∆

∆+

∆

∆

−−

−−=

∆

∆

0

0

)00

00

'

'

/1(

)(/1. (3.14b)

Where 0ω is the synchronous speed, 0rω is the speed about

which the equations are linearised, deλ and qeλ are

flux at the point of linearisation, deλ∆ and qeλ∆ 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:

( ) ( )qededeqeqededeqe λλλλλλλλε ∆−∆−−∆=∆ 0000 (3.15)

46

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:

( )( ) 2

0

00

2

20

.1

.1

λ

ωω

λ

ωε

ωε

sG

Ts

Ts

rr

r

rr

=

−+

+

+

=∆∆

=∆∆ (3.16)

where ( )02

022

0 qede λλλ += and it is assumed that 0'

0 λλ = and

roro ωω = . The linearised flux 0λ is determined from the

synchronous frame model of the induction motor. The

speed at this point corresponds to roω .

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 3.10

47

The characteristic equation of the closed loop system

is given as

1 + K.G(S) = 0 (3.17)

where, G1(S)= G(S)

+

SK

Ko2

12λ (3.18)

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.,

0=dsdK (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. 3.11

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

48

operation. The values for K1 and K2 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.3.12.

Close loop root locus

-15

-10

-5

0

5

10

15

-30 -25 -20 -15 -10 -5 0

Real Axis

Imag

Axi

s

Figure 3.11. Root Locus of the closed loop observer

49

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5

Time (sec)

Am

plitu

de

0

0.2

0.4

0.6

0.81

1.2

0 0.2 0.4 0.6

Time (sec)

Ste

p r

esp

on

se

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

3.4.5 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.

50

Estimated speed using Closed-loop observer

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6

Time in sec. s

Spe

ed. r

ad/s

Estimated Speed Original Speed

Figure 3.13 : Plot of Speed estimated using Closed-

loop 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 Rs and Rr, as they

are sensitive to temperature variations during the

operation of the machine.

51

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 real-

time, as with increase in operating temperature, the

resistance of stator and rotor vary. The variations

in resistance is given by the relation

++

=a

originaltrise tt

RR5.2345.234

(3.20)

where, triseR is the resistance at a temperature °t

centigrade, at is the ambient temperature and originalR

is the initial resistance at °at 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.

52

( )( ) .1/

222'

constss

RI

r

R =−ω

(3.21)

222 ror RkI−= ωω (3.22)

where, rω is the speed of the machine, 0ω is the

synchronous speed, s is slip, I2 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.3.14. The reason for

this is that, the machine operates at a different flux

level from that calculated by the observer.

Actual

speed(RPM)

Temperature

increase(°C)

Estimated

speed(RPM)

Error in

Estimate(RPM)

1760.5 Ambient

temperature

1760.2 0.3

1760.5 100 1751 9.5

1760.5 150 1745.5 15

1760.5 200 1740.4 20.1

Table 3.1: Effect of parameter variations on speed

estimate – Simulation results

53

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

54

the model. The block diagram with the filters

incorporated is shown in Fig.3.15.

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.

S/(s +1Tr)

AdaptiveMechanism

Adjustable model +

-

West

Ed

Eq

λqr

λdr

λq

λd

Reference Model

S/(s +1Tr)

I'dIq

I'q

Id

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.3.16.

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.

55

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20

Estimated Speed Original Speed

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20

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.

56

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

( )dt

diLRiv

dtd ds

ssdsdsd .. σ

λ−−= (3.23a)

( )dt

diLRiv

dt

d qsssqsqs

q .. σλ

−−= (3.23b)

qrd

rds

d

Ti

dtd ''

' 1λωλ

λ−−= (3.23c)

57

drq

rqs

q

Ti

dtd ''

' 1λωλ

λ+−= (3.23d)

+

−−+=

dtd

dtd

dtd

dtd

Kdt

d dq

qd

qd

dq

r'

''

'1

.. λλ

λλ

λλ

λλ

ω (3.23e)

[ ]qddqK ''2 λλλλ −

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 2nd and 4th order routines were

tested and they performed in a similar manner except

over the region where a change in speed occurs.

58

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.

59

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 4.1

60

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.

61

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

62

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

63

given to the A/D board for use in the Simulink model

and Matlab routines.

4.2.2 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.

64

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 50000

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.

4.2.3 Induction Motor and Load

The experiments were carried out on a 3Ph, 250W, 200V,

1.7A, 1725 rpm induction motor. The parameters of the

65

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

4.3.1 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.

66

The simulations used fixed-step solvers at a sampling

frequency of 1kHz or 5kHz depending on the method of

acquisition. The 5th order Dormand-Prince ODE solver

was used as it gave better results than the 4th 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.

67

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.

4.3.2 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

68

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 4.1

69

Actual

Speed(RPM)

Temperature

Increase(0C)

Estimated

Speed(RPM)

Error in

Estimate RPM)

1644 Amb.temp. 1644.4 0.4

1644 100 1640 4.0

1644 150 1638 6.0

1644 200 1637 7.0

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

70

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 4.3.4.

4.3.3 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.

71

From (2.4), which has been reproduced here for

convenience, the frequency corresponding to this

harmonic is given by

Hzfp

sf sh 11 .

11

−±=

where, 1shf is the slot harmonic frequency, 1f 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

72

no-load applied. Both use 50000 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

73

%0,.1

1 1 =

−= swhenHzf

pfshl (4.2)

%10,.9.0

1 1 =

−= swhenHzf

pf shu (4.3)

where, 1shf is the lowest possible value for the rotor

eccentricity harmonic and shuf 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., 1f 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

Hzfp

f shl 06.30.1

1 1 =

−=

and highest frequency is

74

Hzfp

f shu 06.33.1

1 1 =

−=

From the spectrum in Fig 4.7 it can be seen that the

peak lies at 32.60Hz. i.e.,

Hzfp

sfsh 60.32.

11 11 =

−±=

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

75

of their use in improving the performance of the

parameter based speed estimation using the observer.

4.3.4 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 estω and that from

the MCSA algorithm shω were verified to be the same,

while the machine was running at a specified speed

rω .

estshr ωωω == (4.4)

76

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

ester −ω as expected was different from the original speed

of the machine. The difference ω∆ in the two speeds

shω and ester −ω was used to tune the machine parameters in

the observer, so that it tracked the original speed

.rω .

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.

77

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.

78

CHAPTER 5

SUMMARY AND CONCLUSION

5.1 Summary

Sensorless speed estimation of induction motors using

parameter and non-parameter based approaches and the

merits and demerits of different techniques have been

presented in the literature survey. This formed the

basis for the work on improved observer based speed

estimation developed in this thesis. The earlier work

by Pradhyumnan R.[8] was helpful in developing a rotor

eccentricity harmonic based speed estimation that has

been established as a reliable method for speed

estimation.

The work in this thesis involved developing and

combining two methods of speed estimation. A

parameter-based method using a speed observer was

developed in Simulink. This involved developing the

machine model based on the dq - axis equations

describing an induction motor.

79

The observer was designed both as an open-loop and

closed-loop system. The closed-loop system, based on

the Model reference adaptive system (MRAS) for speed

identification, was found to perform better. The

theory behind the MRAS and its development has also

been dealt in the thesis. The dynamic performance of

the observer and the effect of parameter variations

were also studied. The performance of the observer,

from simulations and experiments, was found to be

satisfactory.

Another method of speed estimation was based on

identifying the rotor eccentricity frequency

corresponding to the speed of the rotor. This is a

parameter independent method and was found to give

more reliable speed estimates than the observer based

method. Here by analysing the power spectral density

(PSD) of the stator current spectrum the speed

information can be extracted.

Data acquisition for both methods was critical in

their performance and hence proper and accurate

methods of acquisition were carried out. This was

80

achieved using a DAS1600 data acquisition board and

200MHz Oscilloscope. The experimental results were

found to coincide with that of simulation results.

The observer was de-tuned to show the effect of

variation of parameters and was tuned back to track

the original speed, making use of the speed estimated

from the MCSA method of speed estimation. The effect

of parameter variations was studied and the results

have been incorporated. The combined approach can

provide a better and faster speed estimate when

implemented in real-time, independent of the speed of

the machine.

5.2 Scope for Future Work

The work in this thesis involved developing an improved

observer based speed estimation and it was used with

real data acquired from the machine. Though a

software routine in C was developed to estimate speed,

it was not used in the real-time estimation. With

some minor modifications this can be incorporated into

a DSP system for speed estimation. This can be

combined with the MCSA method of speed estimation for

accurate, real-time online speed estimation.

81

Regarding the observer, the effect of variation of

other parameter can be studied and accordingly the

model can be improved and modified for better overall

performance.

Further, the tuning of the observer parameters can

also be made online. The combination of eccentricity

harmonic based speed estimation with online parameter

tuning of the observer can be realized in real-time

and can prove to be a effective method of sensorless

speed estimation and control for induction motors.

82

REFERENCES [1]Jiang.J and .Holtz.J,(1997). High Dynamic speed sensorless AC drive

with On-line model parameter turning for steady state accuracy, IN

I.E.E.E. Trans. On Ind.Elec., Vol.44, No.2, April. ( pp 240-246)

[2]Holtz J,(1993). Speed estimation and sensorless control of AC drives ,

Proc. 19th Intl.conf. on Ind. Elec., Nov , pp649-654.

[3]Kanmachi T and Takahasahi I, (1995). Sensorless speed control of an

Induction motor, IEEE Ind. Appln. Magazine, Vol..1, Jan.-Feb., pp 22-27.

[4]Tamai S, Sugimoto H and Yano M,(1987). Speed sensorless vector

control of Induction motor with model reference adaptive system,

Conf. Rec. IEEE/IAS annual meeting, pp 189-195.

[5]AI-Tayie J. K. and Acarnely P. P.(1998)). Estimation of speed, stator

temperature and rotor temperature in cage IM drive using the extended

Kalman filter algorithm, IEE Proc.Elect.Power appln., Vol.144, No5, Sep,

pp 301-309.

[6]Schauder C, (1989). Adaptive speed identification for vector control of

IM without rotational transducers, Conf. rec. IEEE/IAS annual meeting,

pp 493-499.

83

[7]Zibai Xu, (1995). On Line speed estimation of induction motors”, M.S.

Thesis, Univ. of New Orleans, 1995.

[8]Pradhyumnan R. (1997). Real-Time DSP Implementation of Motor

Current Signature Analysis for Induction motor Speed Estimation and

Control, M.S. Thesis, Clarkson University, 1997.

[9]Williams B, .Goodfellow J and Green T. (1990). Sensorless speed

measurement of inverter driven squirrel cage induction motors, in Proc.

IEE 4th Int. Conf. On Power Electronic and Variable Speed Drives,

pp.297-300.

[10]Ferrah A, Bradley K.J and Asher G.M. (1999). An FFT based novel

approach to non-invasive speed measurement in IM drives, IEEE Trans.

On Inst.. and Meas., Vol.41, No.6,Dec, pp.797-802.

[11]Blasco R, .Asher G.M., Bradley K.J. and Summer M. (1996).

Performance of FFT-rotor slot harmonic speed detector for sensorless

induction motor drives, IEE, Power App., Vol.143, No.3, pp.258-268.

84

[12]Hurst K.D., Habetler T.G. (1996). Sensorless speed measurement

using current harmonic spectral estimation ion induction motor drives,

IEEE Trans. on power electronics, Vol.11, No.1, pp.66-73.

[13]Hurst K.D, Habetler T.G., Griva G and Profumo F (1994). Speed

sensorless field oriented control of IM using current harmonic spectreal

estimation, Conf. Rec. IEEE/ IAS annual meeting , pp.601-607.

[14]Hurst K. D and Habetler T.G.( 1997). A comparison of spectrum

estimation techniques for sensorless speed detection IM, IEEE Trans.

Ind.Apps., Vol.33, No4, July/Aug , pp 898-905.

[15]Landau Y.D. (1979). Adaptive Control – The Model Reference

Approach”, Marcel Dekker Inc., New York.

[16]Chee-Mun Ong (1998). Dynamic simulation of Electric machinery –

using MATLAB/SIMULINK , Prentice Hall, New Jersey.

[17]Nise Norman S. (1995). Control System Engineering”,

Addison-Wesley publishing company, California.

lxxxv

APPENDIX A A1. 1 Details of Induction motor used in Experiment 3Ph, 60Hz, Squirrel cage Induction motor Rated voltage-208 V Rated power- 1/3 HP Rated current –1.7 A Rated Speed- 1725 RPM

A1.1.1 Determination of Motor parameters A 1.1. 2. DC Resistance test

DC V (V) 1 (A) RsΩ 5 0.31 15.9 10 0.63 15.87

A.1.1.3 No load test

V (V) I (A) W1 (Watts) W2 (Watts) Speed (RPM) 200 0.85 110 -52 1789

A. 1.1.4 Blocked rotor test

I (A) V (V) W1 (Watts) W2 (Watts) 1.7 59 88 8

From the above tests and using the standard machine equations, the equivalent circuit parameters of the machine were determined. The inertia of the machine was determined from the manufacturer’s details on the rotor. Using DC resistance test: Stator resistance Rs = 15.9Ω directly from DC resistance test. Using no-load test results: P1nt = Pnt/3 = 19.33 Watts V1nt = 200V =Qnl = = Xnt = V1nt

2 /Qnt = 415.5 Ω = Lm = Xnt/ 2∏ƒ =1.102H

A1

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Using Blocked rotor test results P1br = Pbr/3 = 32 Watts V1br = Vbr = 59V I1br =Ibr 3 = 0.98 A ⇒ Qbr = ⇒Xbr = Q1br / I 1br

2 = 50.14Ω ⇒Xs =Xr =Xbr/2 =25.08 ⇒ Ls = Lr

` =0.0665 H ⇒ Rbr – 1br / I 1br

2 = 33.32Ω ⇒Rr` = Rbr - Rs = 17.42Ω

From above equations the parameters of the machine are given below, Rs- Stator resistance –15.9Ω Rr- Rotor resistance – 17.42Ω

Ls - Stator self inductance – 0.0665H Lr- Rotor self inductance – 0.0665H

Lm – Mutual inductance- 1.102 H J- inertia pf the rotor –5.8e-3 Kg.m2 Lss = Lrr =Lm +Ls =Lm +Lr = 1.168H

A.1.2 Parameters of the machine used in Simulation

These parameters were used in Simulation of the observer and ware taken from the

reference book [16.]

3Ph, 60Hz, 4Pole, Squirrel cage induction motor

Rated voltage- 200V Rated Power-1HP

Rs- Stator resistance –3.35Ω Rr Rotor resistance –1.99Ω

Ls- Stator self inductance –6.94mH Lr –Rotor self inductance-6.94mH

Lm- Mutual inductance –163.73 mH J- Inertia of the rotor-0.1Kg.m2

A2

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APPENDIX B

/*This program is to solve the speed observer equation*/ /* using the 2nd Runge-Kutta method for integration */ #include <stdio.h> #include <stdlib.h> #include <math.h> #include < string.h> #include <malloc.h> double ids [3], iqs [3], vds [3], t [3]; double sigma_ls; double rs, rr, lr, lm, ls; double lds [3[, lqa [3], ldest [3], lqest [3], w [3]; double K1, K2; double T2; double *k1, *k2, *k3, *k4; int n; void main () void getK (double *y, int j, double *kx); double d1 (int j); double d2(int j); double y [6], y1[6] double I; int j, m; FILE* vds_p, *vqs_ p, *ids _ p, *output_p; char vds_file[]=”f_vds.txt”, vqs_file[] = “f_vqs.txt”; char ids_file[] = “ f_ids.txt”,iqs_ file[] = “ f_iqs.txt”; char output_file [] = “ out.txt”; vds_p =fopen (vds_file, “r”); vqs_p= fopen) vqs_file, “r”); ids_lp = fopen (ids_file, “r”) iqs_p = fopen (iqs_file, “r”)

B1

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k1= (double*) malloc (sizeof (double)*6); k2=(double*) malloc(sizeof(double)*6) k3=(double*) malloc(sizeof(double)*6) k4=(double*) malloc(sizeof(double)*6) output_p = fopen(output_file, “w”); rs= 3.35; rr= 1.99; ls =0. 17067; lm =0.16373; sigma_ls=9lm*lm/lr)-ls; K1= 152.3; K2= 2284.8; T2 = lr/rr; /*read data and call rk2 function */ /*read data*/ i=0; while (feof( vds_p)) fscanf(vds_p, “%le%le”, &t, &vds[0]); i++ printf(“total no of points is %le/n”, I); fclose (vds_p) /* counted the total number of points. Going back*/ /*to the beginning of file by closing and opening*/ vds_p =fopen(vds_file, “r”); /*Initialize y values*/ for(j=0;j<6;++j)y[j]=0 /*end of initialization*/ /*Begin j-loop*/ for(j=0; j<(I-3)/2;++j) /*reading first 3 points without invoking rk2*/ if(j<3)

B2

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fscan (vds_p, “%le%le”,&t[j],&vds[j]) fscan(vqs_p, “%le%le”,&t[j],&vqs[j]) fscan(ids_p, “%le%le”,&t[j],&ids[j]) fscan(iqs_p, “%le%le”,&t[j],&iqs[j]) else tdiff=t[1]-t[0] /*invoking rk2*/ getK (y1, 1,k2; for (m=-1;m<6; ++m0

B3

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APPENDIX C Table : Estimated Speed Using Open Loop Observer (fig 3.5)

C1

Time (secs) Original Speed Estimated Speed0 0 182

0.1 0.7 1820.2 12 1820.3 22 1820.4 30 1820.5 35 1820.6 46 1810.7 55 1800.8 62 1800.9 73 179

1 84 1781.1 92 1771.2 103 1761.3 115 1751.4 123 1741.5 135 1721.6 145 1701.7 155 1701.8 164 1741.9 173 178

2 182 1822.1 184 1842.2 186 1862.3 187 1872.4 189 1892.5 190 190

3 190 1903.5 190 190

4 190 1904.5 190 190

5 190 190

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Table : Root Locus of the Closed Loop Observer (Fig 3.11) Axis Real Imaginary Axis (neg) Imaginary Axis (pos)

-11.8 -4 4-12 -7 7-13 -8 8-14 -8.5 8.5-15 -8.9 8.9-16 -9.2 9.2-17 -9.3 9.3-18 -9.2 9.2-19 -8.9 8.9-20 -8.5 8.5-21 -8 8-22 -7 7

-23.25 -5 5-24 -3 3

-24.5 0 0-25 0 0

Table : Root Locus of Closed loop Observer and Step response for different values of Gain (Figure 3.12)

C2

Time (sec) K2/K1 = 15 K2/K1 = 13 K2/K1 = 170 0.002 0.02 0.02

0.01 0.2 0.2 0.20.02 0.4 0.4 0.4

0.025 0.6 0.6 0.60.035 0.8 0.8 0.80.05 0.85 0.84 0.860.06 0.91 0.9 0.920.07 0.95 0.94 0.960.08 0.975 0.974 0.9760.09 0.975 0.98 1.050.1 1 0.98 1.03

0.15 1.02 1 1.040.2 1 1 1

0.25 1 1 10.3 1 1 1

0.35 1 1 10.4 1 1 1

0.45 1 1 10.5 1 1 1

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Table : Plot of Speed estimated using closed Loop Observer (Fig 3.13) Time (sec) Estimated Speed Original Speed

0 0 00.125 7.5 110.22 19 18

0.375 27 27.50.5 35 38

0.625 45 47.50.75 52.5 62

0.875 58 67.51 72.5 82.5

1.125 82.5 97.51.25 92.5 110

1.375 102.5 122.51.5 112.5 132.5

1.675 115 147.51.75 117.5 160

1.875 122.5 1702 125 175

2.125 126 182.52.25 127.5 184

2.375 130 1852.5 132.5 186

2.625 134 1872.75 137.5 187

2.875 140 1873 142.5 187

3.125 145 1873.25 147.5 187

3.375 152.5 1873.5 155 187

3.625 160 1873.75 165 187

3.875 172.5 1874 177.5 187

4.125 185 1874.25 187 187

4.375 187 1874.5 187 187

4.625 187 1874.75 187 187

4.875 187 1875 187 187

C3

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Table : Effect of including a Low Pass filter in the Observer (fig 3.16) Time (sec) Estimated Speed Original Speed

0 0 00.5 40 40

1 80 801.4 90 135

2 125 1802.5 135 190

3 145 1903.5 160 190

4 190 1904.5 190 190

5 190 1905.5 185 185

6 185 18510 185 18515 185 185

C4

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B3

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c

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