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DETERMINATION OF PERMANENT MAGNET SYNCHRONOUS MOTOR PARAMETERS

St.John's

by

@Li Wang

A thesis submitted to School of Graduate

Studies in partial fulfillment of the

requirements for the degree of

Master of Engineering

Faculty of Engineering and Applied Science

Memorial University of Newfoundland

December 1989

Newfoundland Canada

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Abstract Permanent magnet (P.:VI.) cxcit. .. d synchro11ot:s machines h;l\'e shown i11crcasing pop­

ularity in recent. years for industrial drive applications. Jlowe\·c~r, the limitatio11 im­

po~ed hy permanent. magnet. <"'Xcit.ation Colrscs sonw difficulties in cval11ating the ma­

chine parameters. This thesis pt·cscnts the expcl'inwnta.l procedure of the dct.crmina­

tion of t.hc P.~l. synchronous 11\ol.or parameters. The work i~ intended to evaluate

the d- and q-axis reactance::; nnd the permanent. magnet. gcllcratPd voltage unclc!r load

condition.

The ;:lir gap fields of the r.~l. sync!tronOIIS motor ill'e evolllal.ed for specific rotor

and magnet configurations. An analog simulation of the interaction of these field

quantities in the air gap is developed. A practicill implementation of the digital

rncnsmcmcnt technique for determining motor t.orq11c angles is realized. A number

·of different test methods arc developed and applied to a 3-phasc, GO Hz, ·1-polc, l

horsepower P.~l. synchronous motor. Con\'crrtional I.C'st methods are mo<.lificd for use

on this type of the motor. :\ senrch coil t(' '' is deve\op(~d to locnt.e variation of the

gencratc<.l voltllgc with increflsing load. Tile infhH'IlC'C or the ill' lllil.tnre reaction is also

in\·cst.igated.

A nux linkage test is developed to mcasme the static inductance of the machine

and the search coils inserted in the stator slots of the motor arc used to measure air

gap nux distribution. Calcula.tion of the d- and q-axis rcactrtnccs is based on the flux

and the current measured. The measurement mc!.hods come from the transformation

of the two-axis theory.

II

Contents

Abstract .. 11

List of Tables vi

List of Figures .. Vll

ACKNOWLEDGEMENTS . XVI

1 INTRODUCTION 1

2 BASIC CHARACTERISTICS OF P.M. SYNCHRONOUS MOTORS 5

2.1 Introduction .... , ....... .

2.2 Conventional Test for Synchronous ~lot.ors

2.~.1 Open-Circuit. and Short-Circuit Test .

2.2.·_) Sl' T t _ _ tp es ... ..... ... .. ... .

.)

6

6

s 2.:3 Basic Characteristics of The P.M. Synchronous :\lotor and Its Parameters lO

2A Review of Literatmc ............... .

3 FIELDS OF P.M. SYNCHRONOUS MOTORS

:l.l Introduction ..

3.2 Air Gap Fields

:3.2.1 Magnet Fields .

Ill

12

15

15

15

16

:3. 2. ~ :\ rmalurc Fif'lcls .

:t 2.3 :\ i r Gap Fields .

:t:l Analog Simulation of The Fit'lds . 28

:J..t Digital ~leasuremcnt. of The l\lolor Torque Angle :n

4 DETERMINATION OF P.M. SYNCHRONOUS MOTOR PARAM-

ETERS - PART ONE 39

·1.1 Introduction .....

·1.2 ~fodificd Conventional Test

-1.2.1 Open-Circuit and Short-Circuit. Tests .

-1.2 .2 No-Load Test.

-1. :L3 Lond Test ..

·L2.'1 Discussion of T<'st. Results

•t.:t Search Coil Sirnulat.ion ... . . .

·lA Influence of Armature Reaction on C:(•nerat.ed Voltage.

5 DETERMINATiON OF P.M. SYNCHRONOUS 1\tiOTOR PARAM-

:J!)

·10

·10

.t:J

·I ;J

H

·18

ETERS - PART TWO 64

5.1 Introduction . .. .

0.2 Flux Linkage Test .

.5.2.1 Theory of Orwrat.ion

.).2.3

D · and Q - Axes Connection

Discussion

.).:3 Search Coil Test .

5.3.1 Theory of Opcr<ltion

.5.:1.2 Installation of Flux Coils .

.5.:3.3 l\Tcasuremcu! of Current l'ompotwnt.s .

;j,3.4 Calcult~tion of Rf"aclancc .

5.-l Discussion of Tests Results . . . .

IV

o4

6·1

67

67

73

75

77

84

6 CONCLUSIONS AND SUGGESTIONS

6.1 Conclusions .. . ...... .

6.2 Suggestions for Further Study

References

Appendices

89

S9

!J2

93

I Circuits of Analog Simulation 97

II Circuits of Digital Measurement of Motor Torque Angle 99

III Rotor Geometry and Parameters 102

IV Calculation of Reactances 105

v

List of Tables

:l.t HPsnlts of nnalog simulation .... . . . .

·1.1 Op<>n-circuit. and short-circuit test n•sults .

·1.2 No-load test results

·l.:l Load !.est

l..t D-axis and Q-axis react anccs .

. 1,!) Search coil simulation - load test

·1.6

l.i

!) . l

Determination of generated voltage from SC'arch coil simulation .

D-a~ds and Q-axis rNictanccs . . . . . . . . .

Rcsult.s of nnx linkage t!'st: d-axis reactance

Results of nux linkage test: q-axis r<•acl.atH'('

Jlf:sults of search coil test . . . . .

\ 'ariat.ion of generated voltage£;

i\lcosmcmcnt of axis reactance .

.1.6 Summary of t.hc t.cst methods .

IV.! Values of the ordinate for d-axis flux of one cycle

1\'.2 Values of the ordinate for q-axis of one cyck . . .

\'I

45

52

63

70

il

8:3

8.1

86

88

108

lOD

List of Figures

1.1 Principal magnetic nux paths: a) d-axis, h) q-axis

~-1 Rotor structure of t.hc salient-pole machine .

2.2 Open-circuit and short-circuit. test .

2.3 Slip test. . . . . . . ... . .. . .

~.1 Rotor and magnet geometry: a) rotor nnd magn~t configttrat.ion b)

dimensions ........ .

:3.2 Flux paths of the magnets . . . . . .

:3.3 13II characteristic of t.he magnet mflterials

3.·1 Flux density due t.o the magnets .... . .

3 .. 1 Direct-axis field due to the armature current

:1.6 Dlock diagram of (lnalog simulation ...

:3. i Results of air gap field simulation: a) light loi1di11g. b) medium loading,

i

9

li

18

18

24

24

29

c) high loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :30

:3,8 Flux waveform distribution obtained from search coil: a) light loading,

b) medium loading, c) high loading . . . . . . . . . . . . . . . . 32

3.9 Simulation circuit ......... . 3·1

:3. LO Results of simulation: comparison of different amplitude of trapezoidal

components- a) smaller amplitude, and h) larger amplitude .... . . 35

3.11 Block diagram for torque angle meitsurement . . . . . .. . .

3.12 Results of the torque angle measurement: line-Oph, symbol-8

-Ll Air gap voltage w;wcform at open-circuit test. ....... .

VII

36

37

·ll

. .. '

·1.2 Filtered air gap rollilge wa\'cform at short-circuit t c~t. . . . . . . . . . t~

-1.:1 Result of simnlat.ion: (a) input of t lw diffrn~uliill.or. (b) oul.put of the

diffcrcntiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,19

·lA Search coil simulation- open circuit l<'~t aud load test.

-1.5 Oct crmination of gcnrratcd voltilge

-1.6 Phasor diagram ( dJ = 6) .. .. . .

-l.i Stator current at different load condition a) li~h t. loading: magneti;.:ing

.jQ

51

!d. h) 6=¢: ft~=O and I,= Ia, c) IH•av~· lo(ldin11;: d(•magndi:dng l,J . • .5:3

-I.S A rma t me reaction a) magnetizing lr~, h) /.1=0. c) dc·rnagnct.izi ng lr1 54

-1.9 Operating point at different load . . . . . . . . . . . . . . . . . . . 60

-1.10 Variation of the generated voltoge(a):nwasmc<l Ei- including the ar-

ma.t.nre reaction component (h):cstimaled act.ual Err separated from

the armature r<'action and considcr<'d lcakilge filct.or . . . . . . . . . . . 60

·1.11 Reactance parameters a) d-axis a.IHI b) q-;n:is

.s.l flux lin kagc t.cst circuit . . . . . . . . . . . .

;j,2 Results of nux linkage test: a) q-axis h) d-axis: symbol-mod i fil~d con-

62

66

ventional test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

?i.:J lmplcmcnt.ation oft he measurement reo li;.:at.ion i·l

.:;..t Search coil arrangement . ... . . 7·1

5 .. 5 Flux coils arra.ngcmrnt using one coil per axis i6

TJ.G 1\teasmcment of current components iS

5.7 Flux waveform of la=6 . .5 A, chl: rPd ch2: rPq so .j,S Results of search coil t.cst: line l - X" line '2-.\'.1• symbol- modified con­

ventional test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

1.1 Analog simulation circuits 98

11.1 Circuits of digital mcC~suremcnt of motor torque angle . 100

11.2 Circuits of digital mcasmcment of mo tor t.orqnc angle -cont'd . 101

... \ ' Ill

II I.l Rotor magnet geomet.ry for calculi\ I i ng l<'<l kage f M:t or a) d i mcnsions b)

flux paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

111.2 ~lachine and rn<lgnct parameters for calculating leakage factor 104

IV.l Flux component: 111 = 2.85 ,.\ chI: lt'd2 ch2: r:•.12 106

JV.~ Flux component: 111 = :1.:1 A ch l: rL•,~;1 ch'2: t/•.1:~ 106

JV.:J Fluxcomponent.: !.:= ·t.;JAchL:~\11 ch2:r:·1.1 lOi

IV..t Flux component: / 11 =·5.8:\ chl: r!'o/!i rh:!: 1.\.; IOi

IV .. i Cmrent. component: 1.1 =2.S.i:\ chl: 1.1J rh2: !.,2 Ill

IV.G Current compon<'nl: lrt =·i.SA chi : 1.1.; ch2: !.,.·,. Ill

lX

LIST OF SYl'ABOLS

Fourier coefficients

Area. of air ga.p

Arr" Ami, Amz Area of magnet section

n;,

d

B' 2

Air gap flux densit.y due to magnets

Air gap flux density

Fundamental component of B 1

Residual flux density of magnets

Air gap flux density due to armature currents

on the direct axis and quadrature axis

Operating flux density on magnet

faces due to armature cmrcnt only

Length of iron leakage paths, used for

determining of armature reaction influence

Direct axis, used as subscript

X

D.

E. g I>

Eo

.f

g

If

Rotor dimensions

Voltage measured from air gap

at different loading

Perm a nertt magnet generated vol tagc

~laximum and minimum armature

voltage in conventional test

Open circuit volta,ge in conventional test

Frequency

Direct and quadrature armature mmfs

Equivalent air gap length

Magnetizing force

Coercive force of magnet material

Air gap magnetizing force due t.o nwgnct s

l\lagnctizing force across air gap

Field strength in the bridge

. XI

!I~.

I

1\,

11' 2

lo

Operating magnetizing force on m<lgnet

faces due to armature currents

Magnet dimensions

Current flow through the bridge circuit

used in flux linkage test

Phase current

Direct. and quadrature axis currents

Field current induced by magnets

Maximum and minimum armature

current in conventional test

Sho1't-circuit current in com·cntional test.

Armat.ure current used in two-axis theory

Axes current used in two-axi., theory

Leakage factor

Gain of measurement circuit used in

search coil test

XII

L

I~

m

N

p

R

Stator winding factor

Waveform factor for open circuit flux density waveform

Static inductance of the machine

Length of the magnet section 1 and 2

in direction of flux

Length of the stator core

Number of stator phases

Number of stator series tums per phase

Nutnber of poles

Quadrature axis used as subscript

Resistance or reluctance

Reluctance of air gap

Reluctance of magnets

Reluctance of leakage paths

X111

fl

s

0

Stator resistor

Resistor of measurement bridge

in flux linkage test

motor torque

Leakage dimensions

rms value of applied stator ,·oltage

Voltage potential across the bridge

used in flux linkage test

Stator voltage used in two-axis theory

Axes voltage used in t.wo-axis theory

Direct a.nd quadrature rcactoncc

Direct magnetizing r·en.dance

Permeability of magnetic ma terials

Number of stator teeth

Arbitrary angle between the two set of axes

for two-axis theory. Angular di ,~placement

XIV

<I>,

~'

u:

Angular displacement bct\\"cen trapezoidal and

sinusoidal components in analog simulation test

Geometrical factor for magnet leakage

Torque angle

Init.ial torque angle

Power factor

Air gap flux

Peak amplitude of the flux

Leakage flux

Flux genemted by magnets

Remanent flux

Flux linkage

Direct and quadrature axis flux linkage

Angular frequency of alternating signal

XV

ACKNOWLEDGEMENTS

The author wishes to express his sincere grat.itudc and appreciation to Professor

T.A. Little for his valuable guidance, advice and financial support throughout the

preparation of this thesis.

financial support from the Memorial U nivcrsity of Newfoundland Graduate Stu­

dent Bursary is highly appreciated as well.

Furthermore, the author wishes to thank all the friends of the Faculty of En­

gineering at Memorial University of Newfoundland for many useful discussions and

suggestions. Appreciation is also expressed to the technicians for their help.

finally, to my wife and daughter as well as my parents, without whose support,

encouragement and understanding, this work would not have been completed, my

sincere appreciation is expressed.

XVI

Chapter 1

INTRODUCTION

A significant technological change has occurred in motor drive systems in recent

years, arising from the confluence of permanent magnet materials and semiconductor

switching device technologies. One type of machine often used for these systems is

the permanent magnet (hereafter abbreviated P.~L) excited synchronous motor. The

P. l\1. synchronous motor offers several ad\'antages over the d.c. excited synchronous

motor. A primary advantage is that the P.l\L motor does not need an external supply

to excite the rotor field. Hence, the field winding and slip rings are eliminated. The

absence of the field winding reduces the cost r~nd eliminates the power loss associated

with the field winding [1,2]. i\[oreover, the P.M. motor occupies less space than field

windings for a given size, which leads to more compact designs [3]. It also minimizes

maintena nee.

Recent research has indicated that the P.~l. synchronous motor has also become

a serious competitor to the induction motor for high performance servo applications

[4-7]. The P.M. synchronous motor is more efficient and has a larger torque to inertia.

ratio and power density when compared with the induction motor. In addition1 for

the same output capacity, the P.~l. synchronous motor is smaller in size and lower in

weight which makes it preferable for certain high performance application like robotics

and aerospace actuators.

1

However, the P.M.synchronous motor has some disadvantages like other type of

motors. One of t.hese is the existence of a braking torque generated by the magnets

during the starting period. This decreases the motor's ability to synchronize a load [8·

9). The problem can be reduced by designing a cage winding having a high resistance

to provide sufficient accelerating torque.

In order to understand the basic characlcristicof the P.M. synchronous motor, it is

necessnry first to appreciate the distinguishing configuration of the interior P.M. motor

itself. The followir.g description is appropriate only to interior P.M. motor where

magnets are mounted inside the rotor. Oy using the sample four-pole rotor geometry

shown in Fig.l.l, the magnetic flux produced by magnets defines a direct-axis radially

through the centerline of the magnets. An orthogonal quadrature-axis is defined

through the intcrpolar region separated from the d-axis by 45 mechanical degrees

(i.e. 90 electrical degrees for a four-pole design). As sketched in Fig.l.l, the magnetic

flux passing through the d-axis magnetic circuit must cross two magnet thicknesses.

Since the incremental permeability of P.M. materials (ceramic, rare-earth, etc) is

nearly that of free space, the magnet thickness appear as large series air gaps in the

d-axis magnetic flux paths. Since the q-axis magnetic flux can pass through the steel

pole pieces without crossing the magnet a.ir gap, the q-axis reactance is noticeably

higher than that of the cl-axis. This introduces a saliency into the rotor magnetic

circuit in which the q-axis reactance is larger than the d-axis reactance. In contrast,

the conventional wire-wound synchronous motor has a value of Xq which is nflrmally

flO - i07o of t.he value of Xd for the salient pole rotor machine. Different design of rotor

configuration and geometry provides different values for t.he ratio XJ/)(7 , which will

alter the torque production and pcrform11nce of the P.ivf. synchronous motor during

both steady state operation and the run-up period [10-13].

The values of X a. and X9 for the conventional synchronous motor can be measured

by standard test procedures (14]. However, these procedures are not suitable for P.M.

synchronous motor because of its permanent excitation. Thus, available experimental

2

(I) (b)

Figure 1.1: Principal magnetic flux paths: a) d-axis, b) q-axis

3

methods by \vhich the P.~f. machine parameters can be evaluated accurately have

been a subject of many investigators. This research is intended to determine the d­

and q- axis reactances and permanent magnet generated voltage of P.M. synchronous

motor.

An outline of the remAining chapters of this thesis is given as follows:

Chapter 2 presents com·cntional tests for synchronous motors and int.roduces the

basic characteristics of the P.l\1. synchronous motors and three of the important motor

parameters. A literature review dealing with the measurement of these parameters is

also presented.

An evaluation of the air gap fields of the P.~t. synchronous motor follows in

Chapter :J. An analog simulation of the interaction between these field quantities in

the air gap is shown and a digital measurement technique for determining the motor

torque angle is also developed.

The measurement of the machine reactance parameters is presented in Chapter 4

and .s. A nmnberof different test methods arc shown and the analysis for ea.ch method

is given. Conventional tc::;t methods are modified for use on the P.M. synchronous

motor.

Finally, Chapter 6 presents the conclusions of this research work. Suggestions for

further study arc also given.

Chapter 2

BASIC CHARACTERISTICS OF

P.M. SYNCHRONOUS MOTORS

2.1 Introduction

A general introduction to the P.M. synchronous motor was presented in the last

chapter. It was shown that permanent excitation of the P.M. synchronous motor

results in some differences when compared with d.c. excited synchronous motors.

Consequently, conventional tests for the synchronous motor cannot be applied to the

P.M. synchronous motor and alterna.te test methods must be devised to determine

its characteristics and parameters.

In order to understand the difference in test methods between the conventional

and the P.rvl. machines, the t~sts for conventional synchronous motor a.re introduced

first. Then the reasons of why these tests cannot be applied to the P.M. machines

are presented. Finally the literature covering the determination of P.M. synchronous

motor parameters is reviewed.

5

2~2 Conventional Test for Synchronous Motors

There arc two different types of synchronous machine due to different rotor structures,

one being a cylindrical- and the other being a salient-pole synchronous machine. A

typical salient-pole rotor structure is shown in Fig.2.1. It is clear that the air gap is

much larger on the quadrature-axis (i.e. in the interpolator space) than on the direct­

axis. Since the air gap is of minimum length in the direct-axis, a given armat.ure mmf

directed along that axis produces a maximum value of flux. The same armature mmf

directed along the quadrature-axis, where the air gap has its greatest length, produces

a minimum value of flux. The synchronous reactance associated with the direct- axis

is therefore a maximum and is known as the d-axis synchronous reactance Xd, the

minimum synchronous reactance is called the q-axis synchronous reactance Xq.

The conventional way of determining machine parameters is defined by "Test

Procedures of Synchronous Machine~" (14], which includes the open-circuit and short­

circuit test, slip test, etc and will be introduced as fo1lows.

2.2.1 Open-Circuit and Short-Circuit Test

This is a way of determining Xd from the measurement of the steady-state armature

open-circuit voltage and short-circuit current [15-16}. The rotor of the machine is

driven at rated speed with the ficid winuing excited, and the open circuit armature

voltage measured. A three-phnse short circuit is applied to t}•e armature terminals

and the sustained armature current is measured, keeping the field current constant.

The direct-axis reactance may be found from the ratio of the open-circuit voltage

to the short-circuit current. If the machine is saturated, the voltage read from the

open-circuit line should be used instead of the air gap line. The result is the saturated

value of Xd, see Fig.2.2.

D-axis

Figure 2.1: Rotor structure of the salient-pole machine

.. ~ : 1 :! '3 .. ~ = ( e t ;)

0

I

0('('

;: .. 1 sec ; I v .. I ,-l-- ''] I--·.-----,. i

~

- -!-J- . --- " £ I I ; i: I ; "' I "

~ :; II:

h t fl

Figure 2.2: Open-circuit and short-circuit test

7

or

2.2.2 Slip Test

v oc ( d) J\d = . Fl , unsal-.Lrate y3o d

X -~ (saturated) d- M I

y.JO e

(2.1)

{2.2)

The slip test is carried out by applying a reduced, balanced, three-phase voltage at

rated frequency to the stator, while the rotor is driven at a speed very slightly different

from synchronous speed with the field circuit open [14-15]. The phase sequence of the

applied voltage must be such that the i'l.rmature mmf and the rotor travel in the same

direction. The d-axis and q-axis of the rotor thus alternately slips past the axis of the

armature mmf, causing the armature mmf to react alternately along the direct and

quadrature axes. The armature current, and voltage, as well as the voltage across

the open circuit field winding arc observccl The measurement method and a typical

oscillogram are shown in Fig.2.3. The minimum and maximum ratio of the armature

voltage to the armature current arc obtained whP.n the slip is very small. For these,

approximate values of d-axis and q-a.xis synchronous reactance can be obtained by

the following equations:

xd Emar per unit (2.3)

lmin

Xq Em in per unit (2.4) = lmar

The reactance Xq is less than the reactance Xd because of the greater reluctance

of the air gap on the quadrature axis. Usually X., is between 0.6X,1 and O.iXd.

8

I

I

VOLTAGE ACROSS OPEN FIELD

ARMATURE CURRENT I .

t;/CI) w -a: )( a <

Figure 2.3: Slip test

9

2.3 Basic Characteristics of The P.M. Synchronous

Motor and Its Parameters

There are three critical parameters for P.M. synchronous motor from which the

steady-state performance of the machine can be determined [11,17-18].

3 Eo V . \12 1 1 . T. - -[--s1n8- -(-- -)sm28] e v · •J \r v w -"\d ~ ·'d ·'q

T1 sin 8 -12 sin 2 8 (•) . ) -.. ')

In equation (2.5 ), Eo is the permanent magnet generated voltage and is equal to

the product Xmd * ljm 1 where Xmd is the direct axis magnetizing reactance and ljm

is the field current induced by magnet flux, while Xd and Xq are the synchronous

reactance in the d-axis and the q-axis which are similar to those of conventional

synchronous motors. Testing and chatacteri?.ation of the P.M. synchronous motor are

in principle similar to the procedures applied to the conventional synchronous motor.

However, there arc three characteristic features of the P.M. motor which result in some

difficulties in using conventional test procedures on the P.ivl. synchronous motor [l!J).

The tests which are particularly affected arc those from which Xd and Xq can he

obtained. These relevant features of the P.M. synchronous motor are:

• The motor is permanently excited;

• Xd is generally appreciably less than Xq;

• The reactance parameters arc more sensitive lo saturat ion effects than those of

the conventional machine.

Firstly, permanent excitation makes it difficult to employ the conventional slip

test as a means for determining Xq. Open-circuit. and short-circuit t.csts have t.o

be modified as well to be applied to the P.M. synchronous motor (hereafter called

10

"modified conventional test"). Secondly, since Xd is less than .\"9 , the second term in

equation (2.5) is opposite in sign to that obtained in the conventional synchronous

motor. Because of this, the peak torque for the P.M. motor occurs at an angle larger

than 90°. If the difference between d- and q-axis reactances is large enough, the

torque at small torque angles becomes negative. This means that the torque angle

at no load is larger than zero (This will be verified practically dming the no-load

test). Finally, t.he effects of saturation cause the machine parameters to vary with

load. Subseq11ently, alternate test methods must be devised to locate the machine

parameters.

Simply xd and Xq can be found from the steady state \'Olt.age equations of the

P.M. synchronous motor [18], which arc given as follows:

xd - V coso -Eo R1 I a. cos(¢- 8)

fa .sin(¢ - 8) fa .sin(¢- 8) V sin8 + R1 lasin( ¢- 8)

fa cos(¢ - 8) laco,q( ¢- 8)

where:

V: applied voltage

Eo: open circuit voltage

Ia: phase current

R1: stator resistance

¢: power factor

6: torque angle

(~.6)

(2.7)

It should be noted here that the determination of xd involves intemal generated volt­

age which is assumed to be equal to the open circuit value Eo and remains constant.

However, Eo in practice, is a quantity which varies with the load. The open circuit

11

voltage may be founJ from the expression l20]:

E~ 2 iT' f ~ V ', X 1 o-s 0 == . M 'l' t 1 I\ 1V

v2 (2.8)

or

(2.!))

The following expre!'sion can be obtained by substitt1ting D1 = H•f1~(;tg) int.o

Eq.(2.9):

Eo (2. 10)

Since the open circuit voltage Eo involves the leakage factor A" =l +f3K, which

varies with the stator current under load condition, Thus E'0 is a machine-satumble

parameter which depends on the saturation of the steel leakage bridges. Consequently,

Xd is also a quantity affected by saturation. To determine E0 , it may be measut'ed only

as an open circuit quantity. It is impossible to measure t.he open· and short-circuit

characteristic in the 11sual wa.y because the excitation cannot be varied. Neither can

xd and x'l be measured by the slip test, because this requires the excitation to be

removed.

2.4 Review of Literature

\Vith t'egard to the detel'mination of P.M. synchronous nutchinc parameters, the lit­

erature of the recent years has been divided into two different types. The first. and

largest group has been concerned wit!. compnta.tiona.l methods for determining the

significant parameters of the machine. The second group, which has a much smnllcr

following has dealt with the test aspect of being able to measure the machine param-

eters.

12

The computational methods use some numerical iterative procedure to sol\'c the

magnetic field of the machine gi,·cn some specific geometry. Of the val'ious methods

available the finite-element method is the most commonly reported, see references

(22-27]. Other methods for solving the nonlinear magnetic field equations by iter­

ative methods have been presPnte<L sec references (3,li.21,28]. Among them, V.B.

Honsinger (21] has presented a critical work. The method presented allows the de­

termination of the saturated values of the macl.ine parameters a11d the work is quite

well for predicating motor performance at entire range of loads. These methods which

provide relatively accurate models at the expense of significant computational effort,

are primarily aimed at design work and are less '=Onccrncd with the actual measure­

ment of parameters. Moreover, for these kind of methods complete knowledge values

of internal construction of the nwchine is rcquir<'d and is usually not available.

Without the ability to remoYc field excitation from P.i\1. synchronous machine, the

conventional methods of electrical parameters identification cannot be utilized. Con­

sequently, a quite limited reporting of rncilsurcmcnt. of machine parameters is available

in the literature. The most significant works dealing with the actual mcilsuremcnt of

parameters have come from General Elcctrics motor development laboratories and of

whom T.J.E. ~·tiller and V.D. IIousingcr [19.21] have been the principle characters.

Honsinger has presented several modified conventional tests which can be used to

measure motor parameters. This method allows the measurement of both unsatu­

rated and saturated values of m::tchinc parameters and work is suitable for predicting

motor performance at no-load and heavy load conditions. ~liller has presented a

method for static determination of the machine axis inductances. His methods are

based on an earlier work hy C. V . .Jones [2!J] and incorporate the in tera.ct.ion of d-q axis

flux. Miller also presented a lond test to measure t.hc Xd and X?. His results clearly

show that there is a region ncar ?.ero d-axis cmrent where the d-axis inductance varies

significantly.

In load test provided by ~Iiller [19], a set of data has to be measured including

13

machine t orq u£> cmglc. ~lost. (•xist.i ng mcasmcmcn t dc,·ires for nwast 1 ring torq' 11~ illlgiP

usc a stroboscope and a graduat.ed disc affixed on the stator frame. This kind of

measurement has poor rc~olution and is not convenient to use in practical tests.

~loreo\·eJ-. it cannot meet the requirement of the machine driw~ system where nil t.he

p<~ramclers have to acquired clect.ronically and digit.cdly. rcfc•rences [:JO-:Jl].

rn end of the rcfNences cit£>d above, although mention was made of the val'iat ion

of th1! genera ted \"Oit.ilg£>, £0 , a II of the ca lcnla lion were clone und<'l' the assn m pi ion of

consl<lltl. £0 • Consequently, saturated value of Xr~ presented by Honsinger [~l]must.

be con:-idC'rcd as a approximate value. The agreement. bet.wc•cu load lest and static

inductance mcasmcmcnt pt-e!:ic'nted by ~[iller [l!l] is illso less t han sat. i~ factory. It is

the objective of this work to investigate I his variation more t.horonghly, to validate

the va rions test methods and to i ncorporat.e a digital tecll!l ique for JllCMillri ng torque

angle.

Chapter 3

FIELDS OF P.M.

SYNCHRONOUS MOTORS

3.1 Introduction

In order lo d<:'termine the tnilchine parameters, the air gap fields of the machine m11sl

be known. The objectives of this chapter is t.o develop air gap field descriptions

based on the configmation of the motor with t'adi;dly oricnt~d and inset magni'IS.

~lorcon~r, the interaction of thr. field quantities in the <lir gap is verified by nnalog

circuit. simulation. The results of this simulation is employed to ll\('asure the motor

torqnc angle by digital techniques.

3.2 Air Gap Fields

Air gap field of t.he P.~l. synchronous machine can be descrilwd by a con~idt•rat ion

of tlw configuration of t.hc rotor and magnetic circuit. of the Inac!.ine. There arc

t.WO riiiX SOUrCeS COntributing lO the air gap field in the P.~(. synchronOUS lllilChine.

One is produced by the magnets in the rotor. The other is generated by the stator

current .. The flux density distribution in the air gap is the resultant of these two flux

15

cornpo11P!lls, each of \\'ltich will be discussed scparat.cly in th'-' fnllo\\'ing sed ion:-;. Th,•

field models and anal~ ,is bilscd on Honsinger's research [:!l] <ll'<' gi\'(lll flrst. Tlu~ work

is c·xtcndcd to der('lop an ana log circuit to simulate t lte fi<•l r1 models nnd a digit a I

circuit to measure the motor torque angles. Figure :J.l ilnd fig11rc 3.2 show rotor 1md

m<~gnet. configur<~lionused in this study and the various paths ;1\·ailahle to the• 1nnguet.

flux. Since the magnets in I lw rotor lie in the direct-axis of I lu• m;u·hinc, the tlltl~tH~Is

will influence the d-axis field. without affecting t.hc q-axis il<•ld.

3.2.1 Magnet Fields

Six cqunt.ious ilre used to describe the magnet fields <Ill<' tot l1c P.~l. acting alone:

Bt -JI1

t flt + /J,., (:U)

n2 = -jl; J[2 + /Jr~ (:~.:2)

<l>m = (~/ + <l>t (:L:l)

lit Lt - 2ll2 Lz = 0 (:L-1)

'2gf!J - Ift Lt ::::: () (:L1)

~~ = ·) <fl 1 H.? I - Rt

(:U) )

Efptations (:3.1) to (:LG) <~rc based on the rotor configmatiott shown in fig11n~ :l.)

and flux paths shown in Figure :3.2 as well the magnd. l'hara,~krist. ic <ts slto\\'11 hy

figure :J.:L In this work. the analysis is applicable for \'ario11s magt~P.I. materi;-ds ilnrl

requires a rotor constmction with radially oriented and insl'l tllil~rtcts. This is the

most common dc~ign of P.:\1. synchronous motor~. The dcri\'at ion of I his set l'qunt ion

is shown as follmvs.

1. ~[odd of The i\[agnet.ic ~-latcrials

Considering the rotor const.ruct.ion shown tn rig.:L I' it is dear that t.IH•rc ilrC

two !llilgnet. sr.ctions in t.hc rotor contributing t.o the lot<~l ;lir g;1p ll11x. Th<) hasic

16

a)

t2

~------~1~------~

magnet section # 2

b)

Figure :3.1: Rotor and mi\gnet geometry: a) rotor and magnet configuration h) d i· mens10ns

17

I

<I> I I "----"'---·~-._, ... ,I

:1 :, ~------~,--~~--~~~ ~------------

' I .. ~/ ) .,...""" / ......... / .,...... _ ....

.,.. .... .., _,..

Figure :3.2: Flux paths of the milgnets

He (COERCIVE ,:-.JRCEl

~ H

B, :RESIDUAL INDUCTION! B --

0

Figure :3.:3: B [{ characteristic of the magnet matcriills

18

dcmt~gn<'l izat.ion Cit I'\'(' of 1 he tlf'rmancnt magnet mall'rit~l in I lwsc sections lllil_\' IH'

moclt•l(•t I as:

( :t 'j)

wlwn' slope tc' is equal to: +Br /He. Therefore, the recoil Blf characteri:-lic of

the mngnet is written ns follnws for magnetic st~ction I and sedion ::!:

-~~~ fit + Brl

-11~ lf2 + !Jr;!

~. Flux flalance Equation

(:LS)

( :l.! I)

Th~ flux 1>m created hy the two magnets is <'qnal to I he snm of the mngtu•t air

gap nux 1l>j and the steel hridgcs leakage flux cf>,, ancltnay he \\'rit.ten as:

<I>m ::: <I> f + 1~/ (:1.1 0)

or

(lll)

- Bt tlml + /J2 .'\ml

- 21&( h, Bt + h2/.?~)

a nrl (fl1 will bt~ gi\'en latr.r.

:L .\pplying Ampere's circuital law to the two flux paths: pt~th .l'!l= cros!'«>s the

air g<lp into the stator to link the stator winding. while path ahc is located t'lllin·ly

in tlw rut or and links only 1 he rotor magnet.s. Thus. t.lw equal ion ht~s<.•d on r\ tnpc·re's

circuital law may be written as (for paths abc and .ry: r<·~pcct.ivcly):

II, Lt - 2!12 L2 = n

2gHJ-!ItLt 0

HJ

( :L I :J)

( ;LJ .t)

Till' \'('Ctor dirc•ction for I he II ilnd n fi<'lds is npposit<' in Sl~n fnr I hP tlla!.!,lll'l

material hut. is of t.he SiHIH' sign in the air g(lp.

·1. Lt·akagc flux in Tlw Steel Bridges

The lcilkagc nux in the :;t.<'d bridges i:; dependent. upon t.hc comhinatin11 of t IH'

lllfl_!!;tll'l nunf and the armnturc mmf. In ord<'r to citlr'nlate th<' lt•akag;<' ll11x. tlw

following is ass11mcd.

• :<I ;!for winding is opt•n-cirruikd, in which cfl;o;e sl ill or cmn·nt is z<·ro. t ltns rlwrP

is no H rma Lure reilct ion;

• llCI'!llPability in t lte iron is iufinit<' rverywiH'rc' <'XCPpt. in the stl'ei hrirlgr•s sm-

ro11nding the mngnPt~.

By con~idcring t lu~ onl<•r hridgc, Ampere's L<tw r< 's lllts in:

C'ondtining the ahm·e c•qnations. gives.

II - '!.y llj

tl-

"· Tlttts. the bridge n11x density is:

B11 = l't !Itt ,,,'!.gill

/,,

Tlw leakilgc flnx for t.wo such bridges (per pole) is:

(~It = 1311 2/ 1 1.~

= 2<P1 n,

Rtt

20

(:l.l.))

(:\.1 fi)

n.1 I)

\\"ltcrt' (I> f = 13 f :\1

R-L 9- ,.\!1

R - r., 11 - 2ut It I .•

Tit<~ tot a I lerl kagr. flnx JH'I' pole is :

<J> I - (~{\ + (~/~ !?.,

= 2 ~~ rn; w lwr<'

.) . Tltc Flnx Dr'n~ity in Thr. :\ir Gap

:-\ ~et of equations Eq.(:J.I) - Eq.(3.6) has been set. up to gi\'e a description ror

t.hr nwgnet. fields. The flux dC'nsity in the ait· grlp prodnccd hy the magnet:-; will be

derin~d from t.hcsr. NjUations. The derivation is flS follo\\'s. Substituting Eq.(:U) <~nd

Eq.( :t:!) into Eq.(:t:l) and re<nranging. gires,

2,qlff /.1

'2g If, '2 L2

S11hst iluting Eq.(:J.G). Eq.(:1.22) and Eq . (:l.~:J) in F.q.(:t'21 ). yields.

') I I I •) I I I

<fl - •) ( II ( - s It fit - ·' 12 Jl.l ) T - !J I l + •) l :;:::

~ I - .I'J

21

( •) o')o')) .,, __

( :1.2·1)

•) { I I

- ·' It / 1·1

L'}.

nrrd ~nbsr.itut. ing rlwm into Eq.(3.2·l ), thr followi11g is oht.;li11cd:

1 1 tfl - •) <'> 1 !? ( - + --)

r - I :J j> •) R· lt - -~

/?, = <fl 1 ( I + :! -

1 ;' ) II

Soki11.~ Eq. ( :1.27) for t lw f1 i I' g« p flux tf> 1, gi n•s.

Th11~.

1 + •) &_ + •) j> ( I + I ) - n, - 1:1 Til 'lll1

<!> r

U m = -r-...:......,,--'RI + 'i7il

1 2 R2 I = H,

1\'t =I+~

<ll 1: air gap rt ux

( •) ')-) ·l.-·)

( :L2G)

( • ) .,-) •l.-1

Tlw t•qnation (:1 .2D) states t.hal the total r£'sid11al ll11x <•f the 111ag11d cf>r is n•dlJn•d

hy t.lw 111achine g<'OJJH'I.I'Y .:3 and the kakagt• flux l\'t · lkarrangiug of Eq . (~.'2!l) with

22

fl, .-\rn

t + ",\., = B 4 /\ "' I .-\7

r • :J l + :'! {\'I

llr•wt•. flux dcnsit,\· in the air gap is gi\'cll b.'·

= Br(:\.,.j:LI)

l + .; ,,-, (: l. : II )

This is the fl11x dPnsity caused by t.hc 1!1i1gncts net ing illottc. The flnx dt•nsi ty

\\'a\-cform df'scrihed hy Eq.( :J.:n) is a trilpC'zoid;d ,,·;m·fnnn and shn\\'11 in Fig.:L I.

3.2.2 Armature Fields

The r1ir gap field caused by the armature currcn t ilCI ing ;tlnne hils n ti-n xis con1poncnt.

nnd i1 q-nxis cotnponcnt. As stated at the beginuing of tl1 is sPc t ion. the Jll iiP,IIf'ts in

the rotor arc locaH•d nntltc d·axis of t.hc macltiuc. th c•n•l'nt'l' bot.h the m;,gtH!ts itlHI

the fll'ltlillllrc wiuding contribute to the d-a~ds ll1tx . Tlw :--tator r.nrrenls arc tlw sole

~ourcc of nux on the q-a;.;is. This spcciill fctlt.nn· will IH' !'OIIsidcn·d in t.ll(• l'ollnwing

ana lysis of the arrnr~ t 11 rc field.

l ). Dirf'ct-,\xis Field Caused h\' Stator CurrC'nts

.-\ ~d of eqnat.ion are ddinC'rl to dcscrilH' direct axi~ <ll'lllillnre fi1•lds:

I •) I //1 Lt - _lf'1 L2 0

2rfll,r~ + 11; L1 0.8 rr F.1m m ... ( ~:) 0

u; = 1t' 11;

13.~ = ,t' 11~

( :L:J2)

( :l,:J:q

(:Ll·l)

( :L;Fi)

( :1,:3())

B

I ~ad I

fl--~-- .....

I

I

~ ' \

'

' \ \

\

' \ '

Figure :t .l: Flux dcnsit.y due to the magnets

..,. ____ _ one pole pitch

Figure :3.5: Direct·axis field due to t.he nrm<ll.ure current

2·1

t

Eq.{ :1,:1:2) to Eq. ( :1.:1()) are ohta i ned from Eq. (:L l ) to Eq. (:U)) wit lr so rue exn'p t ions

r~s follo~rs:

• The circuit. equa1.ion for path .t'JJ= encloses all arno11n!. of t.hc stator cmrc111 gi\'Cil

by {2 Fr1M. cos( p/2)0) which is sole source of the flux.

• The magnets arc \lllllHigtH?tizccl. in which cas<> tlw nwgnct. has zero rc•tn<Hlcnt

mogn<:'lisrn.

• The nux balance rqttation rhangcs its fonn to:

ln the obovc cquiltions. (~n·l is the flux which cros.~rs I he ilir .!;ilp illlrl ran be

dct.rnnincd by integrating the d-nxis flux field n."L o,·,•r one pole pitch:

llc11ce, I he flux hal a nee equation is rcwt'it.tcn as:

1\' here

The rlnx generated by t.he two milgnct. sections is gt\'l'll pre\'iously hy:

''<~~ = 2 L5 (h,,t' n; + 112,,. 11.;) I l l

II, L, ( R, + 2 fl~ )

= u; L, Rm

25

( :L3S)

( :t:l!l)

(:3..1 0)

\\'ll<'re Rt. fl.2 and R m hn\'C bern given pre,·iottsly. Th(• iutt•grat ion of eq11at ion

(:J.Ti) i!' performed by sttbstit.ut ing the \'nine of /l~rl from Pquation ( :I.:J:3) \\'IH'rt' /1"d

= 11o /1,1'1.

ln)p

/) L., 13ar/ d (} )

1?!1 = f;: <~ir gap reluctnncc

A9 = nl.: nrca of air gap

Suhsl it.111 ing cqttllt.ion (:L.tl) nnd (:3.·10) in equation (:1.:18), gives,

\\'here: 8 = 2 If:, gi\'(~11 prc\'iously.

(:Lll )

The ilir gnp flux clt·nsit.y 13"1 c·<ul now be found frotn Eq.(:3.:3:1) J,y climinnt.ing 11;

\\'hidt is gi\·en in Eq.(:1.·12) , nnd results in :

[' J _ O.·ll'o 1r F1m [ • P ,1 .J.1rl - ro." - r. -

9 :! '2/'o I ~ I I + iJ I\ I

I I i:-. observed t hilt t.lt<' cl- i1 xis lid <I <1<-scri bed hy Eq. (:L I :q i ndndl'~ t wo COlli ponent.s.

otw h .. ing a sinusoidal INill fllld I he ol hr.r lwing a I l'il!Wzoid;d t 1'1'111. <llld the rolllpositc

\\'an.fnrnl is shown in Fip;.:L.).

'2). Q11ndrat nrc-Axis Fic•lrl CattsC?d hy :\rmat ure <.'11rrcnl.s

~(j

Tltt~ q-nxis flnx density nrl? is similar to I he d-CI xi:-; nllx density nf!d except for

conqHlflt•nt.s related to thr. magnets and is gi,·cn hy t lw t'Xpre~sion:

3 O..lpo 7r F'lm · P 11

/1rt;::: 31/1-11 I g •)

3.2.3 Air Gap Fields

lip to now, three primary f'qll«tions ha\'C been ohtain1'd which tiO\\' can he ll!'il~cllo

describe the ilir gnp fields of t.ltc P.:\l. synchronons tnntor. The three cquiltions ilre

H'Pt'itiPd here for com·Pnic-nn~:

~l'o / 17 J 1 +.a r\·t (:UG)

Ha~ed on the anr~lysis gi\'C'll ahovc. it is clc~ar that tltc flux dL'Ilsity in t.he nit· gap

IS ('OflljlO~Cd of a lllrtglll'l. flc]d and Clll al'lllCltlli'C C'IIII'<'Jit r11 ~Jd w!Jil'lt ntny he \\'l'ittl'll

as:

11 Ira-; hcen strll.cd that. of these components the rn;1gnet. COIIIIHltrent. is l'OT1St.i1nt

on~r tliH' pole pitch and lras Cl t rapPzoidal \\'n\·cforlll. The \l'a.rcfornJ of d-axis ilrmnt me

Clll'l'l'lll. component is the resultant of sinusoidnl atHl t ropczoid;.t lr'rm~; while the q-

axis ill'lllil tlll'c current cornpol!l~nt may be dcscrihed by 1-1 pmrly sinu~oidol \1'11\'<'form.

The anillysis of the air g11p fic,lds as described ilho\'e results in following comments.

Fir:-;tl,r. ,,n cHwlog simnl11tion <"an he implcmctttcclto :-;itnlllilte ,·arion:> compont:uts in

the air ~ap and their interact ion from which it will be sho\\'n wheth1!r this ficltlmodd

gin•s il I'CtlSOililblc rJpscripf.ion to ildllcd fiPld of the madrine. s~xoml\y. it. jli'IJ\'idi~S illl

analytical expressions to ill\'C~:-;tigiltc inlhtcncc of t.he Cll'lllill.urc react io n on gerH'l'ill.()d

\'Olt.il~t~ at. lond couditions which will he of signirinlllt. llll'alling t1:> predicat ing st Pady­

sta te pt~rformilnce of tlw mach irlf' hy usc of the lond tc:-;t., Both oft hc~sc willlw I'Pa li:-:ed

in this work.

3.3 Analog Simulation of The Fields

B<t!-ic•rl orr the rc•srdts of the f>l't'ft'ding antllysis. <I simrrlnlioll circ11il rn;ty lw dt•\"<•lnpc•d

to lwlp nndersli\nd tl11' nir g;flp ri(•ld df'scrilH'd in thP last sc'ctioll. The simnl;ttion

ci I'C'llit is composc•d of il sign nJ ~011 I'C:C'. \\'ii\'C•fOI'Ill sha pi 11g eire IIi t illld p hoSf' shirt <'J', :\

blod; di;1grarn of lhc circnit is shown in Fig.:l.G and tit<' circuit is ~drown irr :\fltH'Irclix

I. The signal gerH'rator pro,·irlcs two types of signals, sin11soida 1 aud t riangnlar where

the the t.ri;wgulr11· i::; tr:<c•d to form a traprzoidal signo I \\'hich reprt•scnts the tn<l~nct

gcnnated \'oltngc hy n~e of a ll'ardorm shapiug circuit. The output. uf the stt bt ract

cirr.:11il. i11 which inpnt,;; 11re t hf~ siuusoidnl nnd t rnpczoidid signa 11i, is used to reprc~cnl

the rlnx in the d-axis. Tit~ onl.pnt of t. hc phase sltiflc~r is us(•d to rcprc~sPnt. t.he

flttx iu fhc q-ilxis. Tlw~c· tllJ'('t' signals, ('('JHcsc•nling; IJ.,.,. ll1 <111d /3., ,1 rrspt•ctircly,

arc ~yut!tcsir.cd in tlw Sllllllllllfinll cirr11il. to sinlldit!f' tlw ilir g<~p flrtx ll:/1 in \\'lrkh

OJWI'afinn is hascd on lhP itSSIIIllplions ~irc•tt h.r:

\\"ith different anglllardisplarPillentlwl.\\"<'c~nlf~r/ and IJJ. tlu•rcsnltsofsinurlation

arc sllfJ\\'11 in Fig.:L7. Stricti.'· ~pPilking, the fic·lds in tltc air g<lp c'illll\01. lw ol,t;tiru~d

hy il lirwa r synthesis oft lw 1 ltn•r comporwnt s. Tlw l'<'ilSilll for this is t.hal. tit•~ rot or

irou i~ m:lgnctic:<tlly uonlin«'<ll' and t.lH! IPnka~f' ll11x in tlu! slf'<'l ilridg;<'s SIII'I'Dtlllllin~

tIre nr;rgrtcf.s is nonlinearly dc•lu'ndcnt orr t.he ln;trling mudition.

llol\'1 ~ \·f~l' upon compnrinp; I lie rPstrlt.s of 1 he• sirnulFII.inrr and 1 hl' actnnl ~ir .~i r p llnx

"·m·c.f,,r"l or Ogurc :LS. it is cl•·;u· t.hat:

IV\ Wavefonn _f\_f\_

Shaping - I--

Bad

B1 Ba -BGq

Phase Shifter ~

Figure :3.6: 131ock diagram or i\nalog simuiRtion

2!)

Bad-

(b)

B -g

- B-g

( C:-}

(c)

Figure :.L7: Hesults of air gap field simulation: H) light. loading , b) medium loading, c) l1igh loading

:w

• t.hc three air gap components used in then llillysis su!Dcimt ly describe the <le t 11al

conditions.

• s11pcrposition is vol id for general operation and t lac nonlinear effects are only

~'wen at high loading.

3.4 Digital Measurement of The Motor Torque

Angle

In ordr>r to calculate the lllilrhinc p11ramct.crs rrornload h ~st it is neccs~a ry t.o nH~ilsnrc

the I o rquc a nglc 8, il t. <'ilrh load step. The con \·ent.iorud 111ethod for mcasmi ng th is

qua11l i ty is by usc of a st roboscope and graduntcd di~: k afHxed to the motor frame.

The st rohoscopic method pro,·idcs only ap1>ro:dmat.c angular quantilie:; since shaft

oscillatiolls and flash irrq~ularit if's result in a rcl<1tirrly poor mcasmement il C'IIlracy.

An alt.crnal.c digital t<'cltniqnc which is both silllplc and pracl.ical for clect.ronir illl­

plcmcnt ;d.ion has hr.cn dc\'<'lop('d and is present cd in this scclion.

Tire torque angle of t II!' P.:\ f. synchronous naotor i~ d<'fined as a angular rlispl <~cc­

nwttt. hctwec~n t.lrc flux c·ollljlOll<'lll.s generated hy t.lw nwgnd. <~ nd ilpplied roll age.

Therefore. t.he torque <111gl<' lll<'ilSIII'I!Illcnt rcqttin's the air gap flux wareform as an in­

put and relics upon I he fact. thAt. the llux componel!!. d 11e to tlu~ nw gnd. is t.ra 1wzoi da I

in sh<~ p<~ and tha t the a pplic·d \'oltnge compone111. is sinusoid;tl. A sir ll llla t ion hy nl'e

of 1111 <llli'llog circuit is C;tl'l·ir•d ont to in\'esligate the I'C'l<lt.ionship and is dc~snihrd in

fig.:L!l where 139 is the' ~i.'~'"'' d(•ri\'r.cl from t.lw <lllalngsinnrlntinn output , ~ee Fig.:J.G.

The sirnnliltion is perfornwd 1,,,. adjusting the anrplitllrl(' nf the trapt•zoidnl r'Oillpo­

nellf. £; illld the phil~e o,.~,. T\\'o statements Cilll IH' dril\\'ll from simtdat.ioll J'I~Sillts

which is shown in fig .:J.lO and T<1hlc :3.1, as follo\\'s,

• poi11t P mo\'es directly as t.lrc phase Op/1 ch<tngr.s:

:J I

/"\

I 1\ v \

\

{\ I (\ (\

I \ 1/ \ 7 '\ \ I 1/ \ I i\ I

\ I \ I 1

!\/ \ 1/ \ 1/

lr' \ r \ ( ~ 1\ . vt \ h

[7 M I 'v, ;, ;

't \J \ 1/ I i\ I : \

v ! ,v--!t \ i l

I

_j i j

•) b)

I (\ {\ r~

\ - I I I \ r \ I rv 1\ \,} \

~ ~ 7 v 7 \ f\ ll 7 \ I

!V v

i I i c)

Fig11rc :LS: Flux waveform distribution obtained from search coil: a) light loilcling, b) medium loading, c) high loilding

32

• amplitude fl varies directly as the amplitude of the trapezoidal compoJu~nt is

raried.

The former is useful to measure the torque angle which is given below and the

later is helpful to analyze variation of the generated voltage which will bP. gi\'cn in

t.he twxt. chapter. The block diagram of th~ nwasmemcnt is shown in Fig.:Ut ctnd

the circuit is shown in Appendix I I. The circ11it consist.::; of a f!ip-nop and a phase

dC'tcr.for ClS Well SC\' Cl'ill schmitt !.riggers and phase shifters. }npul or this !lip-!lop

comes fi'Otn the trapezoidal component in the a it· gap measured by search coil and the

sinusoidal component mensurcd by a voltage transformC'r. A eli ppcr is used in this

measnrement circuit l.o shape the wavdorm and a voltage follower is ttsed to buffer

the output. The schmir.t trigger com·erts a sinusoidal warcform into a rcctnngular

one. At the beginning of the measurement the phase shifter is used to adjust the

initi;tl phase angle to zero. ,\s the motor is loaded, the flip-flop prod11ces a signal

nH~ rl:--lJred by the pha~e det.ector using tcrmin<'ll rolt.age of the machine ns a n.fcrence

~ig:na I. This angniRr di::;placcmcnt between out put oft he flip- !lop and the rdcrellcc

is "· reflection of the motor torque angle. Fig.:J.l2 presents a compara.tire rc~11lts

bct\\'(.'«'11 measured phase (}I'" and t.he torque flllp;le b me<Jsmcd by the :.;trohoscope. [t

is clcill'ly shown thnt they hi-1\'C a good agrcC'ment. This technique pro\'ides bet IN

rcsolntion and is more conn~nicnt t.ha.n the stroboscope method. Fmtltcrmorc, it can

be <tpplicd in machine di\'C"!systems which rcq•drea fast-response digital tneaslll'C' IllCllt

oft he torque angle.

Tl1e test methods for dclerlllining machine part~met.crs will be gh·cn in t.h(~ 1wxt

two chilpt.ers.

R

I

Bs

Figme :3.9: Simulation circuit

a) b)

Figure :l, 10: Results of simulation: comparison of different amplitude of trapezoidal components- a) smaller amplitude, and b) largP.t' amplitude

35

s h . ~arr..

Ceil Filter & Cli;Jper

Differentia tor Circuit

Terr:anal

Voltage Phase Schmitt ~

Flip-Flop -Shifter Trigger r--- Ul ,. ~·

0 1!1 :J

0' Ill ~

,.... Phase Schmitt ::1

~ Shifter Trigge r tJ

Ul c: '1 (1) a.

Phase Schmitt Reference signal ~ Phase 8

f--Shifter Trigger Detector

Figure 3.11: nlock di<~gram for torque angle measurement

:36

Torque Angle [Elec. Deg.]

50

45 / 40

35

30

25

20

15

10

5

a 1 2 3 4 5 6 7 8

Stator Current [A)

Figure 3.12: Results of t he tot·que angle measurement: line- Oph , symbol-o

Tnhlc :3.]: Hcsults of analog simulal ion

Ej 1/,, .,.r,rJc = J. :t2T In this I <':--I • \ • = 8 \ '

E ( V)

2..1 :l.O -1.0 :) .0

o.o / .0 :-\ .0

11 ~~·/II (V)

1.8 J. :~:l

2.·1 !.~;)

:3 .0 l.t3 :3 .8 l. :t~

.I,.) I . :~:l ;},() 1.·10 6.0 I .:!:!

Chapter 4

DETERMINATION OF P.M.

SYNCHRONOUS MOTOR

PARAMETERS- PART ONE

4.1 Introduction

Three P . ~l. synchrono11S tnotor par·am<'I.I'I'S \\' f'l'f' dc~rrihd in lhc first two cht~pt P.rs

a nd it was shown thflt, the 1Hll'olllcters of the lllilchiuc can he deter111ined by use

of cornpntntiorwl met hods or !!~s t. mc•t hods of \\'hich the fo rmer is pt·irnar·ily aimed

at dcsig11 wot·k . The \\'ork prc•sc•nted here> is cotll'l'l'lled with the actual measurement.

Thcrdorr.. a mtmher of tliffcreut t(•st methods an! clc\'dopcd to determine the machine

pilranwt crs. The tnrtchinc used in this study is a :J-phase.·l-polc. 60Hz. Y-conncctcd

pcrmilnent magnet synchronons motor. fls rilll'd speed. \'olt.<~gcand current a rc 1800

rpm. :!O~F ilnrl :l.O:\ I'I'Siwctin·ly. Jn this chaptc•r. the rnodificd coll\'l'llliomd tes ts

and sr~nrrh coiltl'st i\1'(~ pn'!WIJII'd. Thr lllilill nhjc·cti\'C! of this chapt.1 ~ r is to dt•l c! l'lllinc

tlw \'itl'i<lfionofgcncri11rd \'nllagc·nnder londin .!!, \\'lticl1 will be used l.ocompntc, l-axis

nnd q-11xis rrMinrwrs of IIH' llliH'hinc ~.

:l!J

4.2 Modified Conventional Test

Since the special cha.ractcl'ist.ic features oft lw PJ.L synchronous motor are difff!rcnt

from the conventional synchronous motor. I his results in some difficulties in using

conventionn.l test procedures on t.he P.i\1. motor. Therefore, the tests including open­

circuit ancl short-circuit. tests han~ to be modified. These tests will he referred to as

the '' modified con,·ent.ional test.'' and are rarriC'd out as follows.

4.2.1 Open-Circuit and Short-Circuit Tests

The tests performed on t.hc P.~l. synchrono11s motor <He similar to those normally

performed on a conventional \\'011110 rotor synchrono11s motor, wi th t.he exception that

the fleld excit.o t ion con not. he rnried . Both open- a Jl(l short-circuit characteristics arc

measured while the machine is operating as a gf!IICrator driven at ro.t.cd :-;peed. Dming

the open-circuit test, the nux in the the oir gap is produced solely by the magnets

which induce a \'oltClg<' in th(~ st ;tfor winding called the open-circuit voltngc E0 • The

short. circuit cttrrent. 1 .•. : is usC'd to dcl!'rmine the motot·'s satur<~ted d-axis rcncta.nce

.\.1, which is il(>(>l'OXilllil1.Cd hy t.hc ratio f~'o/ f.,c. \'oJtagc waveforms llll'iiSI!rC(l from

t.hc air gap or the machine undf!l' hoth t.he open- and short-circui t condition:-; are

presented in Fig.-1.1 and Fig.-1.:2 which <~n' rcconh•rl by using sensing coils as machine

is dri\'cn al raiC'rl speed nnrl also COIIsistrnt. with that of the air gap field described

in pl'<'ccding cl1<~ptcr. It is rlf•arly shown that the short-circuit Clll'rent contnins sig­

nificant. harmonic component. i11 which tile third lt;mnonic is domin<Jie. Therefore,

t.hc innllcnce or the third harmonic 11111~1 IH· consirl('l'f'd when CilkHiating ['('ilci.C\llCC

paranwtcrs of t.hc machine by liSt' of opt·n-rircuit. 11 11d short-circuit. t (~st.s. The t.f'~t

data is list ed in Table ·1.1.

10

~-,-AI .AI 411 ~ ... , I" . ., ., ... .,

) ~ J

~ l.t .... •• L.a..-....t .A l.t •••. a r• , .. ~ ~ ... ,

I

Figure 4.1: Air gap voltage waveform at open-circuit test

·ll

Figure 4.2: Filtered air gap voltage waveform at short-circuit test

42

4.2.2 No-Load Test

This test is performed on the P.~l. synchronous motor while it is runmng syn­

chronously with no attached load. Table 4.2 shows the test data for the no load

cmrent lnt and applied voltage \/. Because of the charac:tcristics of P.~.f. motor, the

totaltorqtlc of the rnot.or can be nega.t.i\'e bcl.\\'<'c'n 6 = 0 and b = 60 at normal supply

voltage. Since fJ = 0 is grtH'rally an unstable point (wi th 0 > dTfd8), the no-load

operRtion of the motor is at iuitialtorqnc angle 80 which is greater than zero. In order

to determine the performance of the motor. the initial torque angle musl be consid­

ered wltich can be obtained by use of no-load tf'st.. In Eq.(2 .. j), T1 is proportional to

V and 72 to V2 • therefore H reduction in the terminal \'Oitagc causes a much greater

reduction in the reluct rtnce torque than in the excitation torque. This results in a de­

crease ,·alue of b. It. can he oh!-iN\'<'d that the rotor position goes t.o a minimum angle

posit.ion which is assumed t.o be zero torque augle. This angular difference obtained

is considered t.o be Approximately 80 and will be used latet· in the load test. For t.he

tllilchine userl in this work. 60 = 22°. Up to now, two values of direct-axis reactances

have been obtained from these two tests (see Table ·l.l and Table -1.2). These resul ts

will he cli~russed later.

4.2.3 Load Test

The load test provides a mc<H1l' for determining the st.eady state machine parameters

X.1 • .\",1 and £0 which can he nscd to c<1kulate motor performance charactet·istirs.

The test in\'olvcs loilding the P.~L motor hy coupling a synchronous machine or a

dynamometer as near al' possible lo t.hc pull-out valll('. In this test, sufficient data

11111~1 be raken to define the phasor diagram uniquely, which includes the applied

\·olt age and cnrrcnt, power factor and torque angle 8. The value of the torqnc angle 8

is <•it.heJ' ohsf.'rvcd hy 11sing a st.rohoscopc. ot· lll<'asured from search coil by using the

digitnl circuit which has lwen gi,·Pn in last chapiN. The initial value of torC)ue angle

comes from no load test, sc0 sect. ion 4.~.2. The power factor can be measmed by

two- or three-wattmeter mct.hods and ot.hcr qunntilics including supply voltage nnd

currC'nt iHC measured by con\·ent.ionalmet.hods (i.e. voltmeters and ammeters). The

loiHI teRt dat.a is recorded in Table •1.:3 and t.hc values of reactance calculated from

slcady-strd.e voltage equat.ion of the machine [:1:!] are shown in Table ·1.4.

4.2.4 Discussion of Test Results

The I \\'o points of d-nxis n•;H:tance nl. unsat urntion and sa.tnration conditions ha\·c

been del rrm i ned hy 11se of no-load test. open-rircu it and short-circuit tcst.s. Two

\'alucs rHr! shown in Tahl<' -1.1 <1nd Table -1.2. llowcvcr, since the open-circuit and

short-circuit I.Pst. measurcnwnts invol\'e the qn<1ntitics £0 and /.,c. which arc mea­

sured at different states of satmat.ion. thr. direct-axis reactance obtained from this

measmcmcnt must be considered as approximate value. While measurement of Xr1

hy no-loild 1 ('St is a good fl ppro:d ma I ion at light cmren t level because sa.tttra lion ef­

fects <tl'e mi 11 i mum. Tlw t'(•sult s oht.a i ned ft·om the~ loa.d lest a t'C ('Onsistcnt with ot. hcr

puhlisltcrl results whC're I he generated voltage is assumed constant at a le\'el of opcn­

circttit \ 'illllc. l'Jote that. tlwrl' is il lo<ld range where the \'alue of xd becomes llt~gati\'e.

Itmcil lls I hat. I he machinf' is lwha,·ing <ts I hough the d-axis armature rcact.ancc were

c<1pacit.ire with coltst.ant Eo hehind it.. :\!'~illi\'1~ \'rtlues of X., a.re entirely conscqttcnt.

upon the assumption of c:oust ant. ,-.;,>· llm\'('\'cr. t lw gcncrilted \'olt.agc is dependant

on the Sfll nration of t.hc steP) leakage bridges <lt\CI thus is a load dependent \'ariablc.

Therefore. the react.anc:e pararn(•ters raknlated hased on assumption of the constant

ge11er<llcd \'oltage at loadiug a11d shown in Table ·lA lllil)' rrsnlt in some error when

tts1•d In pr«'dict pNfonnatl!'l'. It is tii'C:I'ssary llo\\' to d('tcrmine how tlw r1ir gap ,·oltagc

chatt.t;Ps a11d ·,\'hat fndors itdlttClH'I' I his \'<trial ion.

-1·1

Table ·l.l: Op<~n-circuit and short-circuit test results

Eo /!C \1 • d

(V) (A) (0)

l!S.O ·1 .. 5 1.5.1

Table -1.2: ~o-loacl lest results

E'o \:' In/ \''l • d

(V) (V) ( ,.\ ) (0)

IIS.O 1!)'/.0 J.S 2.5.:3

where

.\~ .. l=Ai.,: snt.urntcd value

.\ .f = :hrfo; llllSRl11 I' (I 1 eel Yill IIC

3 "' "nd b0 = :!~"

·1.5

Table .t.:l: Load test

v p fa 8 (V) (W) (A) ( Elcc. De g.)

202 3.50 l.SO G.O :!02 .)00 2.10 6.0 :202 600 2..10 8.0 202 ilO 2.70 12.0 202 820 :too 1·1..1 20::! sno :). 25 16.0 202 1000 :1.50 18.0 202 1160 ·1.00 2:Lo 202 1:320 ·1.60 26.0 202 1-100 ·I.S.j 28.0 202 1!>:!5 :>.:l:) :H.O 202 1 6:)0 :1.S:3 :~6.0

202 liOO 6.15 :38.0 202 lS·IO i.OO ·12.0 202 1!)00 8.10 52.0

·I c;

Table ·lA: D-axis and Q-axis reactances

In lr{ lq xd Xq (A) ( :\) (A) (D) (D)

1.8 1.013 1..188 37.040 30.633 2.1 0.688 1.08·1 ·1.5.692 28.24.3 2..1 0.597 2.325 -18.291 25.565 •) ... -·' 0.3·12 2.678 G9.7-12 2·1.58!) 3.0 0.117 2.0!18 175.10-l 2:3.WC

:1.2.5 0.028 3.:Ei G-12.1-15 22.110 :3 .,j ··0.2DO :1 .-l·IS -.) l . 732 21.338 4.1 -0. iOO 3.D8!) -10.217 20.:145 -1.6 -1.0-1:3 ·L-180 -1.7·16 18.910 .J.s -1.:.W3 -l.672 1 .2:].j 18.603 5.3 - l.Si9 .).00!) 6.383 18.600 5.8 - 2 . }();j .j.-135 7.·183 1 7. ·154 6.2 -2.32:3 .).6!).5 S.oSG HL9i·l i.O -2.701 6.-1.18 10.6.')8 1.5.-150 8.1 :3.;')6:1 i .2i·l 1:1.8:3-1 lt.4!)6

-17

4.3 Search Coil Simulation

The iden. of this test. comes from the analog sirnulat ion of the air gap fields, see section

3.:3 and section 3.4, and its results, see Fig.3.9 and Fig.3.10. The simulation has shown

that there is a linear relationship between the discontinuity amplitude denoted by If,

nnd t.he amplitude of t.hc trapezoidal component denoted by Ei· This gives a means

for dct.ermining t.he \'ariation of t.hc gcncral.<•d voltage. The test procedure is as

follows. While the P.~l. machine is operating as a generator· driven at rated speed.

the air· gap flux wa\·eform is rnc<tslll'ed by use of the scMch coils and low-pass nltered

to rcmo\'c tooth ripple lunmonics. The outp11t of the filter is differentiated and result

is recorded in t.hc Fig.·1.3. The P.~l. motor is loaded by coupling a synchronous

machine, while the air gap flux of the P.~l. motor is measured from senrch coils and

is taken as a input of the test circuit. \Vith different loading, the stator current /11

Hncl di:::cont.in11ity amplitude If nrc record<'cl as shown in Table 4 .. ). FigAA shows

both res11lt.s of open circuit nnd lond test..

The generated voltage cnn now be determined from the search coil t.cst results. The

calculation assumes that El Eo = II I 110 which is verified from t.hc <tnalog simulation of

the air gap filed. Rc;;nlt.s which :-;how the \'nriationi.<!JHiency of the gcncrat!!d \'oltngc

wi I. h i ru: reasi ng lond are prC'S('Jl t eel i 11 Til ble .J.G and Fig.L5. In Fig.·L1. however. only

the t.wo \'<tlllt'S shown at point. :1 and point [j can be validated. The reason for l his

is that Ei obt.aincd from search roil test iwlurlcs t.wo components given by til<' field

equations (see last chapter). One is due to the field strength of the magnet sect ions

fJ1 and another due to annatttrf! reaction in din!ct-axis /3~,11 which arc repeated for

conw.'llicnc:e below:

!JT ( ,\111 I :\ ~ ) I + ;J 1\.t

·IS

( 1.1)

(·1.2)

{a)

(b)

------ r H

0

E0

=117. Sv

H0

=2.Sv

Figure 4.3: Result of simulation: (a) input of the differentiator, (b) output of the differentiator

49

8

6 / / ,.

4 ,,.

H (v) / •'

_r' ,. ' -

I

2 ' / ~,/

0 0 2 4 6 8

stator current (A)

Figure 4.4: Search coil simulation- open circuit test and load test

400

.300 _,./ , . ,-

/ ' '

Ei (v) 200 ;~~'·'

/ /

-~/ 100 ./ I

// I y' I

0 I r I;o 4 0 In',Z 6 8

stator current (A)

Figure 4.5: Determination of generated voltage

50

whf!re

( ·1.3)

But, no armature reaction occurs at either point A. which is the open-circuit value or

point B nt which It~ equals zero (ld:=/11 sin ( rfi- 8) = 0 \Vhen power factor¢ equals

torque angle 6). Siner these two points contain no direct-axis armature reaction, they

can be used to calibrate the lc:-;t. procedure. The value of generated voltage at point

/J cnn also bl? obt aincd from the phasnr diagram associated with the moment of zero

d-axis armature rr.action, sec Fig.·l.6. At this moment, ~=8=:38°, Ei = V cos¢ = 159.2\/ ( V = 20::!V). This value is included in Table ·1.6 for comparison. ft is clear

that t.hc values obtained from search coil simulation and from phasor diagram have a

good agreement. The value of saturated Ei c\'aluated from this point is more close to

th11t at short circuit than £0 at open circuit. Consequently, it is used to recomputed

saturated ralue of Xd as: Xr~ = Ed/3* I~c = 20.5 (f2). This value will be used to

compare with that from othrr methods later.

v

E=V cos¢

Figure 4.6: Phasor diagram ( 6 == 6 )

Up to now. two \'a lues of generated voltnge at d)fferent load conditions have been

.51

l '

cntlunted . One is the open circuit \'alue and t.h(' another concsponds to a condition

of zero d-axis current. The two conditions h;wc different values corresponding t.o

different loads. The reason for the variation of generated voltage under load is given

below. l_! pon <'xamination of the phasor diagram and t.he flux linkage paths of the

rotor ttndcr diffc~rcnt load conditions. sec Fig.·!. i - FigA .8, it is ohscr\'l'!d that tlu!

opplil'd \'olt age n·sttll.s in <t map;nct izing CIIITI'nt /,f at. light load, which will increase

silturation of the lll<lin flux pat.hs and reduce saturation of the rotor st.ed bridgl'!s.

:\ positi\'l• !.,. tiH~n ·forr, will n•sttlt in a dern·ase of generated \'oltagc Ei. On the

other hand. nt hea\'y lo<td, the armotmc reaction becomes nlilgnrtizing with respect

to the magnc•ts ctnd dPtlli\gtwtizing with rcsrH•ct to the stc•el bridges which results in

illl increase of gencratc•d voltilgc Ei. In Fig. I.~. <f>.t is referred t.o the flux due t.o the

<ll'llltl I It I'C l"f'il Ct ion, '~m I 0 !.he fl IIX d IIC! to l he rn<~gnct.s and <f>t is t.hc lea kagc flux of the

siPPI hrid~cs. lnflur.nce of the armature r<'ctc·tion will he studied in the next section.

Tilblc ·1.5: s(~arch coil si111ulation - load test

/,, II ( :\) (V)

l.'i O.'i . ) . ) l.ti -·-•)--·1 ~.1

:L2 :u 1.2 ·1.0 1.8 ·1.8 .) .. ) .i.G (' . ) l,_ n .. t

-•) .. ,_

v E 0

•)

v

----------~------•d-axis

b) c)

Figure 4. i: Stator current at different load condition a) light loading: magnetizing ld, b) 6=¢J: IJ=O and lq=la, c) heavy loading: demagnetizing /d

•)

b)

c)

Figure 4 .8: A rma turc reaction a) magnetizing I d, b) f.r:;;O, c} demagnetizing I r1

5·1

Table 4.6: Determination of generated voltage frorn search coil simulation

!'I fl £, E = Vcos<b ( r\) (V) (V) (V)

1.7 0.7 32.9 2.:2 1.6 -- ') I•J .~

') -- · I 2.4 112.8 3.2 3.4 1.59.8 159.2 4.2 4.0 188.0 4.8 4.8 22.5.6 5.5 .5.6 263.2 6.2 6A 300.8

where E E II

= 'o Ho Eo = II 7 .. )\-~'

llo = :UW

55

4.4 Influence of Armature Reaction on Gener-

ated Voltage

The influence of armature reaction on the generated voltage under loading is invcsti­

~ilted on thL' f;perific rotor structure which was dcsigued hy T.A . Little [2]. Recalling

the I hn•c flux density field equations as follows:

131 Rr ( :Ln I .- \.~ )

:: I + :'1 1\.J

H.·l)

0.-IJioiT Fdm p ·) I B-zr1 = [ ('().<; - 0- - ILn rr J

.'1 2 l + d [\'J ( .t..1)

A.,q O.·lftu ;r F7 m . P fJ

:: .~Ill-

!J 2 (·t.G)

The totfll air gap field is giv('ll as:

( -t. 7)

The abn\·c• equal ion shows that there <ll'C t.wo components affected by the sal.ltra­

tiort fcH·tor. one heing !31 cansf'd hy magnet. sections. the other heing 13:a= 0·8:t~;~-: 19

nllt:.;cd by armatmc n·aclion. Both of tlwsc components fombinc to rrodurc t.h<'

\'oltng'' H. that. is lll!'HSill'l'd hy thf' :warch roil and has lweu rrcorclcd in Table ·l.fi.

Tht' influence of the lt•ak<tgc factor,,. = 1 +~IKt in IJ .• .~ \\' illlw invc·sligated as follows

lw~cd ou tlw rotor st.ntct.mc wit.h di1111 ~nsior!s. paramet('rs and flux paths are givl'n in

:\pp(•ndix f f I ilud tr~ken from rdN<'nr.e [4 :\~sumr that the lrakagc factor is.

J - J.[l ,l - H,.,

I . - I . !.bJ.. \J - ,. llt

( ·1.8)

,j(j

Calculating reluctance values in \'arious flux path~. gi\'es:

Ra :::

:::

:::

:::;

R= :::;

:::;

:::

where

=

Rm'1

Thus.

.'1

Jl l\.7

9 Jtl .• rrD

o.oo:3a x ~

1.0 X 7.()2 X :l,l-1 X 6.80

0.000-1{):! .5~12 amp- trml.~/wr:ba

I _1_+ _,_ H..,1 !1,.2

I i-Ll + :!i .. j

(J.OO!l amp -lr11·nsj !l'clwr

0.5.)!)

1.011 X 2 X 7.()2 X O.!l!)()

0. o:Hi:ll 8 0111 Jl - f. ILl' II .~ I I L' t; bcr

L'2 1/2'.!./&h'}.

0 .. 5.5!) 1.01·1 X 2 X i.o2 X 2.{)8

O.Ol:H!l2 amp- lums/weba

;1 2 R:l I R-n

- ().Qf)

,,., I + Rm a,

!LOO!l - t+M

(·I.!))

(·1.10)

(·1.11)

{·1.12)

(·Ll3)

U, is n·luctance of leakage path ;md a key parameter which affects B~d· Since all

the IPakagc paths for the given rotor configura I ion itrc in parallel (Appendix I l 1). Rt

can he written as:

where

Rt = _,_ + ..l.. + _, + _,_ Hn Hr2 H,,, Hr4

Rll

1?12

1?/3

R1.t

1 == ,, + jl + /1 + ,,

1:.!.';" T.:j ~1.!1 :1.1

l .:J.l = -- a11111- ftn'n8/u·cfwr Jl

L, =

:! /3 {,I Jl I .,.I

= !') --·' I ' -- (/rtl/1- IIIT'I/8 U'C 1('1'

I'

'·2 :! 1Jr.21' ~ T~ ·1.·1

= - ((1/lfl- 1111"118/!t.'dJt:l'

I' L'J

= "2 !.:, JI:J ·r; :) . !)

:::; - awp-lunt.'i/ll'cbcr I'

[, ,, -----2 /, ,,,., T1

'1 --·-' a 111 p - lt/7'118 /weber

fl

(.t.l5)

(·I. L())

(-t.li)

( ·l.l s)

( 4.19)

To simplify t.hc problem nrrr<"nt.ly st.udi<•d. I he (><'l'llH'ability of all the l<~<tkagc paths

arc assumed l o he the sa me. a It hough t 1\('y an· not eq11 al in pra ct. ice. The permcabi li t.y

oft he st.ccl hridges JL is a function of tlw lit•ld sl n·ngt.lt 1/. This varies wit.h k:·.ding

condition and shifts the opcrctting point. along the /J II rharac:tcristic shown in Fig.·UJ.

In \'<'SI igation is based on the \ 'ill'iil t ion of liPid sl rcngt.h <tt different loading cond i lions.

Tlw total rlux clensi ty on the milgnel facl'( <HW of t he magnet sections) is described by

13m= Br -1/ If+ 1/ ll'. The ilrmai.1Jrr> rr•a<"tion c·onlponent 1t' II ' crtn hl' magnetizing

or demngnetizing depending 011 tlw sign of !<{,. which is dd.crmined by t.he sign of I.J

gh·c~n in Eq..t.:l. :\t the open cirrnit point whrrP f.t C'cpwls zero. 13111= Br -1/ II. When

loading the machine-, the OJWrat.ing point will 1110\'e up to when• t.he power fac.tor 0

equnls torque angle 8, and after that. ld will change sign and be demagnetizing. tints

the operating point will mo\'c down. Therefore. the permeability Jl is higher at open

circuit condition and lower under load condition. The leakage foetor is:

1\ =

= I t (J.O!J ( l t ~)

l 1.on + n.onoG 1t

:\~suming ,,=.:;ono ill unsaturat.ion and 1'=10 ilt. sillttration. Tints.

ol ... at ural ion

!\" = 0.11.} o I 111180 t u ration

(·1.~0)

(·l.:!i)

It is clear thilt the leakage factor consists of both magnet and armature reaction

('omponents find has signific<wtlr different \'illucs at diffctcnt load conditions. The

contribution of the armature n•action compont~nt to the generated voltage is difficult

to c;dculatc been usc of the nonlitwa ri ty oft lte magnetic materials. The dctcrrni nation

of D,',.1 will not. he considered at thi~ moment.. From consideration of lcabgc factor

l\ .. a general statement can he drawn as follows: At. light loading, B:d has a negative

sign which makes the measured roltagedt'creasr. below the actual generated \'oltagc.

As the load is inn·eascd, B:,1 will change sign. causing in the measmed voltage to

increase t~hove t.he actual generated voltt~ge. Fig.·l.l 0 sho\\'s the effecls of removing

l he ill'tna.t.u re react ion com poltf~nt to ohl il i 11 t h1~ ;p ~ • .•. 1 magnet gmerat cd \'ol tagc E0 •

In order to simply calculations two \'illnr.s of Eo \\' tl ! be chosen t.o approximate the

\'ariation of the gr.ncratcd voltage (see Fig.·!. I 0). Usillg these two ralucs the reactance

fl<HC\111et.crs can now he compttlr•d as shown in Tahle ·l.1 and Fig..!.ll. H. is clearly

shown that. in Fig.·l.ll t.hcre is an in trrmcd in 1.<> range of loads where d-axis reactance

lwcontes difficult to dct.errninc. This is in tltr range where the armature reaction

.)!)

c- : ' . .,·) •. '

:£ ~0 ~:..'i~E'H ;;:£~:0T 'lG ?CXI . ...._

CPE'~A~ ~G ;;·~ ·~T N• r~ ~~'.IA7 . R~ q£.\Cr,CN

H-I

H

, d I

,-_a,

: I I \. ~ · H

I J ---1-a

. L'lo\0 L . .. E

Figure ·l.!J: Operating point at different load

200

0 0

------(b)

-----L-------L------L---,.-, L 4 l

.., •'

Figure 4.10: Variation of the generated voltage(a):measured E,- including the ar­mature reaction component (b ):t'stimate<l actual Eo- separated from the armature reac:t.ion and considered leakage factor

60

in the d-axis is changing from a condition of "nli·lgnclizing!, the magnets to one of

demagnetizing them. The value of x.~ derived from the steady-state voltage equation

of the machine is extremely scnsiti\'e to the accuracy of the experimental measure­

ments. The negative values of Xr1 do not appear in any range of loads as compared

with Table •l..t in which the d-axis reactance is determined using the assumption of

constant E0 • This is consistent with the sl.alcrncnt that negative v:llucs for Xd are

cntirf'ly conscqtwnt upon the assumption of constant Eo and do not imply anything

impossible ahout. the machine.

fil

t • t • L'" • • •.• t

-·

a)

b)

·}(H'i ,--- --·--- ---·,--------,--- ·- · - --1

1~0 I I

I .:'_'If ) // illl /

I .... " ~ \

l:'tf ,I ~~- - · • • ' \

., --- -·- __:~_ 1

n L __ - ··-- ------L --· ----------L- ___________ J -4

ld (.A.}

1.) '.' ·--

•. ·lt" J· T - - - . . - ...--- - -- - ----.--- ----.,.--·---

11') 0

' ..

--~ ·-----.,._ -·-- -.. j -------1-----------L---- ... L ____ --

4 [ :. Q • . 1

Figure •Lll: Reactance parameters a) d-axis and b) q-axis

•- modified conventional tests

('? >-

Table 4.7: D-axis and Q-axis reactances

fa iJ [1 xd x7 (A) (A) (A) (0) (0)

1.80 1.01:3 1..188 37.040 30.633 2.10 0.688 1.984 ·l.j.692 28.243 2.40 0.597 2.325 48.201 25.565 2.70 o.:H2 2.678 69.7·12 2·L589 :LOO 0.117 2.998 l7.j ,l0·1 23.160 3 .. 50 -0.29 :3..188 :w:too7 21.338 4.0.5 ·0.7 :3.989 120.069 20.345 4.60 -1.0·1:3 ·1.48 85.692 18.910 ·1.85 -1.:30:~ ·1.672 71.172 18.603 5.35 -l.SiD 5.009 5·1.909 18.600 .5.8.5 -2.16.5 5..135 ·19.5!.12 17.454 6.15 -2.:J2:3 .s.o9., 4 i.!J:l..J. l0.971l i.OO -2.701 6..1.58 ·1-1..103 15..1.50 8.10 .;j}}6:l 7.2·17 :19..! 16 14.496

()3

Chapter 5

DETERMIN-ATION OF P.M.

SYNCHRONOUS MOTOR

PARAMETERS- PAI~1T TWO

5.1 Introduction

The modified conventional test and the search coil simulation p1·esented in the last

chapter were used to locate two specific values of d-a.xis reactance and to show the

\'n riat.ion of gcnerat.cd \'Oitagc under load conditions. In this chapter, additional test

methods, which employ the flux linkage test and the search coi! test to determine

r. M. synchronous motor para meters cll'C prcs<'nlcd. The tests are performed on the

same mutnr used in the tests described in the last chapter.

5.2 Flux Linkage Test

The flux linkage ted, which is based on the theory of opera.t.ion introduced by

C. V .. Jones [:2D] can he applied l.o the measurement of the inductance of the P.M. mo­

tor: This was first proposer! by Mil1cr [19] and later mcd by Little [2). The mcl,hod

64

is 11~ed for measuring static inductance without. lu•ing affected by the presence of the

magnet flux or hy induced current in t.hc starting cage. The mP.asuremcnt circuit

showr. in Fig}). I satisfies these rcC]uircments. Strictly speaking. t.hc tc>st rncasurc•s the

flux linkage of the> winding rather than the inductance it.sclf. This bridge circuit is

composed of t.wo non-inductive resistors (R.1 and H.1) r~nd one \'ariable non-inductive

wsi::;l or ( R·1 ) usr·d t.o br~lance the bridge circuit .. Also one de power supply ( \-') ctnd

one· on-off switch (S') arc included in this circuit. The /?.1 and I, rc•prf'st>nl th'-' self­

inductance <lnd resistance of t.lw machine wiuding. The \'oltagc: across the hridge is

ohsc·n·f'd and IIH'ilS11red hy usc of au oscillosf'"'"''·

5.2.1 Theory of Operation

\\'hen the switch of Fig .. j.J is closed \\'ith tlw rotor and stator aligned along the

din•ct-axis (or quadratmc-axis). a const.Hrll current I establishes a field in the stator

\\'inding. :\ft.er balancing the bridge (i.e•. \.-;,~ = 0) and then opening the switch the

cmrcnt I t.hrou~h the inductor will fall t•xporH•nt ii'llly to 1.ero. Assuming 1',1 ~ is the

instantarwons \'olti'lge across the hriclgc during this t ransie11t period. Thus.

f! n~--­

. 1{, + £7, I'

= H2[ Ht + /?2

and

Since the bridge is balanced,

Tints,

(i .i

( .j. l)

( - •J) :) ..

(5.3)

(.5..t)

(.5.5)

s cb R4 Rl L

v vab d- connection

R3 R2 C) q- connection

,P R3+R4 ~,

Rl =2. 711 R3

=2oe n L= - = 1. 087 I R4 I

R2 = 2 s ""4o n R4=2.4K X=Ct.IL= 409.67 t

Figure 5.1: Flux linkage test circuit

(i6

Tlw nux linki'l~(' ran he obta.in<'d by int<·grating this transil~llt \'Oltage:

( .),())

or

L 1;'· R1 + H2 I H. ,~, /?:} + /?..1

(5. i)

Therefore. integrating the voltage change across the bridge gives an indication of

t.he ind 11 dance <~long the axis of the align men I.

5.2.2 D- and Q - Axes Connection

Once t.hc bridge circuit is connected to the power supply, a de field is established

in the stator winding. For the machine used in this work. t.hc rotor has four poles.

Thus. the mechanical angle between d-axis and <1-axis is ·15 degrees. The d-axis of t.he

rotor will align with the axis of the phase of stator winding automatically when the

power supply is connected to the bridge circuit. The q-axis can be found by rotating

the rotor through 45 mechanical degrees from the d-axis position. The results of

measurement are shown in Tables 5.1- 5.2 .

5.2.3 Discussion

The !.est results, including modified con\'entional tests and nux linkage measurement

are shown in Fig .. 5.2. This figure shows a reasonable correlation for the d-axis re­

actance. Since this test is a static measurement, it provides a means for separating

()7

gcncnttNI \·oltagc E, and d-nxis rcartnnre .\".1 in whid1 the diffic11ltics of measuring

.\.1 is a\'oided. Therefore, t.he r<'sult.s from 1 hi:; lf'st is more nccuratc. Secondly. it

prr·sc•ut s the \'it Itt<'~ of xrl at entirP. loads .. '.nge. llowew~r, the thermally induced

resistance \'ariation in the bridge• compoHc·nts lewis to a rliiTicuiLy in maintaining a

halancl'd condition. This factor pn·,·<·nts this tc•rhnique from IH'ing used for high cur·

rent \'altt•·s. Tlu·rdorc•. n•sist ancc•s with lm\'1'1' t l'lli!H'I'ature SC'nsiti\'it.y are sttggc•stcd .

Xq(Ql

89.99 r------------

79.99 j 69.9~

I

50.99 J

\ \

\

\

a) 49,Q9 '""· ..__ ---

b)

39.90

29~09

--------------

10.09 ' ' . ' ' ' 2. so 0. 59 1. 99 1. 59 2 I 00 2 I sa 3. oo 3. 59 4. Q9 4~ sa

Iq (A) Xd(Q)

59.9Q -r--------------.

49.09.

39.99.

9~69 +----,------r---r---~-~ -5.9Q -3.99 -1.99 1.09 3~99 5.99

ld (A)

Figure 5.2: Results of flux linkage test: a) q-axis b) d-axis: symbol-modified conven­

tional test results

69

I !., r , ... ~ ;·~d.) l (~·2,) ( .\) ( \.)

-:-). 0 lll.ll I I. I ~ •)

\ ·-- 1..-) !) .. j II. II! l.j K.(i

-·l.ll !J.O 0.11!) !) . :!

-:L.i ·".:) n. n~.-, !l.!);j

-:Lo K.O ll. OS I II .! I ·2 .:) i.G ll .Oifi I:!.:, -2 .0 fi.i O.IHii 1 :L i -I ,;j {i.O O.llli I IU -I . () . I. fi f).fl .lfi 1:-\.~

.() .. j :.!.·1 f). f):,!.t I!). i 0.:} •) --·' o.on 2:.?.1 1.0 6.0 0. ()(i :~.I. (i

1.:} I 0.0 0.1 21 .:1 2.0 J:LO 0.1 :l :!G .fi 2.!) l.i .O 0.1 ;, '2· l.ii :ul 18.0 0.18 '2 1.()

:L.i :,!().f) 11.:! :!:J..I .t,() 2:!.0 0. :?:! •)•) --- ··'

ill

Table• :l.2: H<'Sillts of flux linkilgc l<·st: q-;1xis r<'iH'tilnn·

1., I P.,b I'·' X., 1

( :\) (V) I (whJ on () /) 8.0 O.OK (1.) .:i

1.0 I ~.0 0. I :! I! I.:! 1,.) I ·I. I 0.1 II :J! I. ;J I ~.0 :!0.0 0.:! 10.!1 • ) -_, ; ) :.!:.!.0 0. :!:! :Hi .0 :Ln :! ·1.0 0.'21 :::! .0 :L.1 :!!J.O 0.'2!1 :J:L!l

-l.ll :HJ.O 0.:~ :~11.7

il

5.3 Search Coil Test

J.ipo [:1:q aut! Plunkt'lt [:1·1] ha\'f' pn'SPiltf'd t IH' use of s<'arch coils in AC induction

IIJntors for dri\'1" application. This IllC'thod catJ iJI:..;o he used to measure the n.ir gap

ll11x distrihntiou of tiH' P.~l. motor[~:~]. In this ~wrtion. d- and q-axis n•actance

nwa-;1\t'l'lll<'fll l)y n~w of llnx sc•m:ing coi ls is dt·:·wrilwd.

5.3.1 Theory of Operation

Tlw ft>sf tlll'' hod is ha ... ,.d 011 tlw I \\'l)·axi.-. t hc•ory in \\'b!'h I l11• I illll'·\·arying parant~•lt•rs

arP c•limin<llt>d. Til<' \'ilri;,hlc•s and pariltll<'fns an• c·xprc•ssc•d iu tc·nns of dirc·rt· ancl

qtladratun··axis quantitit's [:J!"l-:Hij. Tilt' d-q clyna111k llll)dc•l of n machine can be

<'XIH't•sst'd i11 Pitll!·r il ~lilliollar,r or a rolilti11g rc•fc•wfl('e frarne. This sf.udy li.S('S the

st nt ionary l'<'f<'l'l'lln' frame·. :\crording tot lw <IX<'s transformation, t ht• phase \'oltagc•s

in INIIl!" of d- and q· \'oltagc•s ran lw writtc•n in lllillrix form as:

l',r

1'1,

t' c

=

('(}.~(} .~i nO ,.,,

co.~(()- 1 ~0°) .~i 11( 0- I :20°) I',J ( ~.s)

l'o.~(O + I :l0'1) .... /11(0 + I :!0°) l'o

Thl' ot.lwr quantili(~S, snrh ns CIIITC'III. anrlllux can he twnsfnrmed 111 a similar

tnanner. The COIT<"sponding inw•rse r<"lntion is gi\'1'11 by:

,.,, ('().~(} C'o.-.( 0 - I ~0°) co.o;( 0 + l :!0°) t'n

2 .~i11(D- I:W0 ) (.i,!)) t'd =- ·" i 11 () 8i11(0 + 120°) l'b :J

''o O.G O.:"J 0.0 ".: \\'here t·,1 is the zero-sequence component.

for hal<tnced three-phase condition, the zcro-scqucHCC component does not exit..

The angle 0 bet.wccn the two sel.s of axes is arbitrary . It is conw~nient to set 0 = 0, so that the q-axis is cnincidcnt with the n-axis. Also ignoring the zero-sequence

comporwnl. I he I rnnsfommt.ion relations ran I)(' sirnplilied as:

')

r·.1 = ~ [ r·,,ro.~O + l'bCO·"( 0 - I ~0°) + r· .. co.-•( 0 + I :.!0°)]

·> I = : [ u + -1' l :J <1 :! <1

I' 'l ru o)

( .i.ll )

Tlwrdon'. I he implc·nwntat.ion of d- and q- \'ollag<•s CCIII lw rcalit:<•cl l,y wii ng tile

I hn·<··phas<' \'olt.ag<' r·'l· r·~ ;wd ''c·· The hlock dia~ram oft hP nwasurl'tll<'lll realiZ<Ition

is ~hnwn in Fig .. i.:l. Til<' til['('<' pha~e ,·nllagf's r·.,. r·~ and r·c arc air gap ,·oltages. V,

r<•pn•sf•nts an intPgrat.or I hrottgh which t lw \'oltagc• quant ily is t ransfonPed i ·1to the

finx rpt<lntity. D-axis quantity can lw t•,·aluat(•d by suhlract.ing the qrtantities of b

phas<' from r phase onrl multiplying C1 coelflci(•tJI. Tlws<' Cilll he impiPlllCtlll'd from a

suht ract ivc and a proportional circuits. hot h of which are np-ampliflN bilse cil'cuits.

5.3.2 Installation of Flux Coils

The motor u~cd in this study has :36 stator teeth and the flux coils arc arranged

concentrically Rround stator tooth as shown in Fig.5.-l. "A'' represents the distributed

11111 i 11 mot.ot' winding for phase a, ., a., rept·cscn t s the sPnsi ng coi I fot• phase r1. In this

cose. the center line of the t.wo coils is nligncd. Similnrly. sensing coils for phase band

pltnsc c can he located. Each coil hns nn equal munber of t.ut·ns <111d one t.um ill this

study.

Fig .. ) .. j shows a riux coil layout for the machine with ~36 stator teeth and ·l-pole

rotor. The armature winding pitch is arranged such t.hat the flux axes of the three

stiltor phase windings are aligned along stator tcc~th. Rclath·cly thin wire is used

y s

y s

'i s

A.b-

i\c

1

13

Figure ,).3: Implementation of the measurement realization

Figure 5.4: Search coil arrangement

74

i\q

for I he s!'arch coil in this cast'. yrt with sufficient st.rrn~t.h to amid brPak~gt~ . \\'irt'

:;i7.t! approximately :30 :\ \VG is t llf' cltoire for this work. The d- ttnd q-axt•s 11111s!

be located first. The nwgnetic axis of t.h<' d-axis rn11 he locat.<·d by rn<'rgizing tlw

phas<-' a winding with a :mud] direct current. By connting forward in the dir<'rl ion

of rotation by S/'2P loot.h pitches the q-nxis can bt' located. Once cl- and q- axPs

iii'C ]orated. coils ran be irnwrtPd into tht• 111olor according to Fig.!'J..l. In this figun•.

tlw \'oltage /',11 I'J, and 1'<. <\I'C' air gap \·olli•gc·s lllf'asurcd hy t hn·~~ scnsin~ roils and

an· t rn.nsfornwd iuto r!·~· r,·,) clll<l 1.',· hy intq.!;rHI ing tlwm. n·pn·sc·nl ing by I /S. Thr•Y

mlqJ;ralor ro<'fficif'llls aw rc'pl'f'Sf'lll<'d hy /\·., l\'~j and /\.b n •stwct iwly. Tlw rwp;ilti\'f• '·".·

ro11w fmm t,\ hy trsc• of a irn·<•rsc circuit c111d is nrldf•cl with 1.\, to oht <lin d-axis llrrx.

Since t.hf' q·<~xis is <~ligncd with phasc• o, q-<~xis f111x i:; ohtairll'd from t;·,, dirc>ctly. In

onler to rnaint11i11 accmacy in the flux nwasurrnwtH. high quality. low"r noi~c and

lo\\'Pr drift. arnplifi<·rs «rc used t.o form the intf'grator. By choosi11g R1 = R, = IOH1.

(' :::: '2.'2p. wlwrc R1 and Care the fC'Pdhack resistor and capacitor rrspcctivdy and

R, is the input resistor:

I \-~~ I l ' . I

..!...

=

:\ '-'

JP+(.:~F :\ t•

s . :~5 1

s.:J.s (5.12)

where Av = !f/; = 1, (.l.'o = n;c = ·l5.45 rad. and r,·'l = l\.b = l\.c for the three phases.

,\ppendix IV sh.'ws the waveforms obtained from the search coils.

5.3.3 Measurement of Current Components

The d-q transformation for the current is similar to t.hat shown earlier for the voltage

equations and is given by:

15

Direction of Rotation

t E~

1Vq=-1J>a va A-axis (

... a-axis 1

~ 2 01. 1 I 3f

[ '*'

I 5

6 'I'd Vb Kb/5

l B-axis

[ 13 7

c 8 ~d =-Js (tf._-l~Jc) 9

D-axis [ 10 I -1

[ 11 1 I 12

Vc Kc/S C-aix

I 13

[ ]

Figure 5.5: Flux coils arrangement using one coil per axis

76

lq CO,<;,/) cos( 0- I :20°) co..-(0 + 120°) I'' 2

sin(O- 120°) .~i11(fJ + 120°) I ,f =- .~ i 1l 0 lb :3 ( 5.13)

'o 0.5 o .. 5 0.5 lc

SimplifiPd as:

lq - '" l ( . i b) lr{ - .;1 1.: -:J

(5.1 ·1)

Arrangcmcllt of current measurement is shown in Fig.5.6. Currents in. h and ic

can be measmed by usc of currcnllrausformer. Cunenl transformer model LE~I SA

Cfl-1228 is used in this task. The summation circuit and integt·ator used in the current.

measmcment is the same as those used for the voltage measurement. Waveforms of

current. components measured arc nlso included in Appendix /\:'.

5.3.4 Calculation of Reactance

Since the fiux \\'aYcforrn obt.aincd from the search coils i:; nonsinusoidal , Fourier series

analysis is used to calculate the fundamental component of the flux. The calcrdation

consists of sc;·cral steps as follows [:n].

• to determine the coefficients of sine and co.;ine terms of the relevant Fourier

series by means of approximate methods,

• to determine the value of t.he fiux fundamental component,

• to determine ld, lq and calculate Xd and Xq.

The general Fourier series expression is given by:

f(x) = l -ao + 2

a 11 co.s n.1' + bn .sin n.t

77

(5.15)

I a • I =I q a

Ib 1

.J3 Id

I =! (I - I b) 1

-I d./3 = --(I · ·I ) I c c J3 b c

c -1

Figure 5.6: Measurement of current components

78

ao ~ (:: .f( .r) d.1· " ./u

1/, = ~ (·~ .f( .l')('o.~IP.l'd.r ,, ./o I '1..~

- { j (.I') .~iII 1P .1' tf.l' 7r ./o

(!).!(\)

It i~ known thilt approxinlillc· intq!,riltion. \\'hiclt is ('qlli\'illt•nt. to !indin~ an an•a.

C'illl IH• Jll'l'fr,mwd by t li<' appliraf ion of t lw I r<qll'zoid;tl rnl1•. fn this study. t\\'1'1\'1'-

point analysis is usr•<l. An {'Xampl!' of d1•tc•rminin~ rodficicnls (concsponding t.o tlw

('ilSI' wlwn /.1 = li.!) : \) is pn'SI'I!Icd hPio\\' . Tlw \\' il\'<'l'or rn ohtaitwd frnm t.h1~ sPm;ing

coils is ~hown iu Fig.:).i. \';dnc•s of I h~· orclinat~· for niH' rycl<• of a periodic: \\'il\'c•fonn

of p<·riCJrl :2:;;- an• givPII i11 AppC'!ldix /\-'. It l'illl IH' ohsPn·c•d I hilt for the llnx wa\'f•form

sho\\'11 in Fig . .i .i. f(.r) = .f(.l' +;;),also f(.r) = -J(.r + r.), i.l' .. l.h<~ funct.ion cont:tins

odd harmonic:-, only. Thus Pomic~1· roefrici1·nl.s an• calndat.<~d as follows.

fit 2 x mwn ro/11 .. ~ ()f f(.r) ro.~ .r Olll'l' o lHTiotl \ II

- 2:: !Jr ro.~ .l'r G r:::O

l _ 2 x - .~"'" or 12 .

l.!Hl!l

I:.! onli11alf'

(!i.li)

Finding nil the cocffidcnts hy t.he proccdmc ~i\'1'11 111 equntiou (.1 . \i) giv<•s t he

fol l o\\'ill .~ res11lt s:

r1 1 = UJ990

h:s = I 8.80i

fl;, = -0.115 b.; = - T.:ll t

i'D

Figure 5.7: Flux waveform of la=6.5 A, chl: tPd ch2: t/Jq

so

TIH• fttrlflillll<'llta! romporu•nt. of tht• llnx ran IH' ralnrlakrl aftr•r tlw FomiN md­

firit•Jtts <tn' dC'Ir·rmiJw(l. It gi\'l•s,

I '· . '/ :.! I.~ llllll'h X 8.:3.5

ll . l ii ll'h

1.'./ f'itll lw r·akulall'd tht• in Silltl<' \\'il~· :

r.'•.J :.! I. i lltll'h x i'\.:i:i

= 0.1.'\1 ll'b

(;).1 ~)

(!). I!))

Sinn• nttTr'tt1. are siuwmidal t'otnpollt'llls. its rms ,·nhw cnn lw dc·tnmincd si111ply.

X,, <111d X.1 are gi \'<'TI from X = w /,=....,• r;• /I, f'il kula tion of r,'•,t and r!·,1 for I he load t f'SI. is

:;!town in :\p]H~tH.lix IV. Hc•cording t.lu• rim: Hnd the c:urn•ttl W111]H>IH'Ill.s ront•sporHiing

to c•<~dt stc>p loild, t.hc d-axis and q-axis n•acfatu·t~ ran he ohtaitli'd and rr•s11 lts arc·

~hown in Table 5.3 illld Fig . .l.8. The tnorlilit·d cort\'r•nt icmaltc•st rc•s1Jlt. is ;dso indudc·d

to pn•sr•Jil. a rotnpari:-;ort. Only a n·;tsonablt• ap;n-euwnt bct.\\'c'r'll 11H•sc two nwthorls

can lw oiJS<'t'\'('d. The rt•a:-;on for this is dtll' to till' dilrindty of sq><trat.ing the map;111'!.

and sfnlor winding flux in\·ol\'l•cl in 1111' nH'ilStii'C'IIlt'nf of tlw sr·nrrh roil tc~sl. :\11 ihc•

tc·st nwt hods and results will IH' rliscttssl'd in t Itt• follo\\'ittp; sr•r·t ion.

~I

·---·-.--- ,---... ---r-·--·,.- ·-- r· ~-- ---,.-- - # \

40 -

.35 .

:~ 5

~0 .

slot or cur r:~nt (II.)

Figure 5.8: Results of search coil test: line 1- Xd line 2-Xq, symbol-modified conven­tional test

82

Table ;),:1: lksulls of st•<trrh coil l<•st

/,, l:'ri 1.'., -'"·1 .\',, ( :\) (wh) (wh) ( rn 01)

I. i:l 0.1 i~!) O.HixO ·I :!.!lG :J().;j!J

:UtJ O.Hiio 0.1 j'!)() '2:J.j,wl ~:L IG

:L:l 0.1 jj'~ 0.1~0!1 '21.!1~ '20 . (i(i

·t.:l 0.1 G 17 0.1 j(J~ !:u;:~ 1 - 1.!1~

5.8 0.1 ·1·10 0.1 j'.):J 11.8!) 11.:10

G .. ) 0.1812 0.1171 I fl.~~ I 0.27

5.4 Discussion of Tests Results

Scn'rill 1<'!'1. methods for drt <~rmi 11 ing t he pa r·n lllc•lt•J's of a P. :\I . ~ync h ronotts mot or

have been presented and discussed in the pl'c'CC'cl i ng c ha pt.cr i\ nd t he preceding sect ions

of I his chCipl<'r. Emphasis t.lwrc has lwf'll plnccd 11pon obt ai ning the t.hi'PI' most

significnnt JHtranwtcrs of the P.:\1. machitl<'. nanH•Iy I hf' din·c·t and q11adrat 111'1' axis

('('ilC'IilllCCS X,[ cllld x.l as \\"('II th(' inll'l'llill gc'tH'riltecl \'oltap,(' /:', . :\ll'<l~lll'f'l l ll' t ll of

tlwsc paranwter:; is dirr.cult sine(' tlw fiPid c•xcitatioll cltte to tlw pPrlll<liH'IIt mau;nds

cannot I)(' n.•morc·d and lwrniiS<' the dfc•cts of tlw gc•nc•raiC'rl \'oltilg;c and t lw din~ct

axis rc·actanr<' nre alway:; colllhillf'd along tIll' salllf' axis and l'annot lw !'-<'Pill'iltc'd.

:\lorC'o\'er. Pach of these paranwtc•rs <tn' shown to IH' load dc•pc•rHII'nt. illlrl not constant

as is often assumed. The following sc·ct ion is prP:-:cuted to cli~nts!-i I he \'Mious lc•st

rc•srdts nnd to show the rdat.in• nwrit of c•ach of the• test. nwthods prc•sc·ntC'd.

:\s prc\'ionsly men tionC'cl I he effects of .\".1 nnd F:, arc combined and I. he len· I of

r:;, is not independently controllable. In ordc•r to ~{·pnrate the <'ffect~ of thc·~C' two

qnantitics thf' machine must lw opemtc•d ll!td«•r rondit ions for whirh E; or tlw l'[fc·rt

oft he reactance .\"-I goes to Z<'l'O . .Just swlt a condilioll exist:-: when I he clin~ct-axb

ClltT<'nt is zero. One obvious opt•rating point wiH'r<' this occnrs is \\'lwu t lu.• lllilrhine

is IH.'ing dri\'f'Jl as au open circ11it. gcn<Talor. Thr• ot hc•r is \\'lwn t lrc• load is StiCh that

the torque angle is equal tot lw pO\\'<•r factor anglr•. In Pach of tlll .'~e ca~r·s t lw nduc of

Ei is c•asily mcasmed with a n>ltmet.<•r. Tr~hlt• .) . I sho\\'s thc• rrsnlts of thesf' l f's ts and

also cl<·arly shows the signi~rant incrf'ase in f, from no load to 1'11llload . .-\ lso sho\\'n

in the table is the results ohtnin('d nsing tlw ~<'arch mil sim11lation. The method was

calihratC'd ill ncnr fulllond anrl shows good Hgr<•rnw111 with t h,, on I ~· other nu•rts tlrahlt•

lrst. condition.

:\ot only the generated \'oltagr•. h11t tlu~ ilXis reilrlttncrs ili'C t1bo load rlt'Jlf'IHIC'lll

qnantities which exhibit salmatiott clrarnctrrist.irs. The inahilit.y to \'il ry the field cx­

citat ion renders the com·cnt.ionaiiC'sts inadC'qunte for dctC'nnining the axi:; reacliliiCC'S.

Condit ion ~lcasurcd raltl<' ( 'omput<'d front sc~a rch <'oi I

[V] simuln I. ion [VJ

o1wn circuit. 1 \~ II~

ZI'I'O d -axis l'lll'I'C'Il f. ( IH'il I' I.:)!).~ I :)!1.8 fnll load)

S<·rcrf\1 lest met hods for dekrTllining the val He of X., and X.1 ha\·c~ !wen prl'sent.f'd Mtrl

arc rompibl top;C'I.Iwr in Table:).;') to show conlparat ivc~ n~:-; ulll' .

Th<! moclifbl conventional l<•sts arc not npplict~hl(' for clc·tc·rrnininp; t.lte \'ill II<':; of .\,1

and g;iw• approximnte VCillles for t.h<· saturatc•d nnd unsaturated cl-<lXis n•act.ann·. Tlw

opr•n Clll d short ri rc 11 it rom pon en ts •uwd I o det.c~nn i llf' .Y.t (sa I.) = /~11/ l.,c a rP OJH•ra t i ng

at diffen•nt len·ls of sat ural ion and the rrns ralue of l.,c has a sip;nifi · . ·;t :Jrcl harmonic

comporwnt. 0\·crilll, tlw rc•sults arc reasonabl<•and \'NY easy to ohti'\in. Since the flux

linkHgc !<'st. is a st.nt.ic t.<•st. this p;in•s another IJH•ans for sr•pnrat ing t.lu~ the <•lfer'ls of Ei

nnrl X.~ . Bera.usc there is no rotilt.ion t.l1erc ranlH' no magtu•t induced roltagr! the test

gi\·es quite accmat.c results for tlw axis n·artance qmllltitics hut n•quires C\ h<llann•r\

bridge \\'hich is very hard t.o maint.ain in t.he h<liCIIIC('d condition due to tl u~ tlwrmnl

lwa t i ng err ret. on the rcsistil.J1C('S. This effect is (':'\ii('('J'ba t.ed (I I. high C\I ITC'III. kvrls

which are near or exceed t.he rated \';dues. Tlw srilrclt coil I est in,·oln·s rln:-: qunnt it.ics

prodnc:Pd by both magnets and stator Clll'l'<'rtt whil<• flll'l'cnl rplantit.y nwasur<'cl is

producrcl hy applied voltage only. It may n•s1tlts in ahnormnlly high cl-axis rc•actanrc

ra:nes at llllsntnrat.ion conditions. The load test pro\'icl<•s rPnsonahiP agrcrnwnt. n•snlts

with the olhcr methods for the q-axis {l'li\llliti<•s hut is in consid<'rahle disagn'<'IJH~nl

fort he high loilrlcd d-a:-ds reactance \'aht<'s. The <~qtwtion used to compnt.e t lw \'ill II<'

Table .i.:i: ~lt'CJSIII'l'nlf' !lf of axis I'I'Rrf;lncc

condition pn ra nwt.l't' nH'thod 1 111<'1 hod '2 uwt hod :J nwl hod 1

X.1 (n) 2?>.:l ~ ·1.8 10 :n unsaturated

x'l (n) xI:\ :HJ :r> :m.o

x., (n) 20 .. 5 IS .S l-1.8 !)()

saturated

Xq(n) ":\f:\ 2:l :12 2:3.~

o ~IC't hod l: ~lorlif1ccl COII\'('111 ionnl t(•st

• ~lethod 2: Flux linkage l1•st.

• ~lct.hod :l: Senrch coil lest.

• ~lt·thod·l : Loadtcst

• Nf A: Not npplicahlc

8()

of .\",1 has a singular point which lit•s in t l11• n').';ion wiH•rl' t hP loitdinp; is hi!!,h t hns

the rt•:-ults art• wry st'tlsitin• to stnall L"llilll)!,t'S in ttH'ilSilrf'd quant.ilit•s. Tllf' load kst

rt•qnin•s n \'illtt(' for/~', in orrl«·r to cillrulatc• the• ntlll«' of X./ which is a arhitrilry way

of s<'pill'fll.ing tllf' I wo t'lfc•cts ;dong t lw d-axis «'X<'I'jlf ing tllf' opt•rat iup; point wht•n!

d-axis curn•nt is zt•ro. :\ Sllllllllill',\' of tht• l<'sl IIH'Ihods is is p;i\'Pll in Tilhl(' .j,(),

Tab!~ 0.{i: S11mmn r·y of tlw te~t met hods

\let hod PMnmetcrs ComnH'n t s ohtilillt'd

\lodiftc•d fo itJJd x.f C:ood approximate• COII\'C'Ill ion ill I'<'SIIit:-: c•asy In

I c•st (H'rfonn

Flux lillkilg<' Xt~ and .\.1 Cood r«'sults O\'t'l'

IC'SI. h11\ lo;HI r<tnge. hard to pnform at hi!!,h cmrent. lc\'ds

Sc•ilrrh roi I X.t nnd .\",1 Rf•rt sona blc n•sults: f<'st. sonw etTor· for 11n~nt ur·a tcd

\'alu(' of X.,: both rnagnel. and winding !lux comporwnts pr<'::;cnt. whir h an' d i fficull t 0 ~('(l ill'ill (' i

Load test \' I ,. • .f <liH • ; ,1 Hcqtli rc•s ,.a lue for f.',: \'C'I'Y S<'llSi I i \'C

t.o IIII.'(!Stll'<'lllell t. ('1'1'0:'

for :-:omr lo;td condition; goorl n•sult for .\1

nnd indicnlcs the \'il rin t ion of xd at. the fttllload range

Srarch coil F-•I G11od rrstd t fot· a simulation sprrific ln<td O(wrat ing

point wlu.·r~ d-axis c tii'I'Pil1 is Z£'1'0

Chapter 6

CONCLUSIONS AND

SUGGESTIONS

6.1 Conclusions

It is \\'(•11 known that. t.IIC' <H'f'lll'ill.e pn·diction 11f tiH~ twrfol'lnaun· of the P.~t. syn­

dmmous machiurs dc•pend:-; mainly on tlw ;u·cnrar~· of tlw c·,·alnat ion of d-q axc•s

l'f'<lclann•. Till~ P\'Riuatiou of d-q i\XPs p i ll'illll<'ll'I'S of this kind of machilll' hy !Is­

ing t.hc t.c·st methods has hc•,•:J carric•cl out. in I his tlwsk Tht• work is iniPlld••d to

drt.c·rmine nxcs ref\danccs awl int<'rnal g«>uc•r;111'd \'olt ilge under load conclit ions .

.-\ general int.rocluction of the P.~l. synchronous motor IIHS hc•c•n prrsc•ntl'rl 111

Chnpkr I, in which the comp;Hison of t.IH• P.~l. synchronous motor and con\·c·nt.ion;d

synchronous motor or induct.ion motor \\'ilS cli~cussr•d. In Chapter 2. I he st an rlind

tests fort IH' con\'cnt ional synchronous motor ha\'c' l)(•c•n prc•sc•ntc•d <111d I lw t hre1~ rii<H­

;I('kristic fc•al11J'rs of t.hc PJ,I. synchronous motor \\'Pre! disrnsscd. Tlw litc•rat11re on

the~ rletPI'IIlinat.ion of P.~I. synrhronous motor pRrnnu·t.,•rs has nlso lwron 1'1'\'ic•\\'Pd from

whic:h the objecth·c of this \\'OI'k is dC'ri\'C•(I. ThP 1'\'aiuation of tltc• air gap riPids of

1 he P. \I. synchronous motor has IH'en de\'<'lopc•d in ('ha pt PI' :~ wh irh sho\\'s t lw air

g'IJ> fields of the P.M. machine consists of both magrwts fldrls and annatnre fi<'lds .

:\n <tnalog ~imulation of the intf'raction of til(' liPid qttantitif's in the air gap has bt'f'll

shown. :\sa rcs\llt of the iltlillog simulation. it hils abo lwrn prc:--cnted thilt torqtu•

angh· of the P.~f. motor can bP nwa:mred by a digital ekctronic method instl'ad of

that from 111otor shaft. Thii'i pro\'idt·s a proct ical and acrurat<· mcasmenwnt oft lw

1 orrl'll' ant;IP. Tlw inllu<>IICf' of 1 he annat urc· n•act.ion ha~ lwcn inw·st igakd and \'a ri­

ation of the gcnf'rated \'oltagf' undt•r load mndition has ht•t•n pr«>scniPd in C'hapll•r

t. It has hf'en shown t. hat t lw g''IH.'ril ted \'olt il ~~· is prorlu n•cl by a mmhi nat ion of tlw

ntagrwt ilnd I he arm at urc components which mnt <lin a lt•akagt• factor. Thi:-: l«>ilkagf'

factor hris significantly diffl'l'f'lll \'illw•s at diffn<•nt load mnditiotls. Const'<pwntly.

it n·sttlts in ch;wgf' of g~'llf'l'illf'd \'oltag<> at htdi11g. Furtlwrmon~ . it n•strlts in th<•

d-axis n~acla!lcc \'arying with tlu! load .. ·\ nutnht•r nf tr:-;t. methods for detf'l'mining

<~xes n·ar1anre:; und(•r Jo;ul conditions han• lwl'll prf'S('Ilt<'d in Chapter I and ChaptPr

.). Tlw l'f'!'lllt of I he various IIH'thocl:-~ ltrts bt•r•Jl di:-~cu~:·wd in last chapt<'r.

In short~ upon th<>rxaminationoftll<' tlwsi~. thr majorcontrihntionsoft.his work

are ]i:-;tf'd i\s follows:

• :\n .~nalog simulation to Vf'l'ify tllf' intt•nwtion of \'nrio11s t·omponents in the air

~·(P oft h<' P.~l. machint• has lw<·n pr,.s<•ntt·d. Tlw ilir gnp lit•ld model has IH't'n

t·,·aluatcd from conflgmation of tllf• rotor and ma,t;tlf'lir rii'C'llit of tlu~ tnarllillt'.

Tlte illlfllog simulation has IH'f'll presf'lllf'd hasl'd on this air gap rield modt•l and

tlu• l'<'t-:lllts of simulation h;:we llf'<'ll shown tltilt. this ilir gap model prm·id<•s a.

sulficit•Jit rcasonahl<• descript io11 for t Itt• art ual t~ir gt~ p fit>lds oft he tn<trhine.

• .-\ digital technique to nwasmt• tllf• t<>I'Cptc• flll~le of the machine~ has been dc­

\'t'lopr·d . In the load tPst. motor's torqllt' an!;l•• must hr. measured in order to

dPit'l'!Hinc the steady-state (H'rforJIJilllcc nf t lw lll<H:hinc. The lf~rhniqllf' prc­

sPntcd in this work has brought :wwrnl ach·<~nt <q.;cs for lm1d test a:; comp<nt>d

wit. It I he existing nwthod currently tlscd. It. is more COII\'enicnt to carry out and

has higher resolution of th<> mcaslll'<'tlll'lll rc•stdts. [t makes a digital dcctmnic

!)()

t!IC'a:mrerncnt possihlc nnd can he' w;('cl in microprocessor-based nH•asur<'IIH'Ill

system iltHI machine dri\·c system whPrc a fast-response digital mcnsurcrncnl. is

I'Pq 1 I j ['(~d.

• The armnt.mc readiou inOucncc on generated voltage hns been analyzed and

t lw snt.uratcd value of I be gcn<'ra.tcd volt.ngc has been expcrinwntally dct.er­

tnitJcd. It. hils been shown that the snturatcd ,·alue of the gencrflt.erl voltage is

not L'C[IIill to t.hat of t.hc open-circuit. vnluc. lfpon examination of the phasor

diitgrnm of t.he m«c\inc and the flux linkag<· paths of t.lw rotor under difren·nt

load rouditions, it hns lwcn obscrw•cl t.hat applied volt.ng<~ rcsult.s in a nwgn<~­

t izing (demagnetizing) cmrcnt clnring t.hc light (heavy) loa cling. Therefore, t.he

armatmc reaction results in a significant. increase of the generated \'ollt~ge over

t.lw rt~ngc of loads. The in,·cstigation of the vnriation of the gencrilt.ed volt:-~ge

ill. loading has siguificant. meaning in determining the satmated value of the

ilXCS reactances hy usc of modified com·ent.ionRI test ;mel load test. By using

the sat. mated value of the generated volt age, the modified convcntion<ll IC'st hr~s

presented a better approximnte saturatc.•d value of the <!-axis reactance t.han

that hy using the open circuit voltage. Secondly! for the load test., the twgnt.ivc

nducs of the d-axis t'Cilctance do not appear in any range of t.hc lot~rls. This

is consistent with the statement. that the negfltivc values for Xr~ are entirely

consequent upon the CISsltmption of constant E0 •

• Sevcrt~llcst approaches to determine axes rcact.anc('s hil\'e been presented and

discussed. Each of the test. rnrthods is compared with n•spect t.o their usefulness

<~nrl range of validity.

6.2 Suggestions for Further Study

The thesis submitted here is focused on the test methods used to determine the PJ.l.

synchronous motor parameters. Thus, fmther work to be done should include:

• ~[easurcment of the steady state performance of the machine. from which the

lorque \'S torque angle cltamctcrist ic can he obtained experimentally.

• Determination of the gtiiCrated \'0ltage 0\·cr the entire rrgion of loads by digit a I

simulation or finite-element techniq11es.

• Prediction of the steady state performance of the machi ne based on the two

axis reactances determined by different test. nwthods. Subsequently. correlation

of the measurement results and calculation results.

References

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[2] T.,\. Little, '' Design and analysis of permanent magnet s.mchronous motor.,, ~[. Eng. Thesis, Memorial University of ~ewfoundland, C<mada, tgs.t

[:Jj Alexander Levrar and Emico Le,·i," Deign of Polyphase i\lotors with P~l Exci-1.iltion ",IEEE. Trans. on !\[agnetics, Vol. ~lAG-20, No.3 , i\lay, l!JS·l. pp .. 507-5l.S

PJ T .. J.E. i\'liller, T.W.Neuma.nn, a.nd E.Richter, ":\ permanent magnet excited high efficiency synchronous Plotor with line-start capahility ., • IEEE I:\S An­nllal iVIeeting, f\'lexico city, Oct. l98:J, pp. ·155-·Wl

[.j] P. Zimmerman, ''Electronically commut.ated de fr.cd dri\'cs for machine tools'', Drives Contr. Int. Vol. 2, pp. 1:3-20. Oct./~mr. 1!)82

[6] 'J'.W. Neumann a.nd R.E. Tompkins,"Litw start. motors designed with :.i'd-Fe-B permanent magnet.s", iu Proc. 8th Int . Workshop Har<'-Eflrth ~lilgnels, Ohio. US:\ , ~la.y l98.j, pp.77-Sa

[i] 11.r\ . Bose, " A High PcrformRncc ln\'crter-F<'rl Dri\·c System of An lntc>rior Permanent ~Iagnet Synchronous ~lachine' ... IEEE IS:\ ,\nn11al ~lectin~. :\tlanta. G:\. USA , Oct. 1087, pp .26~)-2i6

[SJ T.A. Little, ~I.A. Rahman. '' Efrect. of Iligh Coc>rci\'e Force on Stating Pc>rfor­mancc of Permanent ~lagnet ~lotors ., .it h lntrrnnt.ional \Vorksltop on H11re Earth Cobalt ~lagnet. nnd their 1\pplicat ion~. B<•ijing, China, Sept. l!)S:L pp.2!J­:H)

[9] ~I.A. Hahman, A.M. Oshciha. " Efrcct of Param(~f.cr Varintions 011 Tlte P< ~rfor­

mancc of Permanent M<~gnct :vrotors ... IEF.E 1:\S :\nnual ~leeting. Den\'N, Co .. t:SA. Sept.2S-Oct.3, 1986. pp.i87 -iD:J

[10] Thomas M .. Jahns, Gerald B. I\lirn<ln. Thomas \V. i\cumann, '' Interior Permanent-Magnet Synchronous i\lotors for Adjustable-Speed Drives '' 1 IEEE Trans. on Ind. Appl. IA-22. No. 4, July/ August, 1986, pp. i38-i4 i

(11] T .. J.E. 11iller," Synchronization of Line-Start Permanent-Magnet AC ~\lotors ", IEEE Trans. on PO\ver App. and Sys. Vol.PAS-103, No.7, July, 198·1, pp.l822-182S

(12] E. Richter, " Permanent Magnet ~lotors Final Report ., , Oak flidgc l\ational Laboratory report, ORNL/SUB/1 i·l52/ I. DEC. I ~)8~

(13] M.A . .Jabbar. "Analysis of The Pcrform<lnce of r\ Pcrmancnt-:\lagnct A.C. :\la­chine,. Ph.D Thesis, Southampton University, lfants. SO!J .ji\11. England .. June laii

(1~] ''Test Procedures for Synchronous ~lachines'' IEEE Std. 115-1983

(I .5] Leander \V. ~latf,ch and .J. Derald Morgan, ··Electromagnetic and Electrome­chanical :\fachines" (book), New York, IEP, l!)ii

[16] J\lnlukutla S. Sarma, "Synchronous Machine (Their Theory, Stability, and Ex­citation Systems)", Gordon and Breach. Science Publishers Inc. New York. NY, 1079

(1 i] E. Richter and T.W. Neumann, " Synchronous machine designs using di fferent t.ypes of permanent magnets", Paper No.l-1 , fifth Int. Workshop on Rare Earth­Cobalt Permanent i\lagnets and Their Applications; Roanoke, VA, .June 1981, pp.l-27

[18] V.l3. Honsinger," Performance Polyphrlse PNrnancnt i\lagnet Machines., , IEEE Trans. Vol. PAS-99, No..l, .July/Aug.l980, pp.l510-1.516

[19] T . .J.E. Miller," Methods for Testing Pcrrnoncnt i\fagnct. Polyphase A.C. ~rotors ". IEEE lAS Cof. Record No. Slch1678-2, HJSl, pp..l94-·l!Jtl

[20] Fitzgerald, A.E., Kingsley, C .. Jr., 1\u~ko~ A., ''Electric ?-.lilchincr.y'' , (book), ~JcGraw-Hill, New York, NY, toil

[21] V.B. Honsinger," The Fidds a.nd Parnmclet·s of lnteriot· Type ,\.C. Permanent ~lagnct Machines", IEEE Trans. Vol. Pr\S-101, No..l, 1982, pp.~Gi-875

[22] A. Hameed, K.J. Binns, A. Vanclcnput, \V. Gcyscn," The Finite-Element Com­putation of The Field in A Permanent :\lognet. i\Jachine", Electrical i\lachines and Converters, Elseviet' Science Pulishers B.V. (North-Holland)/Ii\fACS 1984, pp. 107- 111.

9·1

[23] I\. i\liyashita, S. Yamashita, S. Tanabe. T. Shimozu, If. Scnto, ,. De,·clopment of A High Speed 2-polc Permanent J\lagnet Synchronous ;\lotor'', IEEE Trans. Po\\'er App. Sys . Vol. PAS-99, No.6, l!lSO , pp.217.5-2183.

[2·q K .. J. Binns, C.P. Riley, T.\V.\Vong, ., Some Design Aspects of High-Output Per­manent Magnet Synchronous ~lachincs with Non-Radial i\.lagncts,, Second Int. Con f. on Electrical i\hchines Design and Applications (Con f. Publ. No.2.1,1) lEE l!J85, London, England, 17-10, Sept. I!JS5

[25] Nady Boales," Field Analysis of P.\1. Synchronous i\lachincs with Buried ~Jag­net Rotor'', Inti. Con f. on Elcct.rical i\lachines, .i\[ nnchen, Germany, Sept. l DSO. pp.l 063-1 (X)6

[2G] 1\ .. J. Binns, i\I.A .. Jahhar, G. E. Parry .. , Choice of Parilmet.crs in the llyhrid Pcr­lllilnen t l\1 agnel Synchronous i\ [otor ... Proc. I EE, Vol.l2.5 ( 8 ), A ug.l !Ji!), pp. i ..f 1-j<( .[

[27] t\ .. J. Binns, i\l.A . .Jabhor, " High-Field Self-Starting Permanent. i\lagnct Syn­chronous i\[otors ",lEE Proc. B, Electric Po\\'cr r\ppl..l98L, 128( :3) , pp.l57-l60

[28] i\Lr\. Rahman, T.A. Little, G.R. Siemon, '' Analytical i\[odcls for Interior-Type Permanent i\.lagnet Synchronous ~lotors ., , IEEE Trr~ns . on ~lagnetics. Vlo. i\IAG-21, No .. 5, Sept. 1985

[2D] C.V .. JiJncs,"The Unified Theory of Electrical i\fachinc:;" , 13uttcrworths, London, lD67

[30] S.l. Ahson, M.H. Ali, ., :\ ~licroproccssor-Based Scheme for TMqlle-Anglc and Speed ~lcasurement '', IEE.E Trans. on Industrial Electronics, \ 'ol.IE-:J.J. No.2, i\lay,l987, pp.l3.5-1 :3S

[31] T.I\.~1. Babu, Denis O'kelly, '' A ~Iicroproccssor- f3a:;ccl Load Angle ~lct~smc­ment System ", IEEE Trans. on Industrial Elcct.ronics, Voi.IE-34, No.2, i\lay. l !J87, pp.l29-l34

[3~] S.F. Gorman, C. Chen, .J . .J. Cathey, "Determination of Permanent ~lagnet Syn­chronous i\lotor Parameters for Use in DC nrushlcss ~lotor Drive Analysis", IEEE Trans. on Energy Conversion, Vol.:l, No.:l, l!JSS, pp.fii·l-681

[:l:lj T.:\. Lipo,"Fiux Sensing 11nd Control of Stat.ic AC Drives Ry t he Us<' of F lttx Coils'', IEEE Trans. on ~fagnetics ,VoDIAG-l:J,No .. j , Sept..l!)77, pp.l·lO:l-1-108

[3·lJ :\. 8. PI unkett." A Cmrcnt-Controlllcd P\VM Transistor l nv('rt.cr Dt·ive", IEEE lAS Annual Meeting, Conf. Hccord, Cleve! and Ohio, USA , Scpt .. 30-0ct..t , 1 !Ji!J, pp.iS.J-792

[35] R. H. Park, "Two-Reaction Theory of Synchronous Machine· Generalized ~let hod of Analysis-Part I, ·• AII~EE Trans., Vol.-18 . .July,l929. pp. il6-i2i.

[36] B.l\. Bose,''Power Electronics And r\C Drives'', (book), Prentic::c-Ifall. Engle­wood Cliffs, NJ, 1986

[3i] 1\: • ..-\. Stroud,"Fourier Sr.ri<>s and Harmonic Analysis" . (hook). St11nlry Thomes, (Publishers) Ltd. 198-l

96

Appendix I

Circuits of Analog Simulation

!)j

Figure I.l: Analog simulation circuits

lOOk

IV\

lOk lOk

I

98

Appendix II

Circuits of Digital Measurement

of Motor Torque Angle

!)!)

Figure ILl: Circuits of digital measurement of motor torque angle

From search coil

til

terminal 120•1 6. Jv0 From :JE voltage \::.J

ft2

IS

115

t3

tl6 ~7

tl6 t7

100

Figure II.2: Circuits of digital measurement of motor torque anglr. -cont "rl

lt5

118

S CP R

0 00 #1: Differentiator lt2: Filter lt3: Clipper It~: Voltage follower lt5: Phase shifter

#1: Differentiator lt2: Filter #3: Clipper It~: Voltage follower lt5: Phase shifter

116

lf9 r·--------------------- ---; I

I

I I L_ _____________________ j

101

0Reterence 8

#6: Attenuator #7: Schmitt trigger #8: Flip-flop #9: Phase detector

#6: Attenuator #7: Schmitte trigger lt8: Flip-flop #9: Phase detector

Appendix III

Rotor Geometry and Parameters

102

Figure III.l: Rotor magnet geometry for calculating leakage factor a) dimensions b) flux pil.ths

~T3 T4*~ ~f

a)

- - - .. - ' ,... ......._ ,. ,...

' / -/ , -......

./ \' ..... .-· -........

I ... .,. --/ / ...

' / I <Pg ( ......

/ / g ...... ........ / ....... ,/ I .......

,/

' / ... - .... I -- ' / .... . / ' ' ,/

I ...... ......

.,;--..... ' ...... ' .. , I , ... ' ' I I

, I I I '

~ ~ * I ' t I ' ! i,4 I \., ~ i2 ' I I

/ ~i.m ~ I ,,

~.2.3 / i I t ....... \. / ; '\ ....... _ .... I . \ / ..

. - ~ / I \

' ,., ... / I .,

-/ - ,/ ' ' -· -..... -- /' '· ' , - b) ... •' ... ,-- ..... ..

,/

' --l0:3

Figure III.2: Machine and magnet parameters for calculating leakage factor

Symbol Description Symbol Description

Motor Parameters: Rotor Geometry: G Effective air gap BLl Length of leakage

length including bridge above magnet 0.565 Carter's coeffi- 0.0330 section 1 cient and satura-tion BL2 Length of leakage

bridge above magnet 0.553 D Rotor outside 6.849 section 2

diameter L3 Length of leakage

LS Length of rotor 7.620 bridge under magnet 0.085 core section 1

Magnet Parameters: L4 Length of leakage bridge in middle of 0.553 HMl Dimension of magnet magnet section 2 section 1 perpendic- o. 9906

ular to useful flux Tl Thickness of bridge 0.039 HM2 Dimension of magnet corresponding to BLl

section 2 perpendic- 2.680 T2 Thickness of b1.·i-::lge 0.104 ular to useful flux corresponding to Bl2

Ll Length of magnet Ul T3 Thickness of bridge 03 0.556 0.559 ln direction of flux T4 Thickness of bridge IJ4 0.124

L2 Length of magnet 02 0.508 in direction of flux

104

Appendix IV

Calculation of Reactances

10.5

I t---f--+--t·--t--t--t----r--t-·- .'

I

I _ _j

r1gure IV.l: Flux component: Ia. =2.85A chl:t/ld2 ch2:1/19

2

Figure IV.2: Flux component: I a= 3.1 A chl: t/ld3 ch2: 7/193

106

Figure IV.3: Flux component: la = 4.3 A chl: 1/Jd4 ch2: 1/;q4

Figure IV.4: Flux component: I a= 5.8 A chl: l/Jds ch2: VJqs

107

Table IV.1: Values of the ordinate ford-axis flux of one cycle

X tPrll tPd2 tPd3 1/-1d4 tPds tPd6

(0) (mwb) (mwb) (mwb) (mwb) (mwb) (mwb)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 30.0 15.7 12.2 14.25 H.78 11.5 16.23 60.0 26.0 24.4 23.95 19.3 18.0 23.7 90.0 31.6 27.8 30.45 31.4 29.75 34.2 120 26.0 26.1 24.6 20.95 17.25 23.7 150 12.8 14.9 16.5 17.2 14.25 17.4 180 0.0 0.0 0.0 0.0 0.0 0.0 210 -15.7 -12.2 ·14.25 -14.78 -11.5 -16.23 240 -26.0 -24.4 -23.95 -] 9.3 -18.0 -23.7 270 -31.6 -27.8 -30.45 -31.4 -29.75 -34.2 300 -26.0 -26.1 -24.6 -20.95 -17.2.5 -23.7 330 -12.8 -14.9 -16.5 -17.2 -14.25 -17.4

Note, tPdl, tPd2 .... tPrl6 and tPqh tPq2, ... tPq6 correspond to the amplitude of the d-axis (!"~ q-axis flux waveform when the stator current is l. 73 A, 2.85 A ... 6 .. 5 A respectively.

108

Table IV.2: Values of the ordinate for q-axis of one cycle

."C 1/Jq I I '~''12 l/'qJ ~'?·I 1/,•qs 1/JqG

(0) (mwb) (rnwb) (mwb) (mwb) ( mwb) (mwh)

0.0 0.0 0.0 0.0 0.0 OJl 0.0 :10.0 11.6 lfi.l 1 i.6 :.!1.7.1 28.1 :30.65 60.0 23.17 19.1 21.2 2G.8 ~J2. ji) :W.05 90.0 30.15 25.0 20.3 ll..)i 6.0 5 .. ):3 120 2.5.2 :J:J. 9 3.).3 :n .. ;;; :31 '1 2ti.85 1.50 l·LS0 22.2 25.6 27.2!) •) ....

-I, ;) 2!) .. 5 180 0.0 0.0 0.0 0.0 0.0 0.0 210 -11.6 -!().! -17.6 -:21.7.) -28.1 -:10.65 240 -2:3.17 -1!J.l -21.2 -26.8 -:32.7.1 -36.05 270 -30.1 !j -2.5.0 -20.3 -1 L-17 -6.0 -.5.53 300 -25.2 -:33.!) -35.3 -31.·1.5 -31.1 -28.85 :330 -t ·t.s.s -22.2 -2!>.6 -27.25 -27.5 -2!JJ)

109

Calculations are given below:

l).fa=l.i3A

d-axis: a 1 =0.837 f>t =30.2!)<! ld=l.5iA r/•d=O.l iS!) wb

q-axts: Ut =-1.216 ht=28..122 lq= l. 7:3 A 1,bq=0.1678 wfJ

2). /'1 = 2.85 A

d-axis: a1 =-1.063 b, =28.:361 1.,=2.66 A l/'d=0.16i5i wb

q-axts: nt=-·1.228 ''1=30.016 l -·) ~=- ·\ ,,-- .<..:·J .• ~·q=O.l iSDi tcl1

:3). fa= 3.3 A

d-axis: nt=-2.071 ,1 =:30.049 ld=:1 .05 A r:·.,=O. l i78wh

q-axts: nt=-4.65!) h,=:30.2ii lq=:L:J A rj•'l =0.1 80D wh

.t).lrt=4.3A

d-axis: at=-0.999 bt=2i.373 lr~=:J . D A ~'d=0.16l i wb

q-axts: Ut=-2.363 "1=28.839 !1=·1.:3 A l/'q =0.1 708 wb

.5). la = .5.8 A

d-axis: Ut =-0.669 ,, =24.:384 la=·l.5i A ~'d=0.1440 tt•fJ

. a 1=0.488 ht=29.6Dl lq=·j,8 A ¢•9 =0.1 7.5:1 wb q-axts:

6) . fe~ =6 .. j;\

d-axis: Gt=-0.338 "1=30.688 I .[=(L28 A rbd=0.1812 wb

q-axrs: Ut = 1.99!) 1>,=29.927 / 7=6 .. 5 A 1/.·,=0.1771 wb

110

Figure IV.5: Current component: I a= 2.85 A chi: /,12 ch2: /d2

Figure IV.6; Current component: la=·S.SA chl;/qs ch2:/ds

111