EVALUATION OF EDDY CURRENT LOSS IN SOLID FERROMAGNETIC CORES SUBJECTED TO ONE OR MORE ALTERNATING MAGNETIC FIELDS

sf

Gheris submitted tu the

PONDICHERRY UNIVER8ITY

&or he award {fhe deyree

6 DOCTOR OF PHILOSOPHY

Department of Electronics and Communication Engineering

PONDICHERRY ENGINEERING COLLEGE PONDICHERRY-605 01 4

lND1A

JUNE 2001

C E R T I F I C A T E

This is to certify that the Thesis entitled "Evaluation of eddy current lors in

solid ferromagnetic cores subjected to one or more alternating magnetic fields"

submitted for the award of degree of Doctor of Philosophy in Electronics and

Communication Engineering is an authentic record of the work carried-out by

Mr. B. Rami Reddy. in Electronics and Communication Engineering Department,

Pondicheny Engineering College of Pondicheny University, Pondicheny, under my

supervision and guidance since 1995. This thesis or any part-of has not been

submitted clscwherc for the award of any Degree, Diploma, Associate-ship,

Fellowship or other similar titles. In my opinion his thesis acquired the standard of the

Ph.D degree.

Innau; I-.- Dr. K. Manivannnn 1 '1 @' ' Supervisor Professor & Head Dept. of Electrical and Electronics Engg. Pondicheny Engineering College Pondicheny India

Dedicated fo

My ahgbter Pan'mab

A C K N O W L E D G E M E N T

I would like to acknowledge gratitude to my guide Dr. K. Manivannan,

'rofessor & Head of Electrical and Electronics Engineering for his support

.hroughout this project.

My sincere thanks to the Head of Department of Electronics &

Communication Engineering for allowing me to canyout research in the department.

I am indebted to administration of Pondicherry Engineering College for

providing infiasvuctural facilities.

My heartfelt thanks are due to retired Professor N. Kesavamurthy of IIT

Kharagpur , for his invaluable guidance and moral support throughout the work.

My special thanks to Mr. K. Subbarayudu, Assistant Professor in Mechanical

Engineering of Pondicherry Engineering College. for his suggestions during the initial

stages of this project.

I take privilege to express my profound thanks to my colleague

Mr. A. Muthuramalingam, for his general suggestions.

I wish to thank Dr. (i. Vaidhyanathan. Professor & Head of Department of

Physics, PEC, who has cleared my doubts on few occasions.

I am proud of my fanlily members; my wife Lakshmi, daughter Parimala and

sun Madhu, who have permitted me to devote my time for research.

At last, but not least I thank all those who have helped me directly or

indirectly.

Place: Pondicherry

Datc: 27-06-2001. B . b m i Reddy

ABSTRACT

The work on the evaluation of eddy current losses in ferromagnetic wres was

canied-out for over two decades. The main hurdle was the nonlinear behavior of

magnetic materials. The analytical solutions are available for eddy current losses

based on the approximation of magnetic characteristic (B-H curve) in different ways.

Since the inception of computer, Crank-Nicholson numerical method is implemented

to find the field distribution and hence power losses in ferromagnetic cores.

An alternative method for the Crank-Nicholson scheme is proposed. This

method is called Pseudo-Spectral Method. The features of this are brought-out after

comparing its effectiveness with that of Crank-Nicholson Method. The eddy current

losses obtained by this method are compared with the values of Crank-Nicholson

Method and experimental results for a single excitation pmblem.

' h i s project is also aimed at developing theory for double excitation. So a

closed form solutions are derived for finding the field distribution, power loss, depth

of penetration etc. for two input signals whose frequency ratio is greater than two. To

complete the problem, a graphical method is developed for the frequency ratio

between 1 and 2.

'The validity of dual excitation field theory is verified by comparing the

theoretical pwer losses with the experimental values of two identical toroids. To

overcome the interference hetheen the input signals. the back-to-back connection of

transfom~ers is exploited fbr the mild steel rings.

-lhr dual excitation theory is also used to find the power losses in the solid

iron rotor of single-phase induction motor. 'The experiment is conducted after

removing the capacitor from the starting winding and with no air-gap between stator

and rotor. Ihe two input signals were at 50 Hz and 450 Hz fquencies. In the case of

C O N T E N T S

Title Chapter Page Na

1.0 lntmduction 1

2.0 Evaluation of eddy current loss in solid cores subjected to

single excitation 9

2.1 Introduction 9

2.2 Formulation of Problem 10

2.3 Linear theory 12

2.4 Limiting Nonlinear Theory (LNT) 17

2.5 Crank-Nicholson Method (CNM) 23

2.6 Pseudo-Spectral Method (PSM) 26

2.7 Experimental toroid and its characteristics 3 5

1.8 Evaluation of eddy current losses by different

methods 36

2.9 Comparision of results 41

2.10 Conclusions 42

3.0 Evaluation of eddy current loss in solid corcs under

double excitation 43

3. I Intiduction 43

3.2 l'heor). 44

3.3 Determination of losses 50

3.4 Graphical solution of the problem 51

3.5 Numerical solution of the problem - CNM

3.6 Experimental procedure and results

3.7 Conclusions

A3 Appendix

143-1 Evaluation of fundamental components of

output using M-functions

A3-2 Evaluation of fundamental components of

output using power series

A3-3 The details of experimental toroids

4.0 Double excitation theory-Induction Motor

4 1 Introduction

4 Description of the problem

4 3 Solution of the problem-PSM

3 4 Graphical solurion

4.5 Esprimental procedure. results and discussion

4.6 Conclusions

A4 Appendis

A4-1 Lktails of Single Phase Induction Motor

AJ-2 Procedure to find total losses in the stator

A4-3 'fhe correction factors for curvature and end-

effects

5.0 Performance evaluation of two-phase Induction

Motor with solid iron rotor under unbalanced load

conditions

5.1 Introduction

5.2 Theory

5.3 Procedure to evaluate performance

5.4 Experimental results and discussion

5.5 Conclusions

AS Specifications of 2-phase induction motor

6.0 Performance evaluation of Poly-phase lnduction

Motor with solid iron rotor

6 . 1 Introduction

6.2 Problem formulation

6.3 Analysis of solid rotor lnduction Motor without

harmon~cs

6 4 I)ual excitation theon.

6.5 Performance of solid rotor lnduction Motor with

harmonics h) field theory

6.6 Perfomlance of lnduction Motor by fundamental

and harmonic equivalent circuits

6.7 Multiple excitation theory

6.8 Performance evaluation by multiple excitation

theory and equivalent circuits

6.9 Results

1 .I0 Conclusions

A6 Appendix: Specifications of 3-Phase Induction

Motor

Conclusions

References

Bibliography

Introduction

Ferromagnetic materials are widely used in the branch of electrical

engineering, as core material for simple toroid to giant synchronous generator. It is a

fact that the voltages and currents are invariably alternating because of their inherent

advantages. Therefore, the ferromagnetic materials are subjected to alternating fields.

More specifically, a time-varying magnetic field causes time-varying electric field,

which in turn c a w s a time-varying currents and hence secondary magnetic field. The

total magnetic flux density and electric field intensity distributions are the result of

primary and secondary components. Qualitatively speaking, when a magnetic flux in

a conducting medium alternates with time, an electromagnetic force is generated in a

plane at right angles to the direction in which the flux is changing, and there is a

resultant flow of currents within the material. These currents are called eddy currents

1 1 1. 1be) depend upon the geometry of the material specimen, its conductivity,

permeability. and the frequency of alternating flux. The eddy current density

directions are always such as to counteract the change in the magnetic flux density

that produced them. The net effect of flow of eddy currents is to prevent the magnetic

flux from penetrating immediately into the interior of the material. Thus. when the

magnetic flux is alternating continuously, the magnetic flux density in the interior

may hc small fraction of the magnetic flux density at the surface layers of the

medium. As a consequence, both flux density and eddy current density decreases from

the surface towards the interior of the medium. and the phenomenon is called the skin

eITeft.

The eddy currents have undesirable effects when they are induced in the

magnetic cores of power apparatus such as transformers, generators and motors. Not

only they do absorb power and reduce eficiency, but also generate heat, which

advmcly affect the rating of power apparatus. It is for this reason that the magnetic

circuits of power apparatus are laminated.

On the other hand, eddy currents are usefidly employed, extensively in

engineering applications such as drag-cup, solid rotor machines, linear machines,

repulsion suppofl for electromagnetic levitation, induction furnaces, induction heating

for tempering and annealing of metal parts, eddy current brakes, and non-destructive

resting.

Therefore. it is interesting to analyze eddy current distribution in

ferromagnetic materials. Methods of analyses of eddy current problems depend on the

geometry, type of excitation, and whether the conducting medium is magnetically

linear or nonlinear. Since eddy currents do not follow any prescribed paths, but are

disuibuted over a solid conducting medium in different d i i t i o n s with various

densities. So it is not possible to correlate from the outset a definite resistance or self-

inductance to eddy currents. Hence, the starting point is the formulation of diffusion

type of partial differential equation, the solution of which gives the field distribution

inside the medium. But finding the field distribution is extremely dificult when the

exact nature of the B-H curve is taken into account. To gain an insight into the

complexities of the phenomenon involved. it is worth examining. initially the results

of the lincar theor) (21. based on the assumption of constant permeability. Under this

assumption, the differential equation governing the field quantities is linear.

This is so because the characteristic property of the flux density, defined by

the Maxwell's equation, div B = 0, equally applicable to the magnetic field intensity

H. On the other hand, in the nonlinear range of B-H curve, the ratio BM or dB1dI-I is a

complicated function of H. Consequently, div.B is still zero, but div.H is no longer

zero and the nature of differential equation in H is highly complicated and for a

general case, cannot be solved by explicit analytical methods. However, a solution of

a nonlinear problem by analytical techniques is possible in certain cases by

approximating the nonlinear B-H curve or the hysteresis loop by a mathematical

function.

Before going into the methods of analyses, it is certainly noted that when the

magnetization characteristic is linear, the field distribution inside the medium

undergoes an attenuation and phase shift, as the point moves away from the surface.

However. the waveform of H would not be distorted and has the same frequency as

the forcing frequency. On contrary, the field distribution inside the material would

undergo distonion. besides attenuation and phase shift, when the B-H curve of the

material is nonlinear.

'Ibere exist several analytical expressions to approximating the B-H curve. In

selecting a particular expression. the guiding consideration should be not only that the

analytical curve closely follows the experimental one over the operating range. but

also the expression should offer the additional advantage for field analysis.

One of the ways to represent the nonlinear B-H curve is by relay type curve.

Such an approximation physically implies that the material is magnetized to saturation

and it is possible to change the flux density from +B, to -B, or conversely. only at

1.14. Baxd on this approximation. expressions an deduced for field distribution.

power loss, power factor, depth of penetration etc. This theory is called Limiting

Nonlinear Theory [3].

The B-H curve can also be approximated by the equation B = kH74], where

'k' and 'n' are coefficients derived from the aaual magnetisation characteristic of the

specimen. This type of dependence has the added advantage that when n=l, it

represents a linear characteristic and when n=O it reduces to relay type variation.

The nonlinear dependence of the form B=~(H-EH~) is considered in [5 1, with

s being small. Further, the analysis is modified to take account of hysteresis by

replacing the actual hysteresis loop by an ellipse, when the surface magnetic field

intensity is sinwidal. However, it has been established in [2,6] that the eddycurrent

loss is several times more than hysteresis loss, especially, in the case of thick plates.

Consequently, in the analysis of eddytumnt loss, the exact nature of B-H loop is

ignored and it is replaced by mean magnetization curve.

The studies on the effect of saturation have fvst been based on solving field

Lsvibution equation on the assumption that the harmonics of field quantities inside

the material have negligible effect on the evaluation of wre loss. On this basis new

methods [7,8] have been developed for evaluating core loss for both cases, namely;

( i ) I'hick plates, where both the magnetising force and cumnt density vanish

at some point inside the material.

(ii) Thin plates, where the magnetising force at the centre is finite and non-

vanishing, although the current density is zero.

'I'hc results of such analyses are found to be in close agreement with test

results for the case of thick plates only, the divergence with test results for thin plates

being considerable and increases with reduction in thickness.

In the experimental measurement of core loss in thin plates, a clear distinction

exists W e e n the observed losses, when the current through the winding is

sinusoidal and voltage across the winding is sinusoidal. That is, a discrimination

baween the surface H being sinusoidal and the surface J being sinusoidal. For

example, for the same root-mean-square surface magnetising force, the core loss for a

sinusoidal magnetising force at the surface can be twice as high as when the surface

cunmt density is sinusoidal. Furthermore, this difference between the two cases is

found to vanish with increasing thickness of the plate.

It may be noted that numerical steady state solutions are developed for field

distribution. power factor, iron losses etc, considering the fundamental as well as third

harmonic for a general B-H curve [a].

No doubt that analytical solutions for the field problems are helpful to know

the broad distribution of field intensity inside the specimen. but to visualize the actual

distribution one has to consider the exact shape of the B-H curve. This naturally

requires the problem to be solved by a numerical method. The well-known method for

many years has been the Crank-Nicholson numerical method [9]. Though this method

is a standard one. but it is a local method since it uses local grid values to compute the

derivative of a function.

A best method or atleast an alternative for the Crank-Nicholson Method is a

new classical numerical method called Pseudo-Spectral Method is implemented [lo].

'This is a global method since the trial functions are global in nature.

The work intended to develop:

(a) Pseudo-Spectral Method

(b) Crank-Nicholson Method

(c) Double Excitation Theory

In chapter-11, the field problem is formulated based on the infinite half-spacc

theory. The linear theory and Limiting Nonlinear Theory are discussed in detail. Later

B-H curve is approximated by Frohlich equation so as to incorporate the actual

magnetization characteristics into the field equation. The field distribution is obtained

by the proposed Pseudo-Spectral Method. Subsequently, the power losses are

evaluated. The problem is also solved by implicit modified Crank-Nicholson Method.

Finally, the results of eddy current losses estimated by three different methods,

namely; Limiting Nonlinear Theory, Crank-Nicholson Method and Pseudo-Spectral

Method are compared for the single excitation [lo].

An extensive work has been carried out on single excitation. But little work

has been done on double excitation. Therefore, an effort is made in this project to

contribute for two-field excitation.

In chapter-Ill, an analytical solution is developed for the double excitation

field problem so that iron loss, flux. power factor etc. of each signal can be found

out [I I]. The two-excitation field problem is also tackled by Crank-Nicholson

Method. An experiment is conducted on two identical toroids made up of mild steel.

The input excitations were at kquencies of 16 and 48 Hz. The four windings of two

rings are so connected that the magnetic interference is minimized [12]. The

simulated iron losses of analytical method and numerical method are compared with

the experimental values. l'he developed analytical solution is valid for the ratio of

frequencies of two signals is equal or greater than two i.e. {(f*/fl) 2 2) . To obtain field

distribution. power loss etc. a computerized graphical method is also suggested for the

condition { I 5 (f2/fl) 52). Further, the graphical construction is normalized [12].

In chapter-IV, the Pseudo-Spectral Method is also implemented for dual

excitation field problem. A practical work 1s carried-out on two winding induction

motor with solid-iron rotor without air-gap. The frequencies of two excitations were

50 Hz and 450 Hz. The magnetic interference between two excitations is avoided,

since the electrical angle between the windings is 90'. The theoretical values of eddy-

current losses arc checked with the practical values of rotor core [13]. Using the two-

field excitation theory, the induced emfs in the respective windings are calculated and

hence the applied voltages are evaluated. The voltages are verified with the

experimental values. The infinite half-space theory is adapted for electrical machines,

by modifying the specific resistance of the rotor material, so as to incorporate the

correction factor for cwature and end effects [14].

The chapter-V deals with the performance of two-phase induction motor under

unbalanced load conditions. It has been established through an illustration that, when

a two-phase induction motor operating under unbalanced load conditions, by

neglecting the curvature. the rotor can be represented by an &te half-space of

material excited at the surface by two alternating magnetic fields of different

amplitudes and frequencies. Using this fact, two-excitation theory developed in

chapter-Ill is applied to find the performance of two-phase induction motor under

unbalanced load conditions. For this purpose a two-phase induction motor is wound

and run with solid-iron rotor. The theoretical results are compared with that of

experimental values and found close agreement.

'The chapter-VI is devoted for finding the performance of three-phase

induction motor. If the traveling magnetic field in the air-gap of induction motor is

non-sinuso~dal w~th time and rhe rotor curvature is neglected. Then. it can be

ascenained tha~ the rotor of induction motor would be subjected to pulsating field

consisling of fundamental and harmonics. Thus the performance of three-phase

induction motor is evaluated [IS], using two-field excitation theory when the fifth

harmonic is present in addition to fundamental.

For finding the performance of induction motor accurately, besides

fundamental, multiple harmonics should be considered. Therefore, a mathematical

method is developed successfully for handling multiple excitations. This method

makes a way to develop a tool for more number of excitations. For the purpose of

predetermining the performance, besides fundamental, f@ and seventh harmonics

are considered [ I 51. The results are verified with that of equivalent circuit theory [16].

Chap ter-II

Evaluation of eddy current loss in solid cores subjected to single excitation

2.1 INTRODUCTION

The study of eddy current loss In ferromagnet~c cores subjected to alternating

flux has been the subject over many years The analysls assoc~ated w ~ t h the

penetration of alternating flux and the evaluat~on of eddy current d~strlbut~on In

ferromagnet~c mater~als depends on the determinat~on of the flux dens~ty d~stnbut~on

lns~de the medlum Such an analys~s becomes d~fficult when the exact nature of the

hysteresis loop or the normal magnetlsatlon character~st~c (B-H curve) of the mater~al

IS taken Into account To galn an lns~ght Into the complex~t~es of the phenomenon

~n\ol \ed the rc\ults of thc ilnear theory [2] are exarnlned

Once the magnetic Iluv dens~t) and eddy current d~str~but~ons are determined

bg the Ilnear theor\ the core losses constltutlng the eddy current and hysteresls losses

can be ebaluated In a stra~ghtfonvard manner But the assumption of constant

permeablllt) IS not al\\a)s val~d, because. the magnetlsatlon curve of a material, In

general IS nonlinear In nature So, to obtaln expresstons for the core loss, power factor

etc . In the presence 01 saturation. the magnetlsatlon curve of a materlal 1s

approu~niated h) r rela t\pc cune Thls theon (Llrnltlng Nonlinear Theory)[3] has

\ ~elded gaoJ re~ult\

,11\o 111 141 tlic nunl~nc~lr dependence of flu1 densltl on the magnetlc field

\trcngth I \ rcpl,icsd h\ tht* equatlon H = LH". \+here 'I and 'n' are coefic~ents

der~vcd IIOIII the n~,ignct~\ttlciri Lune Th19 o p e of dependence has the advantage that

when n 1 11 hecome\ a Ilnear theory and when n=O 11 reduces to llmltlng theon

When 'n hd\ 4 \rlue hetween O and 1. it represents a character~st~c kt\veen the two

I~m~tlnp L,IIC\ \ '~th tvpc 01 approulrnatlon. the dlstr~but~on of field. eddj current

loss and power factor are depending on the coefficients n and k. The nonlinear

dependence of the form, B = p(H - E H ~ ) is also considered in [ 5 ] , where, E being

small. Based on this approximation, expressions are deduced for various quantities.

Further, it is modified to take account of presence of hysteresis. Later numerical

steady state solutions are developed by various authors to take exact shape of B-H

curve into account. One of such methods is Crank-Nicholson Method [9]. In this

chapter, a new classical numerical method called Pseudo-Spectral Method is

presented. Upon touching the linear theory, Limiting Nonlinear Theory and Crank-

N~cholson method, a comparison is made among the methods excluding Linear

'I'heory.

2.2 FORMULATION OF PROBLEM

Cons~dcr an infinite half-space of iron subjected to an alternating magnetising

force at the surface. An infinite half-space is defined as a region of a material which,

for example. extends from -m to + oo in the y and z- directions, and from 0 to m in the

x-direction as shown in fig.2.1. 'The surface of iron is chosen as the y-z plane of a

cartesian coordinate system, the x-axis is normal to the surface and extends into the

material. l'he magnetizing field (H) at the surface is in the y-direction and is

lndependcnt of z. Clearl!, at any point inside the material, only H, and J, (current

dens~ty). eslst. and they are purely functions of 's' and time 't'. owing to the

assumptluns ol' large wdth and length of iron along z and y-axes respectively.

Fig.2. I : Rectangular Coordinate System

Therefore. Maxwell's equations for this coordinate system are

The abovc two equations gne rise to equation of field distribution as

(or) simply

Where p is specific resistance of the material.

If it is assumed that the material is homogenous and isotropic, the fundamental

relationship between H and B is governed by the magnetisation characteristics of the

material and can be written as

B = f(H)

The boundary conditions are

(i)Ar x=O,H=H,Cos(T)

dH (11)At x =x, . - = O forallUT"

d s

(or)As X-0 , H = O forall"T"

Where. T = o t and subscript 's' stands for surface value.

2.3 LINEAR THEORY

If the magnetisation curve is assumed to be linear. then the equation (2.4)

becomes

B = p H (2.6)

Where 11 1s the pernieah~l~r) of rhe material. taken as constant.

Nou the equatlon ( 2 ;) can he wrltten as

Assume a solul~on for tl as

(or) H=H eJ"

and substitute in equation(2.7), leads to

where ?. = ,/$ The solution of second order linear differential equation (2.9), simultaneously

satisfying the boundary conditions of equation (2.5) is

Then

aH J = - = -(i, /I{ e"'"' "'I

P x I

And flux

Where

h, = i f i j

The real and imaginary parts of the solution given by equation (2.10), corresponding

to the surface magnerlzlng force being Hs Cos(or) and Hs Sin(ot) respectively. If we

assumc rhdr the surface magnetizing force is 14s Cos(ot).then the solution equation

To see the varlatron of H wth x, from the above equatlon (2.13), cons~der two cases

'I he above two equations reveal that the H-wave 1s propagating In llnear conducting

medlum I he phase and attenuatron constants are equal and glven by A. The d~stance

traveled by the plane electromagnet~c wave In the medlum IS one penod. IS termed as

itr wavelength In other words. ~t IS the d~stance between the polnts of corresponding

phases of t ~ o consecutive waves In the present case, the wavelength 1s glven by

2rr/)., and the phase front 1s propagated at the phase veloc~ty In the dlrect~on of x-axls

I he phase of a ua te IS decrded h! ( a t - hx) Slnce the derlvatlve of a constant IS zero,

the phase \eloc~t) IS grven bu (wlh) In a linear magnetlc material, w~th good

conduct~v~t) and hlgh permeablht), both attenuauon and phase propagatlon constants

are high. where as the wabe length and phase veloc~ty are very low Thus the flux

penetration Into the mcd~um 1s v~ewed more as a process of magnetlc d~ffuslon rather

than ar electromagnct~c uate propagatlon Also. a close examlnatlon of expressions

In lhls sectlon leads to certaln Important phys~cal lnterpretatlons These are

( I ) I he amplltudc ut magnetlvng torce and current denslty anenuates

e\porrentlall! In the u-d~recr~on, bes~des phase sh~ft In ldrt. a constant

, ir- P Ilk). can delinsd n l ~ c r e . ~ = 12 ha\lng the d~menslon of length and

A \ O i l

\lpnllylng thr dcpth ol pnetratlon 01 the tlu\ rns~de the medlum.

(ii) The magnetizing force at any layer below and parallel to the surface is

equal to the total current (per unit length in the y-direction) below that

layer,

(iii) The current density at any layer below and parallel to the surface leads the

magnetizing force, in time phase by 45'(electrical), at the same layer.

(iv) The total flux (per unit length in the z-direction) below any layer lags the

magnetizing force at that layer by 45'. Hence, the power factor is 1/42

2.3.1 Evaluation of eddy current loss

TWO methods of approach are possible for the evaluation of eddy-current loss

In the core

Method 1:'fhc aterage lnss o ~ e r one cycle of time period 'T' is

The ~n~egration IS ober the volume 'V' of the material. For harmonic variation of 'J'.

the ahove eupresslon ( 2 14) s~mplities to

I: ' ~ ~ / ~ , ~ , J ~ + J ~ + J ~ ~ A (2.15) \ 2 ) ,

Mcthod 2: Ihe energ! flab In the ~ntenal of t~me '"I into the material of volume 'V'.

lrnundcd ti! surface 'S'. using ~ h c Poynttng vector IS

If E = Es Sin(rot)

and H = HS Sin(ot-v)

Then equation (2.17) becomes

Where 0 is the angle between the vectors E and H and w is the time phase difference

between E and H

At this juncture, it is worthwhile to note that the power transferred by the

magnetizing winding must appear as the total losses in the core. This power is equal

to the product of the voltage required to balance the induced e.m.f in the winding, the

current through h e winding and the power factor of the winding. In general, the

problem is one of determining the induced c.m.f and the current in the winding. The

e m f induced In the winding per unit length in the z-direction is the same as the

elccmcal intens~ty kits. at the surface of the iron.

1huc

I,:,, = p J t s =pl l . ,~ l l se 'm ' (2.19)

Where J,, is pivcn hy the equation (?.I I ). when x = 0.

'Iherciore, ti,$ is the voltage applied per unit length of the winding. The current in the

wlndine per unit length in thc y-direction is Hrs = HS g'. Thus the average power loss

per unit surfacc area iron1 rhc cquation (2 . IS) is

Alternatively, the loss per unit surface area is given by equation (2.1 s), is

Substituting equation (2.11) for J and integrating results

2.4 LIMITING NONLINEAR THEORY (LNT)

2.4.1 Graphical Construction

l h e magr.e\isation curve of the material is approximated by limiting curve. as

shown in fig.2.2 where it is assumed that the linearity ends at small values of H. So

the initial step for graphical consvuction is to assume a small value for H at which the

lineari~y terminates. Using this H, the corresponding flux $ is found from the linear

theory i.e. from qua6on (2.12). Where as 'x' may be taken as very small value say

I x lo4 m. Then the values of H and 4 at an adjacent layer, at distance (x+bx) are

found by laking (he vwtor sum of ?f and AH. 6 and A4 respectively. Where.

and .W Is, AT

also . i l l , *1u4, (n=I .? .... , , I

. i @ R . ' t o l { n (n=I.:. . . ) i

In this wa) i l and 6 a1 s ~ r c c s s ~ ~ ' c layers arc computed. To reduce the errors. the

incrcmenrrrl deplh L Z ~ should bc as small as possible. it is suggested that for a known

valw of surfag cxc~tatioti tis. the compondiny value of flux density from B-H

curve ma). k chosen as Hm.

Employing the above procedure, the loci of H and I$ are obtained in 131. These

are shown in fig 2.3.

Fig. 2.3:Thc geometry of H and 4 loci

From the gcomnr) of I4 and 4 loci (fig.2.3). it is followed that

. d H " . = w?*\ S l n ~ p

B d on ph! slcal cons~derat~on. h e potter lass per unlt surface area is

Making use of cquerion (2.24) for Siny, and F.=t~i+. the equation (2.25) is modified as

Alw, the total power loss per unit surface area is

Equate equation (2.27) with (2.26) and solve for

Now. substituting for 4 in equation (2.24). so as to get an equation for Siny,

2.4.2 Particular solution of the problem 131

l,ct i.i=H Cos(or+e)

and BIB Cos(ot4)

substituting thew two equations in (2.3) and equating similar terms. yields

If ti assumed to be I I = li B, H'. &en the equation (2.30) can be normalised as

The normalized step-by-step graphical construction outlined in section 2.4.1, is in

effect the solution of nonlinear simultaneous differential equations (2.3 1 a) and (2.3 I b)

respectively.

There appears to be no general solution with arbitrary constants. However, one

could recognize the above two equations as analogous to the dynamics of motion of a

partick with radial and uansverse accelerations. It is well known result that, if a

particle moves in an equiangular spiral and if the radial acceleration is zero, then the

kansversc acceleration is proportional to square of distance. Stated explicitly, if H' is

propontonal to E"". then the above equations (2.31a) and (2.31b) holds

simul~anrousl!~

As a consequence.

Whcrc d IS an wbttran constant.

If11 IS a s s u n d that d-0. thc equation (2.32) reduces to

Once the law of variation of H with x is known, the relevant quantities could be

deduced using the equations (2.26), (2.28), (2.29) and (2.34). The derived equations

an as follows:

Loss per unit surface area

Flux per unit length of perimeter

Power factor

Total depth of penemtion

Wherc h e subscript 's' is an indication of surface value.

2.4.3 Limitations

i I'hc actual distortion of H at each layer cannot he visualized. since it is

a s u d tha~ the magds ing force at each layer is having only

funduncntol component.

(ii) The results of the graphical conswction cannot he used directly for plalcs

of finitc thickness.

2.5 CRANK-NICHOLSON METHOD (CNM)

An implicit scheme for a parabolic partial differential equation (2.41) leads to

a system of algebraic equations after replacing aWh by forward fmite difference

approximation and dw0x2 by the arithmetic mean of its central difference

approximations on the i" and (i+l)" time-rows. The figure 2.4 shows the

discretization. The computation of unknown pivotal values requires the solution of a

system of linear equations. Though the implicit scheme is iterative in nature, but the

mahod will converge for all finite values of ATIAX~. Whereas the explicit scheme is

valid only for (A?'/Ax2 ) s 0.5. Therefore, this restriction, necessitates a very small

time step, making h e method computationally uneconomical.

I. \

0 I t

!I< ,! k t 8 t

* ) I t L t

r - 0 1' 0 0 H - H s C m )

I .T

Fh 2.1: Gnd showing boundrry and initial conditions for psraholic PDF

23

The magnetic curve has been repredmted by the Frohlich cquation, given by

The equation (2.3) can be written as

What aB/&i is the slope of the curve at a point calculated by taking the derivative of

quation (2.39). Hence, the equation (2.40) reduces to magnetic diffusion equation,

P(Y + /HI)" where S(H)=-

a T

In g m d the finite difference scheme to diffusion equation is given by

W k Fn is Ihc second derivative of H at the n* time-step.

'Ihe above cquatlnn (2.42) glvcs different schemes depending upon the value of 8.

( I ) 11'84. Explicit xhcmc

(i~) 07 I . I:ull! implicit scheme. a d

(iii) 0-0.5, Crank Nicholson schm

In the implicit scheme of Crank-Nicholson Meltrod, as shown in fig 2.4. the finite

diffcrmcc appmximntion to quation (2.41 ) is

A h nanangemcnt, the above equation (2.43) becomes

AT' Whm K = --S(H) AX'

I t should be notcd that in equation (2.44), the LHS contains three pivotal values of H,

which are all unknown, whnras h e RHS values of H are all known. If there are N - 1

inkmal mesh points. then for i = 1 and j = 2, 3, ..., N . equation (2.44) gives a system

of (N- I) lincar equations in (N-I) unknown pivotal values of H along the second time-

row. Thex are in terms of known initial values at first time-row and boundary

vducs. Similarly, for i = 2, j = 2, 3, ..., N , equation (2.44) gives a system of (N-1)

lim equations in (N-I) unknown H's along the third time-row in terms of the

computed H's along the second time-row and ~o on. The coefficient matrix formed by

thcx (N-I) linear s~multaneous equations is tridiagonal. Consequently Thomas

algorilhm is implemented to solve the equations. It may be noted that the number of

operations (only multiplications and divisions) with 'N' grid points are 5(N-2)-4.

Hcncc, thc duction in computation time.

23.1 ModiCkd Cnak-Nicbobon Metbod

A modifiwian IS done to the standard Crank-Nicholson Method. The modification is

given hclnw in a scqucncc of steps:

(i) The nodal values are computed for one full cycle of surface excitation

hs = HS COBCT),usiq the initial values ( m s ) at tbe h a t - t h e r o w T = 0

and boundary conditions;

(ii) The initial values at T = 0 an replaced by the nodal values at T = 2x;

(iii) The new values a! the sccond time-row ~ T C computed u s 4 the wised

initial values md slopes. nK slopes arc determined using the average of

n v i d initial values and old values at the second time-row at each layer;

(iv) Thc new values at the second -tow arc compared with the old values.

If h e didiffaence is m m , the old values will be replaced by the new values

a1 each value of x and step (iii) will be hpoated. This is done till the error

criteria is satisfied, say e = 0.1 ;

(v) In the same way the nodal values at all other rows are computed;

(vi) To achieve symmetry of the p e m d q wave, the nodal values at

T = a a r c compared with the values at T = 37u2 at all layas, if the

dilTmacc is more than 5 , h whole pcedure otamng from ~tep (ii) will

be repeated.

2.6 PSEUDOSPECTRAL METHOD (PSM)

A new c l a s of m & d s for obtaining numerical solutions of partial

diffcrcnl~al equations are known as spectral methods (17). Spectral methods may be

viewed m an extreme development of class ofdixrelizalion schemes for differential

equations known genericrily as the Method of Weighted Residuals (MWR). The key

elements of the MWK arc the vial hrnctions (also called expansion or approximate

functions) and test functions (also known as might functions). Ihe trial functions are

wal u the W s f u ~ ~ t i o n s for a truncated scrim expansion of the solution. The t e ~ t

functions are used to ensure that the differential equation is satisfied as closely as

possible by the truncated series expansion. This is achieved by minimizing the

residual. An equivalent requirement is that the residual should satisfy a suitable

orthogonality condition with respect to each of the test functions.

The trial functions for the spectral methods are infinitely differentiable global

functions. In the case of finite-element methods, the domain is divided into small

elements and trial function is specified in each element. Thus the trial functions are

local in character and well suited for handling complex geometries [18,19]. The finite

diffmnce trial functions are likewise local.

The choice of test functions distinguishes three most commonly used spectral

schemes. Namely, the Galerkin, Collocation, and Tau versions. In the Galerkin

method, the test functions are the same as the trial functions. In the Collocation

method, the test functions are translated Dirac delta functions or unit impulse

functions 6(x - xi), centered at special, so called collocation points. This approach

requires that the differential equation to be satisfied exactly at the collocation points.

Spectral tau methods in the way that the differential equation is enforced. However,

none of the t a t functions satisfy the boundary conditions. Hence, a supplementary set

of equations are used to apply the boundary conditions.

The commonly used trial functions are

(a) Fourier e' '" (b) Chebyshev TL (XI

(c) Legendre LL (x)

(d) Spherical Harmonics Ynm(x ,y)

The Fourier functions can be used only for periodic problems, whereas, Chebyshev

functions can be used for the expansion of pcriodic as well as non- periodic functions.

All the trial functions are orthogonal functions. Chebyshev functions are defined in

the interval -1 5 x S 1. However, a function can be approximated in the region

a 5 y 9 b, employing linear mapping.

If a = 0 and b - 1, the span of space is reduced to half the total value and the equation

(2.45) reduces to

x = 2 y - 1

(or) ax = 2 ay

The above equation (2.46) can be written as

Further,

d ! ~ a2u (or) --;=4 - as* ax2

a ? ~ a2u lngeneral = K' - as. ax2

Where K is the factor by which the span is compmsed.

The approximation of a function f(x) is given by a finite Chebyshev series with N

tnmsss

When a ' s are spectral coefficients. These are computed like Fourier coefficients by

making use of the orthogonal property of functions. The orthogonal property of the

Chcbyshcv trial functions is

WhereJd is the Kronecker delta function defined by

1, m = k 8mk =

0, otherwise

and

I 2, m = O C,(or) C, =

[ 1, otherwise

Upon multiplying the equation (2.49) by T,(x)/d(l-x2) and integrating fmm

x -- 1 to l ,yields an expression for the spectral coefficie~its as

2 / fN(x )~m(x)d* a,,, =- nc, , m

The grid points or collocation points m re chosen that the accurate evaluation

of integral of equation (2.51) is achieved. The Gaussian quadratures an very accurate

means of numerical integration, whmin the choice of weights and collocation points

arc found, so as to get the best approximation of an integral. The integral is estimated

by using a weighted sum of the function values at the collocation points. That is

1 with W(x) = -

\ll_X2 Where

W fx) - a non-negative weight function

fN(xj) - the function values at the collocation points xj .These values are

winen as fj,; j = 0,1,2,.. .,N.

aj - the weights assigned to this hction value.

For the Chebyshev, the collocation points (nKse are called as Chebyshev-Gauss-

Lobano points) an defined by

This set has the end points (x = f I), which makes it easier to impose

boundary conditions. Moreover, not only does this choicc of uneven spacing produce

highly accurate approximations, but also enables the fast Fourier msfonns to be

implemented.

For the Chebyshev - Gauss - Lobana quadratures. the weights are

Replacing the RHS of equation (2.51) by the quadratures {equation (2.52)}, gives

Where Tk (x,) are the Chebyshev polynomials.

The equation (2.55) can be evaluated by fast Fourier transforms.

The Chebyshev polynomials are defined by

T, (x ,)= CosF Cos-'(x)] (2.56)

With the uneven spacing, given by the equation (2.53), the equation (2.56) reduces to

The equation (2.55) can be written in the matrix form

Where the elements of A are given by

withcO=cN=2;ci= 1,lSiSN-I.

One of the methods [I 7,201 to waluate derivatives of function f(x) at the collocation

points is as follows:

The second derivative of equation (2.49) is given by

Also

Where the superscript '2' within brackets indicate the second derivative.

By substituting quation (2.58), the equation (2.61) can be brought into matrix form,

where the second derivatives an in terms of function values themselves.

The entries of the matrix B arc obtained by differentiating equation (2.56) with

r e ~ p e c ~ to 'x' twice. The elements an given by

with x = Cos(njM), j = 1,2, ..... N-1 and k = O,1,2 ..., N

And at x = 1 i.e. for j = 0

B (OM = (k4-k2)13; k = 0,1,2,. .,N

At x = -1 i.e. forj = N

B(N,k) = (-lf (k4 -k2)/3; k = 0,1,2 ,..., N

The simplified form of equation (2.62) is

It has been verified that the second derivative matrix 'C' of equation (2.64) with the

square of first derivative matrix given in[ Canuto et.al (1988)l and found correct.

2.6.1 Implicit method

Making use of quation (2.64), the implicit time stepping scheme to equation (2.41) is

above matrix equation (2.65) is written as

nrl 0 (Ha

(2.66)

\ H N J \HN

w h m Cl(i, j) = S(H$xC(i, j), i, j = O,l, ..., N

Finally the matrix equation (2.66) is simplified as

Where the superscript denotes the time stcp. The boundary conditions given by

equation (2.5) arc incorporated in to the coefficient &x 'D', by changing the

elements of first-row and last-row accordingly. That is

(i) D(1,I) = I, D(1.i) = 0, i =2,3 ,.., N

(ii) W , N ) = I, D(N,i) = O; i = 1.2,. . .,N-I

(iii)Ho"is equal to the surface value and HN' = 0

The solution of equation (2.67) with the boundary conditions. is the solution of the

problem. It may be noted here that the derivative boundary condition (Wdx) = 0,

at x = XN of equation (2.5) can also be implemented by making the following;

(i) D(N,N) = -D(N,N-I) = I ;

(ii) D(N,i) = O; i = 1,2 ...., N-2.

(iii) H N ~ = 0

. Since H(N,N)-H(N,N-l)= Ax

2.6.2 Featurea

In most practical applications, the benefit of the spectral method is

(i) Not the extraordinary accuracy available for large N, but rather the small

size of N (necessary) for a moderately accurate solution,

(ii) It is well suited to transform techniques,

(iii) The uneven spacing is not only economical but also helps to study the field

very closely at the surface.

2.7 EXPERIMENTAL TOROID OF REFERENCE 131 AND ITS CHARACTERISTICS

2.7.1 The detail9 o f Toroid

Material

Resistivity

B-H Curve

External diameter

Internal diameter

cross-section

Perimeter of section

Mean circumference

Number of turns of magnetizing winding

Number of secondary turns

2.7.2 characteristic^

The Magnetization curve of toroid is given in figure 2.5 and it is represented

by equation (2.39). with a = 1.803 T and y = 935 Alm.

: Mild Steel

: 18.5 x 10.' R-m

: Fig.2.5

: 0.3366 m

: 0.2921 m

: 0.0222~0.0143 m

: 0.071 7 m

: 0.987 m

: 800

: 500

2.8 EVALUATION OF EDDY CURRENT LOSSES BY DIFFERENT METHODS

For the purpose of cornperison of eddy nvrsnt losses of three different

methods, the miice excitation Hs = 16000 Mm is chosen.

2.8.1 Liming Nonlinear Theory (LNT) 3r

The loss per unit surface area is calculated using equation (2.35)

When B, = 1.705T at Hs=16000 Ah, from the B-H nwe, shown in fig. 2.5

F k Z.S:B-H Curve of the specimen

2.8.2 Modlfied Crank-Nicholron Method (CNM)

The outcome of the sequential p d u r e discussed in section 2.5.1 is given in

Table 2-1. Knowing the grid values for one full cycle of surface excitation, the

Fundamental Components of Magnetizing Force (FCMF) at the specified layers are

computed. Also the current density at the same layers is determined using the finite

difference formula (2.69). Both results are given in Table 2-2.

Table.2-I: Final Nodal values of Crank-Nicholson Method

Table 2-2: Variation of FCMF and current density of Crank-Nicholson Method

When the superscript 'k' indicates the layet number

The subscript 'F' stands for fundamental quantity. SO

H F ~ is the FCMF at x =0, lx10'.', 2x10'~ ,...,

H?' is the FCMF at x = &Ah, lxl@+Ah, 2 x l ~ ~ + A h , ...,

With fi =1x104 m

It may be noted that the nodal values at Ix 10" + A h , 2 x + Ah,..., are not

provided.

The iron loss per unit surface area is calculated using the Poynting equation given

below

P, = O.SJ,pH,Sin~,watts (2.70)

When Sinys is given by the equation (2.37).

2.8.3 PstudoSpcetral Method (PSM)

Having taken zero initial values at the first time-row T = 0, the nodal values at

the second time-row are computed using the equation (2.67). With the computed

nodal values at the second time-row, the nodal values at the third time-row are

computed considering the nodal values at the second time-row as initial values and

using the equation (2.67). Similarly, the nodal values at all time-rows are obtained and

summarized in table 2-3. The field distribution at various layers is also shown in fig.

2.6. The FCMF and the c m n t density at different layers, as mentioned in table.2-3..

arc evaluated. The results an given in table.2-4.

Table 2-3: DiaMbution of field by Spectnl Method

Table 2-4: Variation FCMF and current density of Spectral Metbod

Fig. 2.6: Distribution of magnetic field strength in Space and T i e

The iron loss per unit surface area is calculated using the equation (2.70). The

computed values of Eddy Cumnt losses by three different methods along with

practical value [3] at Hs = 16000 AIm am tabulated in table.2-5.

TaMe 2-5: Comparison of eddy current losses by three different methods with

experimental value (101

9 150 watts 9779 watts 8592 watts 8966 watts

2.9 COMPARISION OF RESULTS

The simulated results of Pseudo-Spectral ~ e t h o d are compared with that of

Crank - Nicholson Method and experimental results of the same toroid. It is clear

from the figures 2.7 and 2.8, that the simulated results of Pseudo-Spectral Method are

close to the practical values compared to that of Cd-Nicholson Method. It may be

noted that the value of N for Pseudo-Spectral Method is 16, whereas for the Crank-

Nicholson method i s 80, but the AT1 A x ratio is same for both methods.

Fig. 2.7: Variation of Iron losses with Field intensity

0 WOO loo00 15000 20000 25000

----+ SurI .a Magnetizing Fom HD (Alm)

Fig 2.8: Deviation of theoretical iron losses from thc experimental values

2.10 CONCLUSIONS

Diffcren~ methods of evaluating eddy current losses have been compared. It is

obxwed that the proposed Pseudo-Spectral Method yields results of reasonable

accuracy with less number of nodes. Moreover, this method is suitable to use Fast

Fourier Transforms. Therefore. h e proposed method is suggested as an alternative to

the existing finite diffennce and finite element methods.

Chap ter-III

Evaluation of eddy current loss in solid cores under double excitation

3.1 INTRODUCTION

The second chapter dealt with the problem on single excitation in detail. In

this chapter double excitation field problem is formulated. A closed form solution is

developad for this field problem. The same problem is also solved by Crank-

Nicholson numerical method. In addition a graphical solution is also provided for

supplementation. Finally an experiment is conducted on two identical toroids.

In electric machines harmonics are due to many reasons like saturation,

irregular gap length, type of winding and slots, unbalanced loading etc. Mainly the use

of switching devices would inject time-harmonics. Thus, the air gap m.m.f of an

electric drive, not only have fundamental component but also harmonics, of course,

here time harmonics are only considered. When, the rotor of an induction motor is

subjected to such an m.m.f, the magnitude of eddy current loss increases considerably.

Induction motors with solid iron rotors are in use. Therefore, it will be useful to know

the eddy current distribution in the rotor. Extensive work has been done on single

excitation. But, for various reasons, attention has not been given to research on dual

excitation, especially in the area of electric machines. Theory of dual excitation is very

well applied in the field of Control Engineering. Indeed, it can also be used to analyze

eddy cumnt losses in electric machines. For the purpose of evaluating eddy current

losses, the input excitation to the rotor can be considered as the addition of

fundamental and the harmonic of highest magnitude. neglecting all other harmonics. In

other words, the non-sinusoidal excitation can be represented as the sum of two

sinusoidal s i g d s of fommmurate frequencies i.e, the hquency of one signal is the

multiple of fnqucncy of other signal. There are cases, where the muencies are

incommensurate.

3.2 THEORY

Let

and assume the following

(i) At any layer, the components of H, of angular frequencies o, and02 alone

exist,

(ii) The flux dmsity at any layer consists of above two components of H only,

and

(iii) The input arc treated as incommensurate.

Hence the flux density at any layer is given by

when BI, Bz are fundamental components of output comsponding to the input

components HI and H2 respectively. The evaluation of BI and B2 are discussed in

appendices A3-I & A3-2 for the given non-linearity (B-H curve), both for

commensurate and incommensurate frequencies.

Substituting equations (3.1) and (3.2) in (2.3) and equating similar terms, yields the

following equations

3.2.1 Solution of Equation (33)

The magnetization curve of the material can be approximated by a relay type

curve such as shown in fig.2.2 @18). For this relay type B-H curve, the formulae for

BI and B2 are given in [21], as

Where Fl(A). Fz(5) are yet to be defined.

The flu density at saturation B,, is the flux density corresponding to the given field

strength (H). Hence, it can be read from the actual B-H curve (fig.3. I).

But in this project, the B-H curve is peplaced by rational h t i o n formula as

Where, the values of = -8.6816~10-'. a1 =1.0213xl@, a2 = -1.1625x10"~.

a, = 5.0086~10"~, a, = -6.5553~10"~. yl = -6.3403~10". y2 = -4.0462~10'",

y, =2.0808~10.~~, y4 = -2.8879~10"~ and H = (HI + H2)n. Still for accurate

representation, the series should be extended.

+ Magactbing Force H (Mm)

Fig3.1: Magnetic c w e

Since it is assumed that the two frequencies are incommensurate i.e. the ratio of

( W ~ I O I ~ ) is an imional number. Then, with h = (H2M1) < I , Fl(h), F2(h) are given by

When E(A), K(A) an elliptic integrals of the first and second kind respectively.

46

It may be noted that the values of F,(h), F2(h) of incommensurate frequencies

would approach their counter parts of commensurate frequencies for large ratio of

(~~2/01)[22]. By expandii the elliptic integrals into power series and dividing by

(41n) (because of first assumption made in section (3.2)), the equation (3.7) becomes

x2 3x4 5h6 F,(h)= 1 ------- [ a 2 5 5 ... ]

If the ratio, (H2Ml) > I , then it can be shown that the above two equations (3.8a) and

(3.8b) must be interchanged for Fl(h) and F2(h.) with I.= That is

with q = (02 /@I), using equation(3.5). the equations (3.3) and (3.4) can be written as,

9= Hl(!!L)2 dx'

dx'

(3.1 Oa)

(3.10b)

(3.1 la)

3.2.1(n): h - (H1/Hl) S 0.4

Now the equation (3.8) would become

Hence, using the above equation (3.12a), equation (3.10) is simplified as

3 = dx'

The above equations (3.13a) and (3.13b) are similar to that of (2.30a) and (2.30b)

respectively. Hence the solution of simultaneous non-linear differential equations

(3.13a) and (3.13b) is given by the quation (2.34). That is the solution or the law of

variation of HI with 'x' is given h e n for convenience,

HI = a x ? (3.14)

Where a = o l ~ ~ t ( 3 ~ d 2 ) .

3.2.2 Solution of equation (3.4)

Making use of equation (3.12b). equation (3.1 1) is written as

&mine a solution of the form

H, =axP

and substituting in 3.1 S(a),gives

Now substituting (3.1 7) for (d02/dx) in (3.15b), yields

After differentiation and substitution of (3.14) for HI, the result is

It may be noted here, ha t the unknown p depends only on hquency ratio 7. By

inspection it is clear that p > 2, when 7 > 2 and

The inference from the above equation (3.19) is that the high frequency signal

attenuates faster than low frequency signal.

For the given value of q > 2. the p is found from equation (3.18) and substituted in

(3.16) so as to obtain solution for Hz.

33a8): h - (HI I Ha) 5 0.4

Now the equation (3.9) reduces to

Consequently, the solution of HI would be the solution of H2 and vice-versa. Also note

that the quation (3.18) would become

Further the statement of (3.19) holds good. It is concluded that the derived

equations pertaining to signal HI would belong to signal Hz and vice-versa, after

replacing HIS and (01 by Hzs and (02 in all the relevant equations of the next section

(3.3). Of course the parameta P should be h m equation (3.21).

3.3 DETERMINATION OF LOSSES

3.3.1 Computation of eddy cumnt leu, flux, power factor, etc., due to HI

The loss per unit surface area, flux per unit length etc., are given by the equations

(2.35) bough (2.38). They are as follows:

Loss per unit surface area i s

Flux per unit leng& of perimeter is

Power factM at the Burfece is

Sin y,, =

Depth of penetration is

33.2 Computation of eddy current loss, flux, power factor, etc., due to Hz

Knowing HI = a xP, the following formulae are obtained, using the equations (2.27),

(2.28), (2.29) and (3.16)

3.4 GRAPHICAL SOLUTION OF THE PROBLEM

The closed form solutions derived for HI and Hz given by equations (3.14) and

(3.16) arc valid for q >2. However, a graphical solution is suggested for 2 2 q 2 1 is

as follows:

00258n

Consider the equation

~ c F I ( ~ c ) ~ F & c ) (3.30)

Whaek,=(H1/H1)<1.

It may be noted that, in the above equation (3.30) the range of T( is 2 2 tl rl, for

0 5 h , 5 1. Moreova, for this range of q, HI and H2+0 as x+O, but the ratio of

starting values (~2'') / HI(') )+li c. Where the superscript '0' within brackets is an

indication of starting value.

In this graphical construction, the values of HI and I$, at x = x are assumed to be

known. Then, the problem is of determining graphically the values of

at an adjacent layer, i.e at x(O) + Ax.

The values BI and B! are given by (3.5).

also AH:"' 1 to 4:"' (n = 1.2, ...) 7

A+:' 11 to H:' (n = 4 2 . 4 J With the above information tbe geometrical construction can be made for and HI,

which will be same as tha~ shown in fig. 2.3 (p19). Similarly, the loci of 41 and H2 can

bc drawn.

3.4.1 PrOadnre

(i) Select a value for q say 1.2,

(ii) Daamine kc using (3.30), so that RHS equal to LHS,

(iii) Choose the starting values as HI(') = 1, ( ~ 2 " ' iHl(O)) = A('), with (0, = 1,

B, = 1 Tesla, and p = 10" R-rn ,

(iv) Using the equations derived in section 3.3, of course with P = 2, got the

other starting values, namely;

W , ( O J = 2(0J sin.! 0 213

(v) Fix the surface ratio k, = ( H f l ~ , ) ,

(vi) Choose I.(')= A , if LS = LC,

h"'= h , + c ~ , if k > h c ,

A'O'=hC-c7u , i f &<LC,

Where E is fractional value.

(vii) Knowing the values of HI, H2. (11 and at x"', find the values of HI,H2,41

and +z at an adjacent layer x"'+Ax, using the equation(3.31). In the same

rnmer vecrorially find out the values of H, 4 and P.F at subsequent

layers.

(viii) Find the power losses PI and P2 employing Poynting theorern{eqn. (3.39))

34.2 Normalhation of graphical construction

The pphical constrution is carried out for col = 1, q = 1.2, B, = 1 T w 4

p = 1 x 1 0' n-rn, and Ax 5x 1 0' m . The results an tabulated in the table.

However, using normalization technique, results can be obtained for any values of col,

Bm and p by adapting the following scale factors;

Scale factor for HI and Hz is

Scale factor for and 4 1 ~ is

Scale factor for PI and PI is

Scale factor for x is

Where Hln,(ol,.Bm, and p. are h e new values of H1,ol.Bm and p respectively.

TABLE:

Rerulb of Computerized graphical construction

3.4.3 Uw of table

The outcome of normalized graphical construction is provided in the table. These

r e s u l ~ an useful to find out power losses for any data. But q should be 1.20. Because

graphical construcrion is carried out for q=1.20. To illustrate.

( i ) Let HIS = 36500 Aim, Hzs = 51500 A/m, wl=261.8 radlsec, or = 314.16

rad/sec. (So 1-1.2). B,= 2 Tesla p =15.lxlO"~-m.

(ii) The scale factor for H using equation (3.33) is,

k(H) =34.68

(ii) The field ahengb of 1 3 ~ row, when they are multiplied by k(H) are e q d

to the given values of HIS and Hzs respectively.

(iv) The scale factor for P is found using equation (3.35),

k(P)=18158

(v) Now multiplying the power losses of 13" row by k(P) gives,

PI = 29172 watts/m2 P2 = 69910 watts/m2

(vi) The scale factor for 4 from quation (3.34) is K(4) ~2.0.

The curves of field strength, power loss, flux etc. of each signal are drawn after

multiplying the table values by the above scale factors. The curves are shown in

fig3.2, so that one can have freedom in choosing either HIS or Hzs.

BOOOO

Fi.3.2: Results of graphical method 5 6

3 5 NUMERICAL SOLUTION OF THE PROBLEM -CNM

Let the excitation at the surface of a material is given by

H, = ~,,sm(u,t)+ H,, ~ m ( w , t )

and the boundary conditions & ~ r u d condlhon are

(I) at x = 0, H = Hs

(11)at x = X ~ , ~ H I ~ X = O foraliN?*'

(111) at T = 0,H = 0 forall "xu

By ernploy~ng the Crank-Nicholson Method described In sectlon 2 5, the field

distnbutlon for the surface values of HIS -7070 A/m at ul=261 8 radlsec and

HzS =I 4 14 A/m at 0 2 -3 14 2rad/sec 1s obtuned and shown m figure 3 3 Havlng found

the field drstnbutlon the fundamenull components of two Input signals are calculated

at each laya Subsequently the power losses are d e t e m e d using the follow~ng eqns,

H 'L' - ~ , t ' ' + l ' where J , , = -L-- ~ = 1 . ? . L = O AX

Where BI, Bz can be evaluated using the equation (3.5) and Sinylt= SinyZr 0.8165

3 Time in mKc.

Fig, 33: Field profile

Along with fundamental components, various harmonics ere determined at

different layers. Consequently, hannonic current densities an evaluated using the

quation,

H!U - ~ ! k + l ) J,, = Jh J h

Ax

For each j =1,2; fix h =3,5,7 ,... and run k from 0 to N.

Where subscript 'j' refers to the input signal number and 'h' indicates the

fundamental and harmonics depending upon its value. The superscript 'k' pointing the

layer number.

But the figure 3.4 shows only the current densities of frequencies same as

those of input signals.

Fig 3.4: Variation of c a n 1 densities with depth

3.6 EXPERIMENTAL PROCEDURE AND RESULTS

An apcriment wrlla conducted on two identical tomids, the details of which

en provided in the llppendix (A3-3). The primary windings are connected in padel ,

whereas the mndary windings an so connected in series such that the emfs induced

in them due to primary ArnpTums (AT) will mce l each other. Therefore, the

interference caused by the primary AT is neutralised at all excitations. The flw

produced by the seconday AT would induce an emf in each of the primary windmgs

and hence circulate cumnt in the i n t d circuit, since the primaries are connected in

parallel. To swamgout this circulating current, two equal high value resistors are

inserted, one in each branch, so that the AT measured on each side would be

comsponding to the respective excitations and loading effect on secondaries is made

negligible.

The primaries, which are connected in parallel, are excited by 48 Hz supply.

Whereas the hese combination of the secondaries are excited by 16 Hz signal. The

low frequency signal is generated by A.C gemtor. Power losses on both sides are

m d . keeping the magnitude of low frequency signal constant 'and varying the

magnitude of the high frequency signal. Two watfmctcrs were placed on the primary

side, one in urch of the parallel branch. The average of these two meters readings is

taken for fmhcr calculations. The iron losses are found out from the measured input

powers. The wults of analytical solution and Ihe simulated values of the Crank-

Nicholsan Method an compand with the experimental power losses. From the figrne

3.5, it is known that the theomical values of power losses are close to that of practical

values.

----+ Surfsee Fkld Strength H=(A/m)

Fig 33: Iron loss Curves

3.7 CONCLUSIONS

An analytical method is devised to evaluate eddy current loss in ferromagnetic

W ~ S , when they an subjected to two sinusoidal signals of different frequencies,

under saturated conditions. To verify the validity of the proposed method an

experiment was conducted on two identical toroids. It is observed that the calculated

p o w r loss of analytical method a d simulated results of numerical method are

asrseing with the experimental values of p o w loss. To complete the project, a

gnphical solution is alsa suggested.

6 1

IU: APPENDIX

A3-1 Evaluation of fnndamentai component8 of output wing M-functions

(a) Commaunrate frequencies

The magMtisation curve of any machine is nonlinear in nature. Whm this

nonlinearity is subjected to two sinwidal signals of commensurate fkquencies, the

output wave is analyscd by Fourier series, to find the response of the element.

Consider an input

The resulting response is given by

B= f ( ~ )

This output will contain fundamental components of frequencies w, and 02, their

harmonics mu1 and nwz and combit ion frequencies (mal f noz), where 'm' and 'n'

arc inlegen.

In order to determine these wmponents, the output wave should be analysed

by Fourier method over a p a i d of time T, during which both signals complete an

integral number of cycles. These components appear as certain harmonics of the wave

being analysed.

Thus, fundamental u l and 0 2 . harmonics ma!, nol: and combination (m cul f n 02)

fmqucncy components in the output arc given by

7 1 B,, = - j f [ ~ , ~ i n ( o , t ) + ~ , ~ i n ( o , t ) ] ~ i n ( o , t ) d t

7.0

2 B,, = - l f l~,sin(m,t)+ ~ , ~ i n ( r o + ) ] ~os(0, t)dt To

What the subscripts p&q stands for in-phase and quadrature components.

Similay, the 0 t h cqdm for hrmonicc ad combitdon fr#luaxy componmts

can k mitten.

0) ~ m m e u ~ t e frqosoeicr

The mahod described above is atmightforward for the case, whca the ratio of

frequencies is a rational number. But one C I I M ~ ~ ~ ~ S m a p p m t difficulty, when this

mtio is irrational. Since for i a c o m t c liqtmcica, tbert will k no finite period

of time, however large, during which the two signals will have an integral number of

cycles. However, this difficulty is overcome by using M-functions (231:

1 " B,, = - ] ~ ( h , , H, ) ~in(ro,t)da,t " 0

So Ihr Dual Input Utmibing F d o n is given by

The cquauon (A3.4) giva the b e f u n d . m e n l p l component of Ihr output of nonlinear

charsctmnics M(hl.H2). whcn its input hi varies sinusoidally with time. In 0 t h

words. in the presence or g i m amplitude ti? of Ihc second signal, the original

d i n c m ~ y hehaves with respect to the signal hl. as if it is modiftoti to

B(hlb M(hl,H2). The cvallution of the equation (A3.4) inwlvu two slagcs. In the

hru we h e dallcnd chrrtaiics arc obtained. Whcnu in rhc second a g e Iht

Dud Sinwoidrl Input chnccuidicr rre fiwd from the Altacd dunctcr i s tk~ .

(c) Dctemht&r of mered mmcrddia

&#d an quation (A3.5), for a givea B-H m, tbe crhierrd d m a e m w . .

cia k obPintd M follow. Suppo~ tk low b q m c y u@ HI Sin (colt) = h ~ , . tbe opnntine point on the B-H curve is at point YI md the oomsponding outpul is MIYI

as shown in figure 3.6. As the high fiupmcy signal HZ Sin((02t) varies, the opmting

point moves on the B-H curve accordingly.

--

~ f a o s a r m g k t s ~ o f h t h c ~ c a u g n t t w i l l b e M ~ D ~ , w h i c h b

di f l6s lrathYY, ,bsPtreoftbs~l l l lorrofthcEHcumrboutthc

point Y1. In r dmilu mmna, the otbapoim qD3 ,... ,DN uc., r q m a h g the

aurpltr CUI k OW when thc S@IBI HI S i m l t ) ~ tk vahrs hla

h~ ,..., ac, with H2 being kept fixed. 'lhur tbc atrrrc OD14 ... b called as AHend or

MdiW Bu. For diffaent vdup of Ha, Modified chranaistic~

h w n uul shown in Iig. 3.7 for the &a~ BH cum (@.3.l.p46).

Fig. 3.7: Altered chactmistics

(d) Dul Mwebdal hprt (DSI) ckmcwwa

B a d on the cqurtioa (A3.4). the tidmmul wqwmi Btr of output

cormponding to HI, is the b d m e d ampomnt of Attaed C-cs (BI 1 hl)

obtained for canstant values of Hz, w h hl V ~ W sinusoidally.

Thc cuwcs BIFVsHl f ~ f differ~nt V ~ W of Hz an call4 DSI C ~ C S .

The DSI curves comspoPding to the altered (figure 3.7) arc shown in

figure 3.8.

FikJ.8: DSI curves

~ * f O r t h e p l r p o w o f ~ e d d y a t n s D t l a s s , t b c E H c w e ~

k c l l l p p r o w i m m d b y ~ ~ p e d i n e a i t y , m ~ l 0 ~ n s d i l y ~ 1 e f o r m ~

to camputc output of each of the input componcot.

A3-2 Evrl~rttoa of fuodrmmtd wmponcne of output wing power w h 121)

For accutetc evaluation of BIF and BZF, npredctlt tbe B-H curve by Frohlicb

equation as

The describing function of the signal H I , when Ht=O, is

AAer integration.

T k describing W o n of signal H I ia the pnscnce of signal Hz, is given by the

power series expansion technique, as follows

Whm V,(HI) is computed recursively ss

with V d H t ) = Nl(H1).

In r h d m way, M an obtain dwaibiag trmction of signal Hz, w h HI

pmmta. Tbe d e a is

and

Similuty, h e tnmcalcd series of equation (A3.10) is

Ihc fundamental component of outpu~ corresponding to the fundamental component

of lhc Input 14, 1s

Similarly.

l l w n d t a of pomr d c s au meaninphJ only when HI >> 7. The above equations

(A3.13) ad (A3.14) for Blr and B p are vdid for(HW~)< 1. If ( H ~ I ) > 1, then the

samt qurtiom must be intmcbqed for BIF ad BZF dong with rcphcuncnt of HI by

Hz and vice-versa.

A3-3 Tbt d&iL of cacb toroid

Material

Resisdvity

B-H Curve

Diameter of cross xction

M m circumference of ring

Primary number of turns

Secondary number of rum

D.C mislance of each winding

: Mild steel

: Fig. 3.1@ 46)

Double excitation theory-Induction Motor

4.1 MTRODUCnrON

In chapter-111, it has been ma~t iod that t& time hrmonics will be g d

in tbc supply due to switching device& Wbcn clcdic motors m givcn wch a wpply,

the iron lows would increase wnsidmbly, especially if the rotor is a solid one. The

iron losses in thc solid-iron rotor caa be estimated by representing thc mn-linear

excitation as thc sum of fundamental and the harmonic of highest magnitude. But thc

test bas been performed on single-phase induction motor by feubg thc fundamental

excitation al 50 Hz and the harmonic excitation at 450 Hz. This type of problem can

be called as rmnxcitation problem or dual excitation problem. One would come

rrrou two excitation pmblem in the feedbeck control systems, where to stabilize the

main signal, a high Freqwncy signal is ycacd in to h e system, of-course the

frequency spatation be~wecn thc IW signals is high. The numerical solution of the

tmrsxcilrtion field problem is achieved by W o - S p t r a l Method [13].

A canpulcnrd gnphiul method is rlso dcvclopd to fd the field distribution.

The theory developed in chap&-Ill, to find h e various quantities like power

Imtu. fluxes CIC, haJ been verified witb tb: prtical results of single-phase induction

motor mth dual exciwon. The 14 lo- on the -or side arr estimated using the

m h i m design thrary 1241. The infinite half-specc h r y is applied to the actual

rotor by modifying thc resistivity of the rotor material with the correction factors for

curvalw and cnd effccls 1141. A wmction factor IS also incarparaled for

~mqmrturc risr. l'hc interfcmcc bctwcen thc IWO input signals with regard to IIIC

windrnga ia avoided. Since thc decaicrl ~k b*wben the wind~ngs is 90'.

4 3 DEMRHTION OF THE PROBLEM

Tbc h c e d h m i i field in the rotor of poly-phase induction motor can be

made two dimensional, if it is Pllrnuacd tha! the i n d d eddycumnts in the rotor arc

in widdirection only. The rotor cm be vicwed as an iron- block, wficn its curvatm

is neglected. Such an iron -block is subjected to travelling field on its surface say in

y-z plane, it is obvious that, thm c x h m altmahg flux through any section

pafallel to x-z plane, d m w i n g a dip frrquency. Conscqucntly, the evaluation of

eddy-cwent losses in solid-iron rotor of an induction machim an be based on the

knowledge of eddycurrefit distribution in an infinite half-space of iron subjected to

pulsating ticld.

Solid-iron mlor induction motors are in use. Due to many reasons, these

motors operate from non-sinusoidal excitation. Such an excitation can be considered

to be the sum of fundamanal and the harmonic of highest magnitude for the purpose

of evaluating cddy-currcnt losses. 'herefore, for analysing cddycumnt losses in

elccuic mhna, the input is taken cu the sum of two sinusoidal signals of

comnaammtc h a q w i a . h f o n . tmwxcitation field problem is simulated using

single-phase induction mtor with two windings.

The pmblem has been formulate4 in section 2.5. But the boundary conditions

arc provided in section 3.5 (qa(3.38)) Remite the equation (2.41 ) for convenience

The consmb arc a = 2.25 Tala and y "787 Mm for the g i m mrgnctiuuion c u m

oftheMnaidurhowninfigurr4.l.

Fi4.l: B-H curve

Tht bounduy and initial conditions sre nmim here as follows:

(iii) Initial values i.c at T = 0, t i = 0 for all "xn

4 3 SOLUTION OF THE PROBLEM-PSM

In d o n 3.2, the dyt ica l solution ia developed for the problan. 'lbe same

pmblan is dso solved numerically by using Cd-Nicholson Mcthod in d o n 3.5.

But in ckptcr-Il. it has been dated that the P&Speceal Method can be a good

substitute for Cnak-Nichoh Method. Heace, in this chqm, the duel excitation

p b l a n is solved by Speclral method.

43.1 Implieit Pseudo S@ Method

In section 2.6, Pseudo Spectral Method or Chebyshev collocation metbod is

discwed elabomtely. The implicit b s l c p p i q scheme IO equation (4.1) is given by

quation (2.67) i.c.,

It may k noted thu to fud t&: values of H at (n+l) timc-step the derivatives ue

mluued at (n+l) timeslep itself. The boundary conditions of equation (4.2) me

implemented by changing the entries of first-row and lapr-row of coefficient maaix

'D' by referring section (2.61). Then Ihe equation (4.3) is solved for Wing the field

d~stnbution. Foc chc sw l rc excitation of H I S = I 1518 A/m, at u1=314.2 dps .

Hn -2303 Nm, at to? 2827.11 nd/sec. the field distribution at various layas is

shorn in figwr 4.2

43.2 Jhahtion of eddy camot ku

Once the field Wbution is abteined thc iron 11x1s ten be c a l M using the

pmculm cncW in section 3.5. In this chapter the effect of harmonics w iron

losses is alao considend.

For this purpose, along with thc f h d a m d compowm, various lmmoaic

cornponenu of field ~IX determined at various laycn, unseqwtly, the cumnt

densities M dusted wing the formula give by equation (3.41) as

Foreschvalueof j =13 : f ix h = 1.3.5. andrunk=OtoN.

Thc figure 4.3 shows the profile of fundmental and harmonic c m t

densities, wbcn the surface excilations me H I S = 1 1518 Aim, at wr=3 14.2 rad/scc; aod

Ha - 2303 A h , at a11 = 2827.8 d s c c .

From Ihc figure 4.3. it is undascnd tbac the higb frequmcy signal mcnuats

fraP tha low frequency signal confuminp the validity of equation (2.38) with rrspea

IO f'rcqucacy. Monovcr. Ihc menuation is nonlinear with depth.

Fowia series is employed to scpmk the fmdamentd components at eacb

laya fmm Ihc resultant wave. Therefore, it has been assumed lhat the two finqumcy

signals uc cammmlurau at all laym. For the incommmsurate signals at the surface.

I! ir diffwult to #pantc thc fundamental components in interior of the materid. In

fm it will bc very intntsling to find the ways to separate Ihc h c n l a l

eompone~b of incommcnsmte signals from the distortad resultant fidd.

Fig. 43: Variation of cumnt densities with depth

4 3 3 Esttmrtion 01 Surface Fkkl Swcmgtlu .ad applkd voltage

For evalunting eddy current-losses by the numerical method aad compare with

h of expcrimartai values, the surface excitation must be the oame as that of rotor.

The procedure follows to estimate field strength at the rotor surface for the given the

stator current.

If D is the diameter of rotor, then the number of conductors on the mtor i s nD.

So h e ratio of vansfonnalion i s

Where N is rhc e f f d v e number of tumr of e& stator windiag.

IftbctWophrrslhvc~urlpumber~ftums*then

~*(NI+N~)/NI fortheihtwinding

5 = ( N I + N ~ ~ N ~ for the second winding

Let Hs k tbc m a p t b i q force at the d a c e of iron in Arnp/m. Ttsen the equivalent

r.m.8 cunmt of the rotor refand to the stator is

I , =- Hs (or) f i K,

Since, it has been observed practically that the magnetising cumnt draw by solid

imn rotor induction motor is as high as 30%. If one assumes the angles between

voltage & I, and vollage & I, arc 36' and 82' rrspactively, the mtor component I,' will

bc 76% of stator cumnt. Hmce

W&rc Is is IIIC stator cumn~.

For diff~tnt values of aator c m t s 11 and 12 of two W i n g s , the

ewrapoadmg velua of HIS and Ha arc found using equation (4.7). Subsequently. Ihe

flux components + I and +Z are determined using the equations (3.23) and (3.27)

mpoctively. I'hw the induced emfs arc calculated using the following equations:

W&rr NI d N2 ur: effcctivc number of turns of rapffCivc stam windings.

L :r Lncn 1 11. h,,L

Now the applied volugea are givm by

V, = E , + I , ( R , + ~ x , )

4.4 GRAPHICAL SOLUTION

In section 2.4.1, gnplaical mahod is d i s c d for single excitation problem.

Whereas in section 3.4, geometrical consauction is provided for double excitation

problem. with the wndition I s (mdm,) S 2. In this section. a similar gqhical

construction wll be explainad for (m2/mI) > 2.Thc graphical construction star& with

h c assumption of values for H and 4 at zeroth-layer. Thm Ihc values of H and 4 at

fim-lycr arc found vectorially. Having calculated the values of H and 4 at first-layer,

the vdues at second-layer arc evaluated. and so on. The above statements can be

wrinm in the mathematical form as.

Lct a! 314.2 nd/aec, nn~= 2827.8 ladlaec, p = 1 6 . 5 8 ~ 1 0 ~ ~ - m , B,= 2.0 T. 'lhe

d e fauor for H, and P M 37.88.2.0 md 23789 rwpcctively, f h n thc equations

(3.33), (3.34) d (3.35). AAa muhiplying the values of the table by these scale

factors. thc curves of field of stmgths, p o w losses, fluxes ctc., are drawn and

shown in fig. 4.4

TABLE:

Reaulb of computerized graphical construction

Fig. 4.4: R e d s of Braphifal mahod

4 3 EXPERIMENTAL PROCEDURE, RESULTS AND DISCUSSION

An experiment wrs conducted on a single-ph induction motor after

moving Ihe capacitor From the swing w i n d i and replacing the wnveapional rotor

by a solid-iron rotor with no air-gap. The specifications of the motor are given in the

rppmlix A4- I . The swing winding was connected lo the 50 Hz supply Where as.

h e m i n g winding was excited by 450 Hz signal. Magnetic intcrfrrrncr is avoided.

since the c l d d angle b?nvcen Ihr two windings is 909. The prrtid eddysurr~nt

loges in h e mamill (rotor) at 1&Jc fraluencies m found-out h m the measured

input p o w . Ir may k noted Iht Ihe iron and stny losses in the stator, including

yoke, are estimmd approximately u 15% of the total ( i ) losses, talring into

lccount the toW mount of irw. lk &tails of calWons an included in the

rppcsdix A4-2. The sirnulaud d meamid iron lor#s an show in fig.4.5

The ulcuiatcd applied voltages RE compllrtd with that of meamred valucs as

shown in figure 4.6.

F i i 4.6: Applied Volges

For p r a t i n g the high frequency signal, oscillator and high capacity (80 Wans)

pow amplifier arc wed. It is taken carc that the two signals s ~ s at the same time

or in ather-words that Ihc phase-shifl is adjusted to zero. But, the phase-shift does not

have much effect on chc power losses. The reason is that, h e lnquency separation

hawem thr two signals is more. Havv thc anrage power loss ur: almost

idcpcndm of phmeshifi.

Thc infinite hdfsprc lhcory is adapted for clatrical machincs. by modifying

thr rpscifa nsislana of mtM &al so as lo incoprue the corrcclion factor for

curvature rad nd d k t a [14]. F i y , a suitable corrcdion factor is ah h q m a t d

into d v i t y to acmmt for tanpsraftue rise. Tk tbaory is provided in appdix

A4-3. Tk effective nsihvity b 1.06 x 1.77 x 1.2 x 16.58 x 10' a-m.

4.6 CONCLUSIONS

A new clusical numerical method called Pseudo-Spectrai mabod is successfully

implemented to find the field disbibution in the solid-iron mtor of induction motor,

when it is subjected to two-fresucncy excitation. It is found that b e simulated powa

loam in the mtor M agreeing with the eexperimentd values. A grapbcal metbod is

also presented to evaluate the field distribution in the specimen.

A4: APPENDIX

ACI: W i h of single-pbuc induction motor:

Rated output pomr : 1.0 )Cw

lnpn voltwe (V) : 220 volts

Effective number of turns of (starhg) winding A , @I) : 334

D.C resistance of winding A ,(RI) : 8.0 Ohms

laducuncc of wrnding A, &I) : 0.045 Henrys

Effective number of turns of (running) wiading 9. ( N 2 ) : 286

D.C resistance of winding 9, &z) : 2.5 Ohms

lndunurcc of winding 9. (L2) : 0.026 Henrys

Diamacr of mtor (D) : 0.106 rn

Length of mtor (L) : 0.109 m

Spccific misturcc of h e rotor material (b) : 16.58~10~ fl-m

B-H curve of rotor material : Figure 4.1(p72)

Length of the air-gap :Zao

End rings : Nil

AC.2 :~mmto i lpd to t8 l~LntbLnta@r

Thcl~hlduItatotm

(a) c o p p t l l o s s g i n t & ~ a n d

(b) I r o n l ~ i n t h e ~

(a) Evrlmtioa of copper lou: Kaowia$ (he nsicrance of each winding, c o p

losses aI any load can be calculated.

(b) Evnlnation of iron lou: Tbest lo- can be further classified as stator teah and

core lows. The calculations of iron losses are based on the total migbt and the flux

density in the material. The values given in the brackets M a t c l y after the

formulae arc rcfemd to the specific motor, whose details an given in the

appendix A4- I.

(i) Stator tu tb loor

T& total weight of all stator leah is given by

(wst-36*5x10" *2x10a * 0.109.7.65xld = 3Kgs)

Whm Ss- numba of stator slots

Ws, dss - width d of each stator slot, metem

LX-le@ of mor con, m a r s

&-specific weight of iron. kgfhn'

T& iron loss in the stslar tath (P,) = a*&,' *WP

(Pal= 6.5*1.5' *3 = 43.87 warns)

Where 'a' is a con sun^. la value is 6.5 for teeth and 4.7 for corc [24].

(if)Strtor are lou

The imn 1- in the stator corc me also &mated in tbc same manna ar tbat of

atator teeth loss.

Thc depth of stator corc is given by

Where D, Do arc inm and outer dinmeten of stator.

The mean diameter of stator wrc is given by

DMU = D O ~ S C

(D-f0.179-0.015 -0.164m)

H a w the weight of che stator corc is,

ww = A.D~~v.~sc .Lsc.~I

(wK= nb0.164* 0.015 *0.109 7.65xld = 6.44 kgs)

The iron loss in the stator con ( P s ) * a * b 2 *WSC

(PLw = 4.7* 1' ,644 1 30.26 W)

Thc total iron loss in rhc stator (P,) = Pia + Psc

(P, = 43.87 + 30.26 z 75 wans)

The won losses and stray losses in stator & yoke an d e n as two-times P, .

So

P - ZbP,

(P ~ 2 ~ 7 5 = 150 watts)

Tbdixe, fortlib Mehhretbetacal iron loucg an the ator side, includiislmy

IoMcSrre ISHof~inpttpoWer.

A C J : T b e ~ 1 h c t o n k r a m h m u d e 8 d c d k c t l [ 1 4 ]

(a) Clll'V8hm

When M infinite half- hwy is applii to r cylindrical coordinate

system, it bccomcs neceJsuy to makc an allowanx fot the curvature to confirm the

physical fact that actual c u m n ~ is reduced. This reduction in- depends upon the

depth of pmetntion. Thc redudion htor given in the reference is (D-2xlTJyD.

W k D is the dimera of the mtor, and xl is the depth of perneation of signal

which goes dapcr thrn Mothn signal. In this use it is the depth of pcmtration of

low frrqwncy signal. For the given surface excitation of the rotor, the stator cumnt is

reduced by the above factor i.e (D2x113)lD. In effect, the d o of transformation is

inuaKd by Ihe ~une h o t . Hence, an alkmmcc for the c u w ~ can be made by

~ncrusing the specific resistance by a hem,

Whcrc the valuc of x, is daenincd using the equation (3.25) for the maximum valuc

0 f I i 1 ~

@)Corrrtb.Cordrlkeb

In the intinite Wf-spueImrlyris, ilhskenlPsMItdthtalltheoumatsin

the rotor flow axially. In other words. ti^^ end e t W s have been ignored. It is obvious

that the end effects would kpend on the physical d i i i o n s o f the mtor, the type of

md- rings used, no end-rings used Md the rotor frequency. An empirical comction

for these cffeas is to modify the specific resistance of the rotor by r factor

When K=l. for the rotor with c~pper end rings.

K = (lhk), for the mtor with steel end rings.

K = 1.77(1+0.49S) with no end-rings,

S. L and t arc slip, rotor active length and pok pitch mpcctivcly.

KZ =K (Since S=O. UIC slip not come into picture as then is no revolving

magnetic field)

So K251.77

(c) Corrcrtk. tor b m p m h n rbe

To consider Ihc cffccl of incrrut in lanpm~c on che resistivity o f dK

mucrd. M appropriate multi f i ion frcm is manned as KJ = 1.2.

To wmmuizc. the effective ooncdion lrctw is K K I K ~ K l = I .06* 1.77*1.2 = 2.25

flmtfore, the m o d i f i nsiwvity is p *p, *~=16.58*10~*2.25 =37.33*14'0hm-m.

Chap ter-V

Performance evaluation of two-phase induction motor with solid iron rotor under unbalanced load

conditions

5.1 r n 0 D U C T I O N

Two-phase induction motor opemtes under un- load conditions, when

it is used as oelvomotor. It is inttnsting to evaluate the pafomuurce of such a motor

with solid imn rotor. Thc analysis developed, bssed on I-D model of tbe rotor. Uadcr

unbalanced conditions the rotor is subjected to two-sinusoidally distributed,

oppositely rotating magnetic fields at the surface. Their sp& relative to the rotor

being dctenninbd by the slip (s) and pole pitch (r). For the I-D model, the rotor can be

consided as M infinite half-space of iron subjaxed to two pulsating magnetic fields

at the surface. So the problem is to damninc the forward and backward rotor

scqcqwce fluxes, the equivalent circuit and hmcc the performance.

5.2 THWRY

Cowidet, the machine o m on unbrlraocd set of voltages applied to the

stmi phrs. This r* can k resolved into two brlmccd ~s of positive and negative

sequence voltages. The positive quare voluges produce a field that trawl in the

sunc direction of the rotation of rotor. The relrtivc velocity of this fonward field with

the velocity of rotor is (scu,rYn. Wherca%, the negative sequence voltages pFoduces a

bsclrward mwlling field which runs in the opposite dimtion at a velocity (2-s) m,rln

uirh respec! to the rotor.

I f the 4~ and +p are the toLal f o r d and backward fluxes per pole, then the

fluxca entering thc mtor surface will rrppou as altanuing fluxes of amplitudes L?

uwl I2 n ticqfrrsuencies sf and (2a)f mpcdvely. However, as we move dong the

pdpbay, the phue &A bctmcn thew flux waves a diffennt adow can be am

to vry unidormly taking all possible d u e s . The proof of this is as follow;

Conrider the instam of time at which the axes of the forward and backward

travclli flux waver coincide. La this instant, reckoned as t=O and it is happened in

section AA in space as shown in figure 5.1.

Fig. 5.1 :Fon+?ud and Backward Traveling Fields

For Ihc dirsc~ion shown, as the forward field moves by s distance y 1 towards

the @I, Ihc kckmnl field moves by a diJuncc yl(2-sys towads the left. Taking Uw

directiom of flux firom left to right u positive, the flux in the ydimtion at M due to

forwud field keasw negatively with time, wUe that due to the t#ohrJard field

hmases positively. Hcace, at lvaign AA, the two components of fluxes, in tbc y-

d i d o n can be acprrrssd as

+b = bb sin((2 -s)o,t] = +, Sin(o,t) (5. lb)

When a, is operating fnquency

Consider next. a M i o n at an arbitrary distance y from the section AA. The two-

components of fluxes at this section ere

Substituting (cur t-n yh)=c~r t' and putting o b 1 cur = (2-s) I s = 7, equation (5.2) is

wrinm,

+, = 3, Sin(@,t (5.3a)

Hencc the tolll flux in the ydircrtion can be exprrssed as

Wh*r a (y+ I ) (n ylr) a ? n ylsr.

Ihw, it can k seen fmm the q d o n (5.4) h~ as one moves dong the

pcriphq, t& phpc shift bcIwccn Ihc hw componenls of fluxes at diff'r sections

variw unifwmly JI @bk valwa. In it cba~gg by 2% ova the

distmee y -sr.

To dacrmine the rotor eddycumnt I-, consider a span dong y-axis

(y=m) ova which the phase angle has unrmcd dl wluw 6om 0 to 2n. Divide this

span into infinitesimal strips, prnllel to the x-z plane, each strip chslaetaizcd by a

definite phase relationship tmvm the two components of fluxes. The total losses in

the span is equal to the sum of the losses occurred in all such strips. Since the

unplitudeo of the hvo components of fluxes remaim the same in all strips, the effect

of phase angle should be summed up, as it varies uniformly from 0 to 2n. In other

words, thc average loss per unit length in this region involves an averaging over the

effects produced by thc phase m@e. when it varies unifonnly fFom 0 to 2n. This

avcngc loss can be seen to bc same as the average loss that would have bax

d. if the two components of fluxes have the same phase relationship in all

strips at any irUtMl but this phase continuously varying with time and uniformly

taking all thc values from 0 to 2n. Now it may be thought of such a distribution of

flux dw would have bun produced by two altanating m&c fields at fiqucncies

d and (23)f Ming simultaamusly at the mhce, dimtul along tbe y-axis, the phase

angle between them being unifonnly wied over dl possible vrlucs.

Thenfore. under unbalanced operating conditions. the rotor can be rrprrsented

by m intinite half-spree of mated excited at thc surf= by two dtemabing

fields of diWennl magnitudes and fquencis along the y-axis. Hmce, the two-

frequency excitation theory developad in chaplw-Ill can k very wdl applied. So the

two fields His and Ha at the surface as mentioned in chapter-Ill will k designated as

f- and Mmnl fields Hr Md Hb ~ v c I ~ .

~ h c f l r a ~ & @ r c m d ( D b ~ t ~ & ~ H b ~ @ v ~ b ~ ~ @ ~

(3.23) and (3.27) rsspectiveiy IB

The rotor power factors am glven by the cquauons (3 24) and (3 28) are

'Ihc above cquauons (5 5),(5 6) 8 ( 5 7) m val~d for y > 2

For y = [(2-sys] < 2. however, the graphicd method prondcd In sectlon 3 4, IS

explo~ted to daam~nc @I and (Ob for the glven values of Hf and Hb at surface of the

mtor The gnph1c.d cons(nrcuon also y lclds phase sh~fts Yr and Vr of Hr and Hb with

81 and mpoctlvely Consqwndy, the rotor p o w factors Cosyr and Cosyb can

be found In the gnpluwl wnsbucuon, 11 IS afhlcwd by mal and aror that the g l m

st tif and would cotx~t

H n v i n g f o d t b ~ f a r w m l I m d b a c h v r r d ~ f l u x e s @ r a n d % , t h e

cquatim of performaace of 2-pbm induction motor u n k unbalanced load

conditions with solid iron rotor M mitten M follows:

(i) Air-gap sequence voltages &ad Et, M given by

E, = @CO,N L+,

Where N is effective number of W p h a p c .

( i i ) Equivalent rotor cumnts referred to the stator 1; and i* an given by tbe

equation (4.6) as

nDH I , , = 4 5 ~

When D is the diameter of rotor and N is effective number of tums of

cech p k *

(iii) S w r forward and backward reqwncc cumnIs Idand la art given by

E I,, = I r , t --L (5.10a) 2.

E I,, = I;, t 2" (5.lOb) XI

M e n 1, IS the magnctidng branch impedance.

(iv) S ~ o r forward and backward scqucna vol~.gcs Vr and Vb M given by

Vt a EI + 1.1 2, (5.1 la)

When 2, is tbe sutor imptdrnce p a phase.

(v) Terminal voltages VI and VZ are given by

V, =iW, -V,)

(vi) Stator phase currents 1, and 12 are given by

1, =i(I., - I* , )

(vii) Toque developed 'T' is given by

Where n, is the synchronous speed in r.p.s.

It may be noted that the analysis so far assumed Hr aad Hb M known priorily.

But in practice, the problem is one of ddmnining Hr and Hb for the given values of

VI ud V1. So M imativc proadwe must k used. The steps involved we illustrated

to tind the pcrfonnana of 2-phase induction motor, whose specifications are g i ~ n in

h e appendix with end rings.

S.3 PROCEDURE TO EVALUATE PERFORMANCE

(i) Lcr S = 0.3.wid-1 the applied volrapes as VI = 230 and V2 = 200 Volts

(~i) Find h e cffcctive resistivity of the mtor material using rhe unphencal

fonnuk provided in the appendix A 4 3 @87),

). Cormtion frnor for curvature Kt = 1.06

). C o d o n i&m Tot md cffeas K2 = I .O(I +0.49S)

(ii)

(iv)

(vii)

(viii)

To r~ with aasumc a arbitrary values for Hr and & as

Hr 2000 AIm S= 500 Aim

Thc limiting d u e of ftux density B, Comspoading to H = $HI&) ,&om

the B-H curve(Fig.4.l ,pn) is Blll = 1.25 Tesla

The ratio is given by

7 = ((ld 01) ' ( 2 - 9 6 = 5.66

Comsponding to this ratio, the value of $ = 2.977 is from equation (5.6)

Now, find the forward and backward flux components @r and @ for the

given values of Hr and Hb using the equations 5.5(a) and 5.5(b)

6 = 0.20 mwb

The air-gap sequence voltages from the equations (5.8s) md (5.8b) are

6 156 Volm J2+= 3.86 Volts

Rotor currents ref& to the stator side arc given by the equations (5.9a)

Ihc equations (5.7a) and (5.7b) gives the positive and negative sequence

mlor powers factors as

Cornyl- 0.8165

Hence I,(= 0.31(-36.26)0 A with mfcnm to I$

Lc = 0.077 (-43.53)' A with refheace to Eb

(xi) T h t r t l t o r ~ ~ w ~ t h e ~ 0 1 ~ ( 5 . 1 0 a ) a n t ( 5 . 1 0 b ) a r e

Ip1.42(-w ) A b=0.15(-61°)A

(xii) The~qsqueaccvo~~the#1~0~(5.11a)and(5.llb)arr

VI = 90.42 Voh Vb= 7.48 Volts

(xiii) The ststor taminal voltage arc from the equations (5.12a) aad (5.12b) are

V13 98 volts V2 = 83 volts

The calculated values of stator taninal voltages V~and V2 are not q d n g

wih the applied voltage. Hence the mw values for Hr md Hb should bc chosen to

~ c u l a t c VI and V2. If one assumes linear variation, then the new values of H

would be given in terms of old values as

Wbac V; . V2' ace he calculated values of Vl& V2 respectively. Now tcpeat the

poadurc until he calculalcd v a l w of terminal voltages tally with the applied

voluger.

Thc final results after eight iterations m as follows:

1, - 3.36 (3.7) Amps I: = 3.25 (3.05) Amps

PI = 352 (340) PI= 235 (210) watts

'F 2.2 (2.35) N-m

VI VIC 230 Volts V2 = V i = 200 Volts

11 my & aolod Ihu the values given in the bmkcts an cxpauncntal results.

5.4 EXPERIMENTAL REIPULTS AND DISCUSSION

A two-ptmac i d d o n mator b wound rrd povidcd with solid iron rotor

h ~ c a p p a c a d - r i a g a . T h e ~ o f d 6 , m o t b i n g i r m i n t h c a p p a d i x . T h e

equinleat circuit of the motor ia obtrrincrl by coaddng suitable tab. The I& test

is also conducted on the same motor unda unballncrd voltsgcs. Thc simulattd

d t s obtained by the dual mcitatiw tbmy dong with the equivalent circuit of the

machine n comprrrsd with the e x p a i m t d values. The performance curves art

shown in figures 5.2 and 5.3. From these figures, it is clear that the theontical values

an closer to practical values. Thc difficulty f a d with the solid-iron rotor is that,

during the process of machng the material became hard. Hence, the motor was

drawing current more than full-load value even on no-load. To overcome this

problem. the makrial is medc to miergo anmaling. It is pmztically observed that the

speed fall is bastic with load. This i s due to heavy losses in the solid iron rotor.

Lqad: A: Ph l 8: Phl Thanml C: Ph2 Thacual D: Ph2. l'mxlal

0 0 3 0 6 0 9 1 2

----b SUP(S)

Fi 5J:Toque-Slip Characteristics

Sj CONCLUSiONS

\The dd excitation Uroq i~tFnP~oYcd to find thc\ippwd and brlwrd

sequmce\uxcs and hence the cquivdcnr &p.t of WO-phase i h t i o n motor with

unbalanced voltages wndpons. For this

wih d i d iron mtohyving copper cnd-ri&nd W

nrulu r cornpA,with

\ vducs. Ir is obscrvd ha1 lid iron rotor induction mot&. w i l copper end-ring 3 produces inore corquc than the 1

AJ: APPENDlx

Spccificdtionr of 2-phe induction motor:

Rated Pow : 1.OKw

Rated voltage and fhquency : 230V, 50 Hz

Number of phases : 2

Number of poles : 6

Name of o p t i o n : Constant rated voltage

Winding factor : 0.88

Effective no.of turns per phase in series (N) : 380.16

Stator impedance per phase : 9.3+j22.5 R

Magmising branch resistawe R, : 306n

Magnetizing branch mctance X, : 47.7 n

Rotor diameter (D) : 0.106 m

Ac~vc rotor length (L) :0.110m

End-ring mamial : Copper

End-ring dimensions : 2cmx l cm

p of the t~~a~rid : 16.58~ lod n-m

Comflion factor for cunqaturc : 1.06

Comxtion factor for tmpcmturc rise : 1.2

Comtion for end cffccts . (1 +0.49S)

Effcct~ve p : 21 .oq i t 0 .49~ ) x 10" n-m

Performance evaluation of poly-phase induction motor with solid iron rotor

6.1 INTRODUCTION

A problem of interest is the preaaennination of the performance of a poly-

phase induction motor with solid iron rotor. Due to the use of non-linear devices, like

transformers, reactors and switching components such as phase controlled rectifiers,

power transistors, SCRs and GTOs, load generated harmonics are injected into the

power system. When an induction motor is operated under non-sinusoidal voltage

condition, besides the fundamental component of current, different harmonic currents

also flow to the motor. It is known that these harmonics produce losses and as a result,

increase of total losses in the machine. Because of additional power losses, the

temperature rise of the motor will be more compared to the perfectly sinusoidal one.

Hence the motor will not be able to deliver the same output for which it has been

designed.

This chapter first discuss a method to predict the perfornmce of the motor

under sinusoidal operating conditions, then presents a new method to predetermine

the performance of induction motor under non-sinusoidal supply conditions and

compares the results of this method to that of the combined equivalent circuit.

Consider an induction machine with solid iron rotor having the balanced poly-

phase winding on the stator connected to a balanced polyphase supply. A revolving

magnetic field is setup, and this, acting on the solid rotor, induces eddy currents

resulting in the production of torque. The rotor eddy cwrents are at slip frequency.

When the curvature of the rotor is neglected, the problem is to determine the current

and flux density distributions inside an iron-block of infinite depth subjected to

traveling magnetic field af the surface with a speed governed by the slip frequency

and the pole pitch of the machine. Strictly speaking, the field distribution is

3-dimensional, having the additional complexity introduced by the prcsen~~ of

saturation. However, the distribution is made 2-dimensional, if it is assumed that the

induced eddy cumnts in the rotor have only one component, this b c i i in the axial

direction. Under this assumption, therefore, it becomes necessary only to determine

the field distribution inside an idmite half-space of iron subjected to a traveling field

on its surface.

To fix the co-ordinate system, let the surface of the rotor be in the y-z plane

and x-direction be measured towards the surface and papendicular to the y-z plane.

Let the traveling magnetic field on the surface is non-sinusoidal with time and is

traveling in the y d i i o n . Clearly, through any section p d l e l to the x-z plane,

there exists an alternating flux, consisting of fundamental and harmonics, altenming

at slip frequency. Thus every section in the x-z plane is subjectad to a pulsating flux

along the y-direction. Consequently, the evaluation of the eddy current losses in solid

iron rotor of induction motor can now be based on the knowledge of eddycurrent

distribution in an inf i te half -space of iron subjected to a pulsating field, consisting

of fundamental and hannonic field. The d t s of such an analysis are extended to the

solid rotor machine to take into account, the physical dimensions of the machine and

also the non-linear magnetic properties of iron.

6.2 PROBLEM FORMULATION

The basic problem has been formulated in chapter-11, the field equation (2.3) is given

here

Where 'p' is the effective resistivity.

Let

Where 8 is a fimction of 'x'.

By substitution of equations (6.2) and (6.3) in equation (6.1) yields,

If the B-H c w e is approximated by the Limiting curve as shown in

figure 2.2 @Is), wherein B - B,. With this approximation the solution of

equation (6.4) is fouad in chapter-11, as given by equation (2.34) is

wh ax* (6.5)

m B m where a = - 3 p f i

Knowing the variation of H with x, flux per unit length and power factor at the

surface of the specimen are evaluated in chapter 11. The equations (2.36) and (2.37)

are given below

Whcn the subscript's' refers to surface value.

63 ANALYSIG OF SOLID ROTOR INDUCTION MOTOR WITHOUT HARMONICS

To analyze the pwfonnnncc of the machine (Whose specifications an given in the

Appendix) unda given induced emf operation, first, it becomes necessary to fix the

operating value for 4s. For this purpose, the equation of induced emf of the machine is

used. That is

Where,N is effective number of turns per phase and L is core length.

For the given value of E, the corresponding value of flux cbs at the surface of

the rotor can be found using the above equation (6.8). Then making use of the

magnetization curve of the material and employing the step-by-step graphical

construction outlined in =tion (2.41), one may arrive at the amplitude of the field

intensity Hs at the rotor surface wmmy to maintain 0s . Of course, the necessary

modifidon is AH..(j2d&i 04.9. The cmstmction a h yields phase shift Ys of Hs

w.r.t %. Consequently, the rotor p o w factor h m e s Sin'&. It may be noted that,

for different values of rotor cltmnt fraqucncy ff-sf, the construction has to be carried-

out.

Knowing the magnitude of H% the equivalent current of the rotor on the stator

side can be found using the equation (4.6). Thai is

a is number of phases

N is the effective nlrmber of huns per phase

Also the stator cumnt per phasc could be dctaminad by adding 1; to the magnetizing

currsnt vectorially. Finally, the torque in ~ y n c h r o ~ u s watts is

T =a E I; Sinv, (6.1 1)

To analyze the performance of the machine under a constant applied voltage

operation, the following equations are used,

If it assumed that the rotor power factor is nearly constant and equals 0.81, then the

equation (6.12) would become,

V = E+I ,X, + I ;S iny , (~ , +X,Coty,) (or)

Substitiaaing equation (6.6) for @ in (6.8) gives

Where 'S' is the fundamental slip.

The foregoing equations of performance will now be applied to the motor, whose

specifications are given in the appendix and the magnetization characteristics of the

material is shown in figure.6.l. This curve is approximated by the Frohlich equation

with a = 2.28 Tesla and y = 1688 A/m.

The procedure to evaluate the performance is as follows:

(i) Assume a value for Hs, say 10000 Alm,

(ii) The value of KT using equation (6. lo), is found as

KT = 5856

(ii) From equation (6.13), daennine the value of E,

E = 363 Volts

(iv) Now using the equation (6.14), obtain slip 'S' by iteration. S i p is a

function of 'S',

S = 0.125

(v) Then find the rotor cumnt referred to the stator b m equation (6.9) is,

I;= l.2OA

(vi) Finally, the torque in Newton-meters is from the equation (6.1 1 ) is,

T = 6.73.

--+ H (AIM)

Fig.6.1: 9-H Curve

It is recalled that the harmonics arc injected into the power supply due to the

Use of nonlinear devices. Wben a 3-phese induotion mom is fed from such a supply,

the field not only hes fundamental component but also harmonics. For the purpose of

pmdctmnining the performance, only S* harmonic besides fhdamental is

considered. It may be noted that, in a 3-phase system, the third harmonic is zero and

higher harmonics have smaller magnitudes.

6.4 DUAL EXCITATION THEORY

Let

H = HI sin(o,t + B , ) + H , Sin(o,t + e l ) (6.15)

Where BI, B2 arc fundamental components of output corresponding ta the input

components HI and H2 respectively.

Substitution of equations (6.15) and (6.16) in equation (6.1) and equating similar

tmns yields, the following equations

The solution of equation (6.17) is the variation of HI with 'x' is given by equation

(6.5). Whereas the solution of equation (6.18) is obtained in chapter-III, as given by

equation (3.16) is

42 where - ( ~ p - l ) J j T o = ~ 3

After having found the law of variation of HI and Hz with 'x', the flux

components +is and 412s cmsponding to the fust signal and second signal are

evaluated. They are given by the equations (3.23) and (3.27) respectively as

Where B, is from the B-H curve (Fig.6.1, pl06) corresponding to H=(Hls+H2s)l;l.

6 5 PERFORMANCE OF SOLID ROTOR INDUCTION MOTOR WITH HARMONICS BY FIELD THEORY

The procedure is as follows:

(i) Assume a value for HIS, say

HIS= 5000 A/m

(ii) Let the fundamental voltage to be of rated voltage and assume a value for

5* harmonic component,

V I = 400 Volts Va = 80 Volts.

(ii) Usiag the quation (6.13). the value of El,

El= 373 Volts

(iv) Evaluate SI using equation (6.14),

SI =0.051 or 5.1%

(v) Now, use the equation (6.26), calculate Sr,

S5= 1.19

(vi) Knowing q = (CO~/(OI) =5 = k, the value of P from equation (6.20) is,

p = 2.8

(vii) To fmd H~s, combime equations (6.8), (6.13) and (6.22) with I, = I, /k and

x2 - kxl,

H~s = 2480 A/m

(viii) Calculate I,I' and In' using equation (6.9),

1,l' = 0.60 1,; = 0.30

(ix) I,' = dfl,l' + In') = 0.67A

(x) I, = I; +Ip = 1.89A

6.6 PERFORMANCE OF MDUCLlON MOTOR BY FUNDAMENAL AND IURMONIC EQUIVALENT CIRCUITS 1161

Prediction of induction motor performance based on fundamental equivalent

circuit is not complete, when the motor operates under non-sinwidal voltage

condition. Therefore, in addition to the fundamental equivalent circuit, the harmonic

equivalent circuit is also used to pre-determine the performance of the motor.

The approximate equivalent circuit at fundamental frequency is shown in

figure.6.2. The bdammtal voltage VI and the parameters of the equivalent circuit

are known. Therefore, assume a value for slip 'SI' and find the f u n a d c m n t .

Similarly, for diRerent values of slip, determine the stator current.

The approximate equivalent circuit for the Krn harmonic fresucncy is shown in

figun.6.3

Fig.6.2: Fundamental Equivalent Circuit

Fig.6.3: Harmonic Equivalent Circuit

The slip at fundamental fiquency is given by,

Whenas, the slip at km harmonic is given by,

From equations (6.23) and (6.24), it is obtained that

I-S, SK

The equation (6.25) is valid for the forward rotating harmonic field. For a backward

rotating field, it can be shown that,

Thus in general,

I-S, SK =I*(*)

It can be proved that the order of most commonly generated odd harmonics are found

by the equation,

When m is an integer.

+ sign for forward rotating harmonic field

- sign for backward rotating harmonic field

Under normal operation, SI << 1. So the equation (6.27) can be simplified as

When k is high, Sr a 1.

For the given value of k = 5, the parameters of harmonic equivalent circuit

(fig.6.3) are known. For each value of SI, find the corresponding value of Ss using

equation (6.26). Then compute the stator harmonic current. Now the total stator

current is given by

6.70 MULTIPLE EXCITATION THEORY

For accurate performance evaluation of three-phase induction motor, it is

necessary to take into account of multiple harmonics, besides fundamental

component.

Let

By substituting the above hvo equations (6.31) & (6.32) in (6.1) and equating similar

tenns resulting the equations (6.1 7), (6.18) and (6.33)

The fundamental components BI,Bz and B3 of output comsponding to the input

components H I , H ~ and H3 are given by the following equations:

1 +" B, = - j f / f ( ~ ) Sin(0,) d0,d0,dB3 (6.34a) 2x1 -,

B, =- ' , (if f(H) ~in(B,)d0, d0,d8, (6.34b) 2f -,

Since, the B-H curve is approximated by relay type curve, the output will be positive

when H >lor negative when H<1. The switching will takes place at H - 0. Thus the

above equation (6.34) is written as

when y~ = Sin-'(k, Sine, - k, Sine,)

H With k , = ~ + l a n d k , = s * k ,

HI HI

The solution of equation (6.35) with truncated series is

It may be noted that the equations for Bz and B, arc interchangeable. That is

B~(HI ,H~,H~) Bz(HI,H~,Hz)

For small values of k1 and k2 the equation (6.36) reduces to

Now refaring the theory given in section (3.2), it can be escenained that the solution

of equation (6.33) is

The constant 6 is related to the ratio (coml) by the equation

From the equation (6.38), it is evident that all equations pertaining to second signal

can be used for third signal by replacing co2 by o3

6.8 PERFORMANCE EVALUATION BY MULTIPLE EXCITATION THEORY AND EQUIVALENT CIRCUITS

Assuming the applied voltages as

Vl = 400 volts, 50 Hz

V5 = 80 volts, 250 Hz and

V7= 60 volts, 350 HZ and making we of the theory of sections 6.5,

6.6, and 6.7, the performance of induction motor is obtained. The results are shown in

figure 6.4 [IS].

6.9 RESULTS

The stator cumnt of an induction motor is computed by the proposed method

as well as by the method of combining the results of fundamental and harmonic

equivalent circuits. The results of both methods are shown in figure 6.4. From the

figure it is understand that the simulated results of the proposed method are w i n g

with that of combined equivalent circuit.

Equi. Circuit

New rncrhod -- X Slip (SI)

Fig. 6.4: Variation of Stator current with load

6.10 CONCLUSIONS

A new method to pre-determine the performance of a 3-phase induction motor

with solid iron rotor is presented. The results of this new method are compared to that

of combined ones of fundamental and harmonic equivalent circuits. To facilitate the

inclusion of two harmonics, multiple excitation theory is developed. This study is

useful, to find the performance of an induction motor, when the harmonics are present

in the supply.

A6: APPENDIX

Specifications of 3-phase induction motor:

-pow*. : 2.2 KW

Rated voltage and frequency : 400V,50 Hz

Number of phases : 3A

Number of poles : 4

Nature of operation : Constant rated voltage

Winding factor : 0.96

Effective no.of turns per phase in series (N) : 426.2

Stator impedance per phase : 8.85+j12.33 n

Magnetizing current : 1.35L81°A

Rotor diameter (D) : 0.139m

Active rotor length (L) to pole pitch : 0.852

End-ring material : Steel extension

End-ring dimensions : 1.5cmx0.81cm

p of the material : 1 8 . 5 ~ 1 0 ~ n-m

Corntion factor for curvature : 1.2

Comction factor for temperatun rise : 1.24

Comction for end effects : 2.174(1 M.2S)

Effective p : 59.84(1+0.28) x 1 o4 n-m

CONCLUSIONS

Investigations en done for a long time on field distribution in ferromagnetic

materials. The analyses were based on the approximation of -tion c w e of

the material. A variety of analytical solutions are available in literature. The Crank-

Nicholson Method (CNM) of numerical analysis is employed with out any difficulty.

In this project an alternative method called Pseudo-Spectral Method (PSM) is

proposed for the Crank-Nicholson Method. The proposed method is a global one

since it uses global trial functions. Momver, it approximates a function by

polynomials that are infinitely diierentiable. So, its accuracy is of order infinity. The

PSM with less number of nodes could yield results that are cornpatable to that of

CNM. This method has been developed using the Chebyshev polynomials. Hence, the

method is also called Chebyshev w11ocation method. The spacing of laym is co-

sinusoidal. In fact, it has been verified that this method qui res uneven spacing

othmvise the method can't produce accurate results. Therefore, the uneven spacing

not only yield good results but also helps to study the field at the surface wy closely.

Basically the span is from -1 to +lm. Howcva, it can be condensed to any rsnge

using the linear transformation. Also the boundary conditions can be easily

incorporated into the field equation matrix. The main advantage of Spectral method is,

yielding acceptable results with less number of collocation or grid points.

The Finite Element Method can also be used for finding the field distribution.

But it requires lot of memory and exccution time. So the Pseudo-Spectral Method

may be preferred among the numerical methods available for solving the field

distribution quation. Of course the medium must be a block material. A cornpacision

is made between PSM and CNM in regard to eddy c m t losses of single excitation

problem.

An analytical sallltion is developed En two-field excitation problem, based on

the infinte half- t b q , so as to find the field dWbution, f l u , iron losses ctc.

of each signal. This theory has been verified on two identical mild steel toroids and on

single-phase induction motor with solid-iron rotor. There is no air-gap between stator

and solid rotor. The application of two-field excitation theory to actual induction

motor involves the practical difficulties of taking care of rotor curvature and end

effects. Since the theory developed based on the infinite half-space. The interference

between the two input signals of different magnitudes and Enqumcies is n d l y

avoided. Since the electrical angle between the two windiigs is 90'. For findmg the

field distribution, besides analytical and numerical methods, a graphical method is

also developed.

The dual excitation theory is also employed to find the performance of two-

phase induction motor under unbalanced voltage conditions. The rotor is made up of

mild steel without any bars. But the rotor is provided with copper end-rings. The dual

excitation theory that has been developed is utilized to hd the f o d and backward

sequence fluxes. Consequently, quivalent circuit and hence the performance of the

machine is evaluated. The difficulty faced with the solid-iron rotor is that, during the

process of machining the material became hard. Hence, the motor was drawing

current more than full-load value even on no-load. To overcome this problem, the

material is made to undergo annealing. It is practically observed that the speed fall

with load is drastic. This is due to heavy losses in the solid iron rotor.

It is proved that the dual excitation theory is also helpful to find the

performance of poly-phase induction motor with solid-iron rotor. The mults obtained

by this method an verified with the values given by the combined quivalent circuit

theory. To facilitate the inclusion of multiple harmonics, for finding p a f o m ~ ~ ~ c e of

3-phase induction motor, multiple excitation theory is developed.

SCOPE:

The closed form solution developed for two-tield excitation is based on the

assumption that th,e frequencies of the two signals are incommensurate. Therefore, the

problem is incomplete in respect of commensurate frequencies.

The analytical solution for finding the field distribution, power losses etc, for

multiple excitations with different amplitudes and frequencies can be a future task.

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