INPERIAL COLLEGE OF SCIENCE. AND TECHNOLOGY
304
INPERIAL COLLEGE OF SCIENCE. AND TECHNOLOGY UniverSity of London Department of Electrical Engineering TRANSIENT ANALYSIS OF SOLID ROTOR TURBO-ALTERNATORS ALLOWING FOR SATURATION AND EDDY CURRENTS by ADEL YOUSEF HANNALLA B.Sc., M.Sc. Thesis submitted for the degree of Doctor of Philosophy in the Faculty of Engineering August, 1975
Transcript of INPERIAL COLLEGE OF SCIENCE. AND TECHNOLOGY
INPERIAL COLLEGE OF SCIENCE. AND TECHNOLOGY
UniverSity of London
Department of Electrical Engineering
TRANSIENT ANALYSIS OF SOLID ROTOR TURBO-ALTERNATORS
ALLOWING FOR SATURATION AND EDDY CURRENTS
by
ADEL YOUSEF HANNALLA
B.Sc., M.Sc.
Thesis submitted for the degree of Doctor of Philosophy
in the Faculty of Engineering
August, 1975
ABSTRACT
Transient analysis of solid rotor turbo-alternators allowing for
the nonlinearities of the iron material and the induced eddy currents
is obtained. The solution is based on the formulation of the electro-
magnetic equations of the machine. Two-dimensional analysis is made to
evaluate the flux distribution in a cross-section of the machine. In
the steady state a discrete form of the partial differential equations
is obtained using the nodal method. It combines the advantages of both the
finite difference and the finite element methods and gives a greater
accuracy than the latter. In the transient analysis eddy currents are
induced in the solid iron and damper circuits producing dissipation, but
the discrete equations are still correctly formulated with the nodal
method. Conductivity matrices are introduced to represent the behaviour of
windings. The coupling between the electric and magnetic fields is clearly
represented in the discrete formulation. Since the problem is nonlinear
because of the dependence of the iron reluctivities on the resultant field,
an interactive procedure is used. The Newton-Raphson method with sparsity
techniques is found to suit the solution of the steady state problem
better than other techniques. In transient problems the Newton-Raphson
method with sparsity techniques and the predictor corrector method are
combined together to form a successful iterative scheme.
The method of solution is applied to two machines; a laboratory
machine and a turbo-alternator. In each machine, flux distributions at
no load are obtained. Eddy current density distributions and flux plots
at successive instants in time are also obtained for field transients.
Eddy currents are induced in the non-magnetic dampers of the microalternator,
and in the wedges and solid iron rotor of the turbo-alternator.
3
. The method is also applied to a loaded turbo-alternator connected
to an infinite busbar. Steady state flux plots are obtained in a pole
pitch of the machine with a suitable model of the stator windings and
motion voltages being included. A field transient on the loaded machine
induces eddy currents. Their distribution during the transient period and
corresponding flux plot are obtained.
Good agreement is found between the computed and test results of
the machines in both steady state and transient disturbances. The methods
used are applicable to any turbo-alternator transient, full consideration
of eddy currents and varying permeability being included.
To
My Family and late Father
ACKNOWLEDGEMENTS
The work in this thesis was carried out under the supervision
of Dr. D.C. Macdonald, B.Sc.(Eng), Ph.D., M.I.E.E., C.Eng., Lecturer in
the Electrical Engineering department, Imperial College of Science and
Technology, London. I wish to thank Dr. Macdonald for his helpful guidance,
constant encouragement and keen interest during the preparation and com-
pletion of this project.
I wish to express my gratitude to Dr. B. Adkins, M.A., D.Sc.(Eng),
C.Eng., F.I.E.E., of the Electrical Engineering department, for his constant
interest in the work, the helpful suggestions he made and the many invaluable
discussions I had during the progress of the work.
I wish to express my gratitude to Mr. C.J. Carpenter, B.Sc.,
M.Sc., M.I.E.E., C.Eng., Senior lecturer, Electrical Engineering Department,
for his many invaluable discussions and the keen interest he has shown in
this work in conferences and to many people in Universities and Industry.
I wish to express my thanks to Mr. A.B.J. Reece, B.Sc., M.Sc.,
M.I.E.E., C.Eng., Research Laboratories of G.E.C., Stafford, for providing
the necessary information of a 333 MVA turbo-alternator.
I am grateful to the Egyptian Government for the financial
support which made this work possible.
Thanks are due to the Computer Centre, Imperial College, for
computing facilities and guidance.
Finally I would like to thank Dr. J.C. Allwright in the Depart-
ment of Computing and Control, Imperial College, and all my colleagues who
contributed directly or indirectly to this work.
6
CONTENTS•
Title
Abstract
Acknowledgements
Table of contents
List of symbols and abbreviations
Page
1
2
5
6
11
Chapter 1. INTRODUCTION 16
1.1 General 16 1.2 Transient Analysis 18 1.3 Application of the Method 19 1.4 Contribution of the thesis 20
PART I
THEORETICAL STUDY OF ELECTROMAGNETIC TRANSIENT BEHAVIOUR OF TURBO-ALTERNATORS 21
Chapter 2. APPLICATION OF THE ELECTRO MAGNETIC THEORY TO TURBO-ALTERNATORS
2.1 Introduction 22 2.2 Basic relationships 22
2.2.1 The relation between field and current 23 2.2.2 Faraday's Law 23
2.3 Application of basic Laws 24
2.3.1 Assumptions 24 2.3.2 Induced voltage in a machine winding 25 2.3.3 Machine equations 25
2.3.3.1. Simplification of the stator equation 27
2.3.3.2. Torque equation 28
2.4 Boundary conditions 28 2.5 Initial conditions 29
Chapter 3. STATIC EQUATIONS OF THE MACHINE IN DISCRETE FORM
3.1 Introduction 30 3.2 The formulation of discrete equations 30 3.3 Finite difference analysis 31 3.4 Finite element analysis 32 3.5 The nodal method 34
Page
3.5.1 Subdivision of the region 35 3.5,2 First order potential approximation 35 3.5.3 Derivation of the nodal equations 36
3.5.3.1. Evaluation of the contour integral of the magnetic field intensity 36
3.5.3.2. The shape of the contour lines 37 3.5.3.3. Nodal currents 38 3.5.3.4. Direct formulation of the elements
of the S-matrix 40
3.5.4 Boundary conditions 41 3.5.5 Finite difference equation from the nodal
method 42 3.5.6 Nodal method and first order finite elements 43
3.6 Numbering System 45 3.7 Choice of the grid 45
Chapter 4. DISCRETE FORMULATION OF TRANSIENT FIELD EQUATIONS
4.1 Introduction 47 4.2 Discrete formulation of the nonlinear diffusion
equation using the method of separation of variables 48
4.2.1 Separation of the time variable 49 4.2.2 Restricted variational calculus and
discrete equations 50
4.3 Variational formulation of the finite element method 53 4.4 Gurtints method 54 4.5 Discrete formulation of the nonlinear diffusion
equation using vector analysis 54 4.6 Formulation of the discrete equations using
Galerkin's principle 54 4.7 Discrete formulation of the nonlinear diffusion
equation using the nodal method 55 4.8 Discrete formulation of the machine transient
equations 57
59
60 60 61
Chapter 5. METHOD OF SOLUTION OF THE DISCRETE EQUATIONS OF THE MACHINE
5.1 Introduction 63 5.2 The iterative procedures for the solution of a
set of nonlinear algebraic equations 64
5.2.1 Basic iterative methods for successive relaxation 65
5.2.2 Successive relaxation methods 65
4.8.1 Formulation of'the conductivity matrix
4.8.1.1. Evaluation of matrix
the eddy current
4.8.1.2. Evaluation of the field matrix 4.8.1.3. Evaluation of the a3-matrix
8
Page 5,3 Methods based on nonlinear optimisation 67
5.3.1 Steepest descent method 5.3.2 Newton-Raphson.technique 5.3..3 Fletcher and Powell method 5.3.4 Conjugate gradient method 5.3.5 The modified Newton-Raphson technique
5.4 Sparsity techniques 5.5 Newton-Raphson and sparsity techniques
5.5.1 Derivation of the Hessian matrix of the static field problem of the machine
5.6 Numerical techniques for the transient problem
5.6.1 Indirect method for the numerical solution of a transient problem
5.6.1.1. Standard formulation of the problem
5.6.1.2. Numerical methods for solving ordinary differential equations
5.6.2 Direct method for the numerical solution of the transient problem
5.6.3 Iterative procedure at each time interval 5.6.4 Initial value problem
PART II
68 68 69 70 72
72 74
74
76
77
77
79
81 81 83
PRACTICAL APPLICATION OF THE METHOD ON MICRO- AND TURBO-ALTERNATORS
Chapter 6. STEADY AND TRANSIENT APPLICATIONS ON MICRO-ALTERNATOR
6.1 Introduction 6.2 Discription of the micro-alternator
6.2.1 Stator 6.2.2 Rotor
85
86 87
87 89
93
93 94 97 97
97 97 99
6.2.2.1. Rotor winding connection
. 6.3 Approximation for the numerical solution 6.4 Choice of the grid 6.5 Grid numbering system 6.6 Modelling of the iron characteristic
6.6.1 B-H curve of the iron material 6.6.2 Methods of modelling the iron characteristic 6.6.3 Methods of modelling the u-B2 characteristic 6.6.4 Modelling of U-B2 characteristic of the iron
material for micro-alternator applications 99
6.7 Iterative procedures and results of their application on steady state magnetic field problems 102
6.7.1 Relaxation techniques 102
6.7.2 Optimisation techniques 102
9
Page
6.8 Computer programs 103
6.8.1 Flow chart of the main program 104 6.8.2 Flow chart of the routine of the numerical
calculations. 104
6.9 No-load flux pattern 108
6.9.1 Open circuit characteristic 109
6.10 Typical results 116 6.11 Field decreement test of the micro-alternator 118
6.11.1 Discription of the test
118 6.11.2 Grid of triangles for the transient solution
118
6.11.3 Determination of the [A6]-matrix• 119
6.11.3.1. Equivalent conductivities of the damper circuit 120
6.11.3.2. Equivalent conductivity of the field circuit 121
6.11.3.3. Approximation for the damper circuit inductances and resistances 122
6.11.4 Unsymmetry of the conductivity matrix and the iterative procedure 122
6.12 Flux distributions during the transients 123 6.13 Eddy current density distributions during the
transient period 134 6.14 Characteristic of the field current 140 6.15 Typical results 140
Chapter 7. STEADY AND TRANSIENT APPLICATIONS ON A TURBO-ALTERNATOR WITH SYMMETRICAL FLUX DISTRIBUTION
7.1 Introduction 146 7.2 Discription of the turbo-alternator 146
7.2.1 Stator
146 7.2.2 Rotor
147
7.3 Approximation's for the numerical solution 147
7.4 Steady state no-load characteristic 148
7.4.1 Choice of a grid for steady state computations 148 7.4.2 Modification of the characteristic of the
iron material 150 7.4.3 Multi-function modelling of the iron
characteristic 150 7.4.4 Flux pattern of the unloaded machine 152 7.4.5 Typical results 153
7.5 Flux and eddy current density distributions during the field decrement test 174
7.5.1 Choice of the grid 174
7.5.1.1. Initial value problem with solid iron and a field winding 177
7.5.1.2. Slot leakage flux and distribution of field nodes 177
10
Page
7.5.2 Conductivity matrix of the field decrement test 179
180 181
181 182 184 184 198 229
229
234 235 235
236 236
7.5.2.1. Field conductivity matrix 7.5.2.2. Iron conductivity matrix
7.5.3 Initial value problem and oscillation in the solution
7.5.4 Modelling of the switiching operation 7.5.5. Overhang leakage. reactance 7.5.6 Flux distribution during the transient period 7.5.7 Eddy current density distributions 7.5.8 Discussion of the eddy current plots
7.5.8.1 Negative eddy currents and the ideal analytical solution
7.5.8.2 Physical explanation of the negative eddy currents
7.5.8.3 Eddy currents in the core and pole 7.5.8.4 Eddy currents in the wedges
7.5.9 Typical results 7.5.10 Effect of the time step on the results
Chapter 8. STEADY STATE AND TRANSIENTS OF A LOADED TURBO-ALTERNATOR
8.1 Introduction 240 8.2 Steady state characteristics 241 8.3 Choice of the grid 242
8.4 Periodicity condition in full pole pitch calculations 244 8.5 No-load flux distribution 244 8.6 Current sheet for a loaded machine 246 8.7 Flux distribution of a loaded turbo-alternator 247
8.8 Typical results for a loaded turbo-alternator 251 8.9 Field transients of a loaded machine connected
to an infinite busbar 251 8.10 Numerical solutiOn using the two axis theory 252
8.10.1 Stator numerical formulation including pip terms 252
8.10.2 Stator numerical analysis ignoring armature resistance and the pIP terms 255
8.10.3 Vector diagram and numerical solution 257
8.11 Flux patterns during transient conditions 8.12 Eddy current density distribution 8.13 Typical results of a loaded machine on transient
Chapter 9. CONCLUSION
9.1 Methods 9.2 Applications 9.3 Future work
References
Appendix A: Steady state finite element analysis using Ritz method
Appendix B: Mathematical methods for changing the nonlinear equation to the discrete form
Appendix C: Effect of the overhang leakage
Supporting publication
270
275
289
299
301
259 263 263
267
267 268 269
LIST OF SYMBOLS AND ABBREVIATIONS
SYMBOLS
Capitals
A magnetic vector potentials (potential)
B . flux density
C constant
D domain, diagonal matrix
E electric field intensity, induced voltage
gradient, error
H magnetic field intensity, Hessian matrix, updated matrix
I current
J current density
L boundary, lower triangle matrix, integrand, differential operator
M moment of inertia, defined matrix, number of triangles
N number of turns, shape function
O null matrix
P matrix
Q matrix
R resistance, region, matrix
S surface, defined matrix
T torque, matrix
U unity matrix, upper triangular matrix, polynomial
✓ voltage
W weighting matrix
Y vector
Z impedance, number of conductors
De region of an element
11
Small letters
a area, coefficient, distance
b coefficient
c contour line, coefficient
d derivative
e induced voltage, element
f function
gradient of a function
i current
k defined factor
length, number of elements in the region
m number, ratio
n normal direction, number
, number of nodes dt
q number of triangles, number of nodes
✓ resistance
s number of nodes in an element, direction of search
t time
u function of x,y,t.
✓ voltage, function of x,y,t.
w defined factor, polynomial, area
x axis
y axis
.12
13
Suffixes
conductor
d
damper, direct
effective, electrical, element
f
field
index
index, inertia
k
index
2, per unit length
maximum, mechanical
0 absolute, initial
q quadrature
r relative
stator , series
t
total
x x-direction
y y-direction
ph
phase
eff
effective
Greek letters
a coefficient, decay factor, matrix, x-coordinate of a point
coefficient, y-coordinate of a point, vector
Y power angle
r contour of integration
6
incremental variation, angle
A.
area of a triangle
14
function, matrix
rL function
scalar quantity
1-1 permeability
U - reluctivity
flux linkage
.111 flux linkage of a winding
a conductivity, matrix
0 angle
T time constant, time
w angular frequency
= 3.141592
energy functional
Ag conductivity matrix. It contains the conductivities and
the areas of the elements.
15
Abbreviations
( ) brackets, vector, function of
I ] brackets, matrix
{ } brackets, column vector
dot product
convolUtion sign, special inversion
integral sign
closed integral
part of the line integral
as a superscript refers to values at nodes and they are used in
preliminary proofs;as a subscript it means vector of several
quantities.
16
CHAPTER 1
INTRODUCTION
1.1 GENERAL
The development and growth of electric power systems has led to a
steady increase in the size of generators, these being economic in comparison
with a number of smaller machines.
At the same time the protection of power systems and the calculation
of stability have required a knowledge of machine transient performance.
These two developments have led to the study of means by which:-
(1) The steady-state electrical parameters of machines larger than
anything previously built may be obtained from design parameters and the
machine may be successfully designed for a particular rating.
(2) The dynamic performance may be predicted.
Steady-state performance has been conventionally expressed in terms
of reactance and resistance. Values of reactance are calculated in the de-
sign office by very approximate methods the success of which commonly depends
on the use of empirical correction factors evolved from experience with
previous machines. These are less reliable when size is greatly increased
in a single step such as when a machine is designed for 11-2 times the
rating of anything built before.
A recent development has been the use of numerical methods to cal-
culate the flux distribution in the machine cross-section. Two approaches
have been made. Erdelyi et al 1'2'3 have approached the field problem
through finite difference analysis, considering conditions at points in
a rectangular grid and solving by successive iteration. Permeability
variation in the iron is taken into account but convergence is only ob-
tained after several hundred iterations, best results being obtained by a
carefully chosen mixture of under and over relaxation. The number of nodes
required becomes excessive when the circular boundaries of rotor and stator
17
are modelled with a rectangular grid.
Silvester et al,4'5'6
have largely avoided the above difficulties.
They have considered the cross-sectional area of the machine divided
into triangles in each of which flux density and permeability are uni-
form. Iron-air-copper boundaries are arranged to fall along triangle
sides and a reasonable model of circular boundaries may be obtained with
many fewer nodes than would be necessary with the rectangular grid. A
solution is obtained by the Newton-Raphson method for the values of
magnetic vector potential, A, at the vertices or nodes. Convergence is
normally obtained in under 10 iterations. This method has gained the name
of "finite elements" but the basic assumptions are very similar to those
of Erdelyi. Each is assuming that permeability and flux density are
constant over a small area. Erdelyi chooses a rectangle, Silvester a
triangle.
Both these methods have been used to study flux distributions in
the steady state. Studies have also been made to obtain sub-transient
reactances from 78 flux plots ' .
Here, the transient performance of the machine has been studied.
Erdelyi's method is unsuitable for this purpose because of its poor
convergence. Silvester's method is attractive but his equations are
derived by the minimisation of stored energy in the field which is difficult
to apply under transient conditions when the field is dissipative.An al-
ternative method has therefore been devised which retains the triangular
elements and the swift convergence of the Newton-Raphson technique. This
"Nodal Method" gives an equation relating current attributed to node a
with the values of potential at that and surrounding nodes by the use
of Ampere's Circuital Law 9. The law is used around each node to obtain
a set of equations very similar to those of Silvester, but having a better
18
current distribution and without any doubt as to their validity under
transient conditions. The Newton Raphson method is found to be the best
for obtaining a solution of the transient equations, being formulated as
an extension of the steady state. A predictor-corrector method is combined
with it for proceeding forward in time.
1.2 TRANSIENT ANALYSIS
Machine transient analysis is considerably more complicated than
that in steady state conditions because:
(1) Currents are not constant but depend on the voltages induced
in windings and circuit impedances.
(2) Currents are induced in damper windings and solid iron. In the
latter the eddy currents and flux density are closely interdependent.
(3) The rotor is in motion relative to the stator, giving rise
to rotational voltages and a continuous change in relative position.
(4) As flux density is varying in time permeability changes and the
new value is dependent on•the previous values when hysteresis is present.
The action of windings in the machine which form closed circuits is
to integrate induced voltage and give rise to a current dependent on the
voltage, the circuit resistance and any external reactance. Thus the field
at particular points is linked together, current at a point being dependent
on the voltage induced at other points. In a cage damper winding the
position is rather simpler, current being dependent on the local voltage
and the end-ring effects which are of secondary importance. The action in
solid iron is similar to that in a damper but the magnetic field and current
distributions are inter-related. All these effects are tackled here with
the formulation of conductivity matrices which enable the correct relation-
ship between voltage and current to be maintained.
19
Relative motion in the machine is considered by the use of two axis
theory, rotational voltages being calculated for the fundamental flux only.
The use of current sheets to represent the stator winding greatly facilitates
the solution.
Hysteresis has for the moment been ignored. The iron used for large
machines has a low remanence and it is thought that this is not a serious
approximation. It is certainly commonly made.
1.3 APPLICATIONS OF THE METHOD
Three applications of the method have been made. Comparison with
experimental results has been possible for two of these.
The field decrement test is the simplest of machine transients, the
field circuit suddenly being short-circuited with the stator open-circuit,
and this was chosen for the initial calculations. It has the advantages that:-
(1) There are no closed circuits on the stator and rotational effects
may be ignored.
(2) The flux and current pattern is symmetrical about the direct axis
and the solution need only be found for half a pole-pitch.
The decay of field current was first calculated for a 3kVA micro
alternator. This was fully laminated and had a cage damping winding. This
made it a good machine for a first attempt, no eddy currents occurring in
solid magnetic material, but the geometry was awkward. Next the same
calculation was made for a 333 MVA 4-pole solid rotor turbo-alternator.
The decay of field current was accurately calculated and the pattern of
the build-up of eddy currents during the sub-transient period is shown.
The decay of flux from the initial steady state condition is also mapped
for both machines.
The calculation for the decay of field current for the 333 MVA gene-
rator when loaded and the excitation supply is lost has been programmed
20
and run for the first two time intervals. Under these conditions the
number of nodes becomes much larger (740 against 192 for the earlier test)
as the solution must be obtained over a pole-pitch. The computing time
required for a solution throughout the decay period would be very great,
and was not available.
1.4 CONTRIBUTION OF THE THESIS
It is thought that the following matters are substantial and original;
(1) The use of Ampere's Law to form equations for the potential at
the nodes of a system of triangular finite elements relating values to
the current distribution.
(2) The use of the above method together with a good numbering scheme
for the nodes, sparsity techniques and the Newton-Raphson method to give
a well conditioned set of equations with a minimum number of entries
located on and about the leading diagonal.
(3) The use of a predictor-corrector method together with the
Newton-Raphson technique to give.a solution of the transient field problem
at successive instants in time.
(4) The development of conductivity matrices to couple together
magnetic conditions in the machine with the electric circuits constituted
by its windings.
(5) The use of the above methods to obtain the flux and eddy current
distributions during machine transient conditions. In particular the
calculation of negative eddy currents associated with a local rise of
flux density in a field decrement test would seem to deserve further
investigation.
21
PART I
THEORETICAL STUDY OF ELECTROMAGNETIC TRANSIENT BEHAVIOUR
OF TURBO-ALTERNATORS
The magnetic field in the machine is treated as a two-dimensional
distribution, which is the same on any section perpendicular to the axis.
Electromagnetic equations of the machine are derived in chapter 2. These
equations are nonlinear and a numerical solution is obtained. The discrete
form of the equations is discussed in chapters 3 and 4. Their iterative
method of solution is explained in chapter 5.
22
CHAPTER 2
APPLICATION OF THE ELECTRO MAGNETIC THEORY
TO TURBO-ALTERNATORS
2.1 INTRODUCTION
The electromagnetic equations of the machine are formulated from
the basic'laws. The approach is not to obtain equations describing terminal
conditions but rather to determine the field within the machine including
the influence Of external loads.
Conventional machine models only attempt to represent terminal
conditions. No attempt is made to correlate the currents and voltages in
electric circuit models with the precise geometry, and the flux and current
distribution within the machine. Here, the machine is viewed as a region in
which flux and current distributions react on each other, the performance
at each point being dependent on the immediate past history of the machine
and also on the loads and voltages connected to the terminals.
2.2 BASIC RELATIONSHIPS
The theory of the electromagnetic field is based on Ampere's law and
Faraday's law, which lead to Maxwell's equations. The changes in the field
considered are slow, so that displacement current can be neglected. Also,
while saturation is taken into account, hysteresis is neglected. Thus the
flux density at any point is a single-valued function of the magnetic force H.
B
prpo H = 1 H
(2.1)
where p is the permeability and U is the reluctivity.Itis convenient to
use the magnetic vector potential A, defined by B = curl A. Since B is a
solenoidal vector (div B = o), this relation together with an arbitrary
choice of div A such as div A = o, determines A completely. In three
dimensions, at any point A is a vector, having three components, each of
23
which is a function of (x,y,z). In a two-dimensional field, however, the
vector A is always at right angles to the plane and its value is a scalar
function of (x,y). The flux density B has components Bx and B given by
DA ay
-DA = ax
(2.2)
The electric field intensity E at any point when the field changes
is
E = - aA
at
(2 . 3)
Both E and the current density J produced by it at a point in
a conducting medium are in the direction perpendicular to the plane.
2.2.1 The relation between field and current
In two dimensions Ampere's law states that
yccH dl = ffsJ dS = I (2.4)
for any surface S bounded by the contour C. I is the total current inside
the contour. Applying Ampere's Law to a small element leads to the partial
differential equation
D DA a DA -57-c U + = -J ax ay ay
(2.5)
The above equation holds at any point in the field. Its solution
depends on the boundary conditions specified by the problem.
2.2.2 Faraday's Law
Faraday's Law gives the relation between the flux linkage 0
in a coil and the induced voltage e between its terminals
d0 e= - — dt (2.6)
When considering a two-dimensional field, the flux cb per unit
°fie s /81
24
length linking a single turn coil having its coil sides located at points
(ctf31)'
(a2'2)
shown in fig. (2.1) is
t = f B dx + IBx dy (2.7)
al 1'2
The interpretation of the above
equations is straight forward and
may be immediately transformed
into vector potential to give the
e.m.f. in a single turn.
e - Al ) e dt (A2
y
/32
0
Fig.(2.1) Single turn coil in x-y plane
(2.8)
when 2,e is the effective length of one coil side and A
l and A
2 are respec-
tively the potentials at points (avy and (ct2,32).
2.3 APPLICATION OF BASIC LAWS
The basic laws of electromagnetic theory are formulated to suit the
machine problem.
2.3.1. Assumptions
(1) In two-dimensional analysis, end effects are subsidiary and are
included separately in the analysis.
(2) The rotor is considered in any position relative to the stator
(3) The rotational voltage arising from harmonics in the flux wave,
are ignored.
(4) Three phase armature currents are replaced by a current sheet
located at the stator slots forming two stationary m.m.f.s in the d- and
q- axes.
(5) Circulating eddy currents in the copper conductors are ignored,
while induced eddy currents in the solid iron are short circuited at the ends.
25
2.3.2. Induced voltage in a machine winding
Machine windings normally have a number of turns distributed over a
finite area. Thus by considering mean values of potential A over the con-
ductor area the induced voltage E in a winding becomes in general
E dt ` fa Al dx dy - f
a A2
dx dy) ] (2.9) m
k
1 ac
where N is the total number of turns in the winding
ml is the number of parallel circuits in a winding
ac
is the conductor area
suffixes 1&2 refer to two numbers of conductors where voltages are
induced in opposite directions.
2.3.3. Machine equations
At any point in the machine the magnetic field is related to the
current density J by equation (2.5). The value of J will be zero at all
points in laminated iron and in air where there is no closed circuit.
Where a circuit exists J will take a value dependent on the circuit charac-
teristics. In a damper bar of conductivity ad short circuited at the ends
of the machine (or in solid iron) where all the conductivity points are
effectively in parallel
v d J = ad ( T - ) 1
(2.10)
where k1 is the effective length and v
d is the voltage drop in the end
ring, the effect of which is explained in Chapter 6. Its value is zero in
solid iron (assumption 5).
In a field winding the turns are in series and all carry the
same current although the voltage induced in each conductor is different.
At the same time, the field terminals are connected to a d.c. voltage source
Ed.c.Thus a field current If can be determined from
26
1 (E
d.c -E
f)
f Rf4.3Z,op
where Rf
is the resistance of the field winding
o is the overhang leakage reactance of the field
_ d P dt
(2711)
Ef is the field e.m.f. which becomes from equation (2.9)
DA2
Ef -r--f I
f /9A1 `at at L=1
(2.12)
where kf
is the effective length of the field conductors and suffixes
1 and 2 refer to two symmetrical positions about the axis of the field
coil. Thus the field current density Jf
can be determined in terms of the
potentials A at the field conductors.
Points on the stator windings should satisfy the condition that
the current is the same in all the turns of a winding and terminal currents
and voltages should also be the same as the external network. Thus for a
voltage v in the circuit which apart from that occuring in the machine
effective core length has a series impedance of Z(p) with the winding,
'ph = (e
ph - v
ph)/Z(p) (2.13)
where Iph
is the phase current
eph
is the phase induced voltage
Z(p) is the operational impedance and can be expressed in
terms of two polynomials U(p) and w(p) as z(p) = w(p)
the current density becomes at any point of the winding Iph J =
ac m1
Inserting these expressions in the equation of potential of equation
(2.5) gives
(2.14)
32A D
2A k
N3 d
U + - C - (f A dx dy -f A dx dy)i o 9
x2 D 2 m a
c dt a 1 a 2
y 1 c i=1 c
vph
1/ acml Z (p)
(2.15)
27
where N3
is the number of turns in series in a stator winding. Since
Z(p) is rational and contains polynomials of the operator p, p= d dt '
the equation (2.15) is an integro-differential equation with order
dependent on the order of Z.
Thus the above equations relate the external network to the magnetic
field of the machine. All the equations are expressed in terms of the flux
distribution and its response in the steady state or during a transient
period. These equations form the electromagnetic equations of the machine.
2.3.3.1 Simplification of the stator equation
The stator windings require special consideration because of the
relative motion between the stator and the rotor which is presented in
equation (2.15) by the total derivative with respect to time. Since the flux
linking the stator windings has not a continuous gradient because of the
slotting of the stator, Faraday's Law becomes more difficult to use. Thus
equation (2.15) cannot be immediately analysed into transformer and rotational
voltages.
Recourse is had to the assumptions made in the two-axis approach
for the calculation of the rotational voltage. Because the windings of large
machines have such good winding factors, harmonic voltages are small and may
be neglected. Fundamental voltage is only generated by fundamental flux and
this may be completely represented by the components along the direct and
quadrature axes. The rotational voltage then appearing in the d-axis coil
of the stator winding is a function of the q-axis flux and vice-versa.
Replacement of the three phase windings by an equivalent current
sheet, which has the same m.m.f., can be done mathematically by considering
two coils with distributed conductors
Zd = Z
m sin 0 (2.16a)
Z = Zm cos 0 (2.16b)
q
28
over angles o to It and -7112
to 11/2 respectively, Currents i
d and i have
constant values all over their windings. The two axis theory can then be
used. The components of the fluxdsap
q linking the d- and q-axis coils are
expressed in terms of the potentials at points on the current sheet as
kZm
f sin() dO
2 = /U
m f A cos dO
-ff/ 2
The value of Zm
is explained in Chapter 8.
2.3.3.2. Torque equation
Following Adkins 10 the electrical Torque T
e is related to the
mechanical a" liec"orquefor a machine with moment of inertia 14.1337
T - T =M. M. p20 m e j
where the electrical Torque is given by
Te = iqd
-idq
(2.18)
(2.19)
The above equation can be expressed in terms of the potential distribution
A at the points in the stator current sheet, atone radius r, by
DA . Te =
m k f . 5(7 (id sirt0 + iq cos()) dO 0
(2.20)
2.4 BOUNDARY CONDITIONS
It is assumed that the flux pattern is limited by the inner
surface of the rotor and the outer surface of the stator. This is not
strictly correct but as long as the back of the stator core has a high per-
meability relative to air the approximation is good. When the flux density
is• symmetrical about a pole axis, flux does not cross the pole axis and is
normal to the interpolar axis. If the axis of symmetry is not known, a
pole pitch of the machine is considered and the conditions on one pole
29
axis are appropriately matched to those along the following pole.
Thus the boundaries are essentially those of Diri.chlet Om = 0),
Neumann DB = 0)and periodicity.
2.5 INITIAL CONDITIONS
The. initial conditions of the machine define the situation of
the machine at the start of the transient. The quantities required depend
on the problem. Normally a current distribution from which the steady state
flux pattern can be determined must be evaluated first. In some transient
conditions it is necessary to know the initial derivatives of the potential
and current functions. These can be obtained with a reasonable accuracy
using a self starting numerical technique.
30
CHAPTER 3
STATIC EQUATIONS OF THE MACHINE IN
DISCRETE FORM
3.1 INTRODUCTION
As a step towards the consideration of transient conditions,
the numerical solution of the field is considered in the steady state. The
approach of Erdelyi is briefly described but is not used. The finite element
method is discussed at some length, as treatments elsewhere are often not
comprehensive. A nodal method is presented as an original method.
3.2 THE FORMULATION OF DISCRETE EQUATIONS
The electromagnetic field equations describing the conditions in
a generator have been given in the last chapter. A solution is sought
subject to the geometry and boundary conditions presented by the machine.
With the irregular geometry and the dependence of reluctivity on flux density,
it is not possible to obtain an analytic solution and approximate numerical
solutions are sought. The variation of potential A over the machine cross-
section is described in terms of values at discrete points, and the equations
giving these values are known as discrete equations. Here a linear variation
of potential between values at nodes is considered.
In all the methods considered the non-linear Poissonian equation
DA D DA
3x „ ax Dy , ay = (3.1)
is not considered directly, but in a large number of small contiguous areas
or elements covering the whole region in which a solution is sought. u is
considered constant and then within each area Poisson's equation
,D A 2
D\ ax 2 3r2
2A ,
(3.2)
holds. The approximation is valid only when sufficient elements are used
and variations in U are modelled by a series of constant values.
31
In the finite difference method the differentials of equations
(3.2) are replaced directly by finite difference approximations taking
into account the variation of the reluctivities in the elements surrounding
the nodes.
In finite element methods, the approach is normally through the
calculus of variations. Equation (3.1) can be shown to be equivalent to a
minimum for B
(A) = f (( f HdB) - JA) d x dy (3 .3) R o
by use of Euler's equation, or may be derived by the minimisation of the
field energy. The minimisation of the sum of elemental stored energy gives
a set of discrete equations.
Almost the same result is obtained by the nodal method, Ampere's
Law being applied around each node.
The finite difference approach is only easy to apply for a rec-
tangular grid although it can be used on a circular basis, the elements
being divided by circles and rays. The use of rectangular elements on the
basically circular geometry is awkward, and the iron-air interfaces present
a problem in representation.
The finite element and nodal methods both use triangular elements
and these lend themselves readily to the modelling of the machine cross-
section. Moreover the size of elements may be readily varied.
3.3 FINITE DIFFERENCE ANALYSIS
Finite difference analysis is a well known technique for altering
the partial differential equation to the discrete form. The method forms a
convenient starting point and often offers the cheapest method of solution
because it is much the easiest to programme. However, the method uses a
regular grid of rectangles or curvilinears. As grid lines coincide with the
32
air-iron-copper interfaces, difficulties arise with. the constructional
details of the machine. Thus the difficulty in fitting one grid to suit the
constructional details of the machine leads to the use of more than one
grid. This increases the number of nodes even in regions where there is no
sharp gradient in the field. The approximation used on the boundary leads
also to a considerable uncertainty in the choice of an approximate difference
equation 11. Unfortunately, the variation of the reluctivity as a function
of the field is unknown. This leads to the use of relaxation techniques, in
which relaxation factors, whose success depends on the experience of the
user, are needed. The finite difference method is not used in the formulation
of the discrete equations of the machine in this thesis.
3.4 FINITE ELEMENT ANALYSIS
The finite element analysis has received considerable attention
in the last decade from many authors. Its earliest applications were in
civil and mechanical engineering. Its use has been brought to electrical
engineering to solve magnetic field problems by Silvester S. In most appli-
cations the Ritz method is used with a first order approximation of the po-
tential in each triangle, although higher order elements have been suggested.
A brief sumfflary of the first order method is given, the mathematics is in
appendix A.
The method established is based on altering the partial differential
equation into an integral form using Euler's equation. The solution of the
field, expressed by the partial differential equation subject to given
boundary conditions, is equivalent to finding the extremum of the integral
obtained. Thus the method starts by using the variational calculus.
The region of the machine is divided into triangles ensuring that
air-iron boundaries only lie on triangle sides. Within each triangle, the
reluctivity is assumed uniform. At the vertices of the triangles, the grid
33
nodes are defined. Thus the energy stored in the field expressed by the
integral obtained is divided between the elements.
A considerable literature now exists on the characteristics
of the elements. Higher order elements, curvilinear elements, elements with
minimum square root error ...., etc., have been treated while values at
nodes have become subsidiary in the solution. The potential at any point
inside an element is expressed in terms of the potentials at the nodes of
the element and their coordinates. Simple relations may be found for first
order elements(triangles). Expressions for the flux density within an
element may be found,from which the reluctivity can be determined.
The relation between nodes is obtained by minimizing the sum of
the energies stored in each element. This does not mean the minimization of
the energy stored within each element alone in contrast with ref 5. It is
following Ritz's method that the minimization of the integral in the form of
functional 9(A) leads to the system of equations 12
a g.(A)
DA (A) = o i = 1 ...,n (3.4)
,
where A. are the potentials at the nodes. Since the properties of each
element are characterised only by the potentials at its nodes this leads to
the formulation of element matrices. The assembly of the element matrices
leads to a set of equations in the form,
[S] {A} = {I} (3.5)
where [S] is a matrix of equations representing the field and {I} is the
vector of the assembly of currents at nodes.
Ways of including Dirichlet's boundary condition are introduced
in the formulation of equation (4.5) 4'13. Neumann's boundary condition is
implicitly satisfied in the functional formulation.
The elements in a region are numbered sequentially and nodes are
numbered independently. The number of the element and the numbers of the
34
nodes at its vertices are described. The numbering of the nodes takes a new
internal numbering system. Accordingly, element matrices are formulated.
The internal numbering system of the nodes in any element return back to
the original node numbering system. The formulated element matrices are then
assembled in a way dependent on the original numbering system.
The finite element method has thus a formulation dependent on
the minimisation of energy. It needs a lot of mathematical explanation
as summarized in Appendix A. The method lends itself to a systematic
approach which is applied without consideration of the shape of the final
S matrix. Thus the relation between the numbering of the nodes and the
positions of non-zero entries in the S-matrix are not known and the matrix
with the least band-width is not normally obtained. As it stands the method
is not suitable for the solution of transient problems as explained in
Chapter 4. An alternative method which avoids many of these difficulties is
given below.
3.5 THE NODAL METHOD 9
In this method the solution of the magnetic field problem, as in
the other methods, is described at a finite number of nodes. The main con-
cept is that if a solution is required at nodes, then equations have to be
derived so that the best approximation at the nodes is obtained. Current in
the vicinity of a node is considered to act at the node. The potential at
any node is related to the neighbouring nodes of the cluster, while the
potential within any element is described by an approximate distribution.
The method brings the field equations in the form of equation (3.5) directly
from Ampere's Law. Thus Ampere's Law is an alternative starting point for
the solution of the nonlinear partial differential equation (3.1). The
boundary conditions are satisfied in the discrete formulation of the problem.
35
3.5.1. Subdivision of the Region
A grid is used to divide the machine region R within its boundaries
into elements so that all air-iron-copper boundaries lie along element sides.
These elements are triangles or rectangles. Within each element the flux
density and permeability are assumed uniform. The rectangular elements are
only chosen in air or non-magnetic regions where the reluctivity is indepen-
dent of the resultant field. Thus the analysis can be done for nodes surroun-
ded by triangular elements while rectangular elements can be regarded as two
specially combined triangles. The points of intersections of the grid lines
are the grid nodes. Any triangular element is defined therefore by the coordi-
nates of three nodes at its vertices. Any node is surrounded in general by
M triangles and M nodes.
The value which the solution has at a node p is assumed to exactly
correspond to the true solution A at the node.
Let the region R contain the domain D surrounded by the boundaries
Li and L2 so that R =D + L
1 + L
2. The number of nodes on the boundary L
1,
satisfying Dirichlet's condition is ni. The number of the remaining nodes, which
lie on D + L2
is N. The nodal values of the elements are identified as A1,
A2, A3 with the subscript 1 always at the node for which an equation is to
be obtained. The subscripts 2 and 3 are used to identify the other triangle
nodes defined in an anticlockwise direction.
3.5.2. First order Potential approximation
The potential A at any point inside the triangle can be expressed
in the linear form as given in equation (A.20) as
3cc 1 A = —2A
ly) (a.+ b.x + c. (3.6) 1=1
wherenecoefficientsai,bi andc.are given by equation (A.18) (in
Appendix A) and its cyclic permutation. B and v are given uniform values in
each element by this expression.
36
3.5.3 Derivation of the nodal equations
Any node p shown in fig (3.1) is surrounded by M-triangles. It
is also enclosed by the contour r marked by the points a,3,y, .a. , such
that these points divide the lines radiating from p in ratio m : (1-m). A
typical triangle of this mesh around the grid node where a part of the
contour line r joins the points a and 8 is shown in fig. (3.2). The coordi-
nates of the points a and 8 are Exi+m(x2-x1), yl+m (y2-y1)] and Exl+m(x3-x1),
Y1 m(y3-y1)] referred to the two fixed coordinates x and y with respect
to the grid.
3.5.3.1. Evaluation of the contour integral of the magnetic field intensity
The integral between the points a and 8 of the components of the
flux density in the directions x and y are
8 f B
x dx = B
x (x0-xa)
a
8 f B dy = B
Y (y -y
a)
a
and the total integral of H.0, from a to 0 is
xf H.d2. = U CB (x - x ) + B (y -y )] x 8 a Y a th
a of thej triangle
Now
B = 1 A.c. x LA i=1
(3.7a)
(3.7b)
(3.8)
(3.9a)
and
?. = - 1 L A.b.
2A i=1 (3.9b)
substituting equations (3.9 a,b) into equation (3.8) yields
8 3 3
4 H. = U
20 C(x
8 -xa
) y A i=1 I1
.c. -(Y(3 -Ya1 ) X A.bAl1
(3.10) a • 1=1 jth of the triangle
The coordinates of the points a and 8 can be expressed in terms of the jth
37
triangle vertices&then equation (3.10) becomes,
0 3 3
2A H.dk= - (x ) 11 A.c. m(y
3 ) y A.b.]
a 2-x 3 i=1 i=1 "
(3.11)
The line integral of the contour I'around the node p in fig. (3.1) is thus
M U. 3 3 H.G12,-= - m X [(x2-x3) X Aici + (y3-y2) y Aibi] (3.12)
j=1 i=1 i=1
The factor m in the above equation defining the intersection of the contour
line with the triangle sides remains to be chosen.
3.5.3.2 The shape of the contour lines
The value of fH.d2. between two points a and 8 is independent
of the path if there is no current and the field obeys Laplace's equation.
When there is a current density distribution in the region of integration
then the governing equation is Poisson's equation. The integral of H.dt
between the two points a and 8 in fig. (3.2) then becomes dependent on
the path between the two points. Therefore, the shape of the contour line
around any particular node' determines the current which may be taken to
act at that node.
Contour lines may be chosen in two ways:
Approach A.
The current taken to act at any node is the integral of current
density over a region in its vicinity, each element around a node making
a contribution. The area of each triangular element may be equally divided
between its vertices and this leads to the subdivision shown in fig.(3.3).
The contour of integration now follows the path a8y and m=0.5. This leads
to a result identical to that obtained in the finite element method.
Approach B.
The best approximation of the field distribution is that which
14 gives minimum error at the nodes. Collatz shows that if a field is
38
approximated over a region bounded by straight lines by a sum of linear
functions, then the point at which_ the greatest accuracy is obtained
(the centre of reference) is that of minimum total perpendicular distance
from all the boundary lines. Thus for a triangular element the centre
of the inscribing circle is the centre of reference.
However, the aim here is to make each node the centre of
reference for the straight lines around it, along which the path of contour
integration is taken. Thus a grid of the contour lines is introduced which
fulfill this condition. These, are shown by broken lines in fig. (3.4) and
form an orthogonal grid to the original grid with m = 0.5.
The part—contours in a typical element are shown by broken lines
in fig. (3.5). The areas attributed to each vertex are not equal except
in an equilateral triangle.
The contour lines of approach A are shown in fig. (3.6) 15. It
will be seen that approach B appears to give a more logical result, the
contribution of triangles at a node being related to their angle at that
vertex.
3.5.3.3. Nodal currents
The currents specified at the nodes depend on the choice of
contour lines around the nodes.
In the first approach, the current specified at a triangle vertex
depends only on the current distribution and the triangle area, but is
independent of the shape of the triangle. It can be seen that the currents
are distributed at the triangle edges and no definite contour lines are
required 16.
In the second approach, the current attributed to a node from
a triangle depends on both the current density distribution and the shape
of the contour line.
39
Fig.(3.1) Typical triangles around a node. Fig.(3.2) Typical triangle in a grid.
P
Fig.(3.3) Part contour line from equal Fig.(3.4) The grid of the nodal method areas to each vertex. ----contours of integration
Fig.(3.5) Part-contour in a typical
Fig.(3.6) Contour lines from dividing triangle. each element into three equal
areas.
40
Each of these methods gives a particular current distribution
specified at the nodes. Approach .A is compatible with finite element analysis,
approach B with finite difference analysis.
th The current supplied to the node from the j surrounding triangle
can be expressed as 3
k k 4c=1
The current at the node p from the M surrounding triangles is
M 3 I = X 131, 4 Jt, 4
j=1 k=1 rs."
(3.13)
3.14)
The value of 0k
in equation (3.13) depends on the approach and can be
calculated as follows:
In approach A, it is calculated as explained in Appendix A.
In approach B, the point of intersection of the lines orthogonal
to the mid points of the triangle sides is calculated. The integral of
the current density over the area enclosed by the contour lines determines
the currents associated with the triangle vertices.
The coefficient (3.k. in equation (3.14) can be calculated from
the predetermined 8k. Its value depends only on the two triangles with the
common side which joins the nodes p and j.
In practice the two approaches are combined together to obtain
an adequate representation with minimum computational cost.
3.5.3.4 Direct formulation of the elements of the S—matrix
The analysis in sec. 3.5.3.1 shows that the contour integral of
the flux intensity around any grid node can be expressed in terms of the
potentials of the vertices of the triangles surrounding it. With m = 0.5,
the integral in equation (3.12) can be expressed with the aid of equation
(A.18) as M U. 3 3
i H.dt = 1— —1- [c XAc+b IA.b.71 (3.15)
2 2 1 i i 1 1 1 I F j=1 Aj i=1 1=1 th j of the triangle
41
Equation (3.15) contains the nodal potential which is always identified by
the 'suffix 1 as defined in sec. 3.5.1. The diagonal element of the S-matrix
at the node p is
Sp'P
7 u. 032 c2 ‘10 j=1 3 1 l j j (3.16)
where b1,
and clj
are the b1
and c1 of the j
th triangle. The general
nondiagonal element of the pth
row is zero except for the adjacent nodes.
Any node k is related to node p only through the two triangles number j
and j+1 around the node shown in fig. (3.7). The entry in the kth column
of the pth
row is
U. U. 1 S = J
1. b3.
+ c1. c3. ) + 4A
J+L (b1 b2
c1 c2 ) (3.17) p,k 4A1 0 J J 3 J j+1 .
j+1 j+1 j+1 j+1
The total number of elements in the pth row or column of the S-matrix is
(M+1).
3.5.4 Boundary Conditions
The solution of the nonlinear partial differential equation (3.1)
is only obtained when the field satisfies the boundary conditions. In the
magnetic field problem of the machine, the two boundary conditions, as
stated before, are Dirichlet's and Neumann's boundary conditions.
Dirichlet's boundary condition can be chosen so that all nodes
lying on the boundary L1 have zero potential and do not appear as variables
in the equations. This means that the contour integral has only to be taken
around the remaining nodes. The zero-valued nodes on the boundary contri-
bute only in the S-matrix. Thus the number of nodes where a solution has
to be obtained is equal to the total number of nodes less the number of
nodes on L1. The size of the S-matrix in sec.(3.5.3.4)is then NxN where N
is defined in sec.(3.5.1.)Any number of nodes can be chosen on the Dirichlet
boundary without affecting the number of equations to be solved.
42
Neumann's boundary condition, at which the flux crosses the
boundary orthogonally, cannot be treated exactly as an interior node. A
typical situation is shown in fig. (3.8) where nodes a, e and d lie on
the boundary on which the A-field is symmetrical. Because of this symmetry,
it is not necessary to make a closed integral about e for the integral pqr
is identical to that beyond the boundary. Thus
I 1 H. dQ = e
pqr
3.5.5 Finite difference equation from the nodal method
With the assumptions of constant flux density and reluctivity
within each rectangular element the finite difference approach is a special
case of the nodal method. Fig. (3.9) shows the rectangular area around a
node divided into 4 rectangles or 8 triangles for which discrete equations
may be obtained by nodal and finite difference methods. In applying the
nodal method a contour line has to be defined. This line, using the second
approach in sec. 3.5.3.2 is indicated by the dotted lines in fig. (3.9).
The potential at node 0 is related to the potentials of the surrounding
nodes, i.e., from 1 to 7 without including node number 8. By substituting
for the elements of the S-matrix in equations (3.16) and (3.17), it is
found that the coefficients of the nodes from 1 to 4 are as given by finite
difference expressions. It is also found that nodes 5, 6 and 7 have zero
coefficients. The current supplied at node 0 is the total current enclosed
by the contour line and it is exactly the same as in the finite difference
method.
Alternatively, one might say that the finite difference equation
could also be derived from Ampere's Law 17. The solution must then be the
same as the nodal method which. is also derived from Ampere's Law. The
important factor is that the choice of the lines changing the rectangular
grid into a triangular one does not affect the shape of the resultant
43
contour line. If approach_ A in sec. 3.5.3.2 is used, then the currents
supplied to node 0 do depend on the way the triangles are drawn. This
means that the equations obtained by the methods are not the same and
consequently the solutions are different.
The difference between the finite difference equation and the
nodal method, where approach B is used, appears in the method of calculating
the reluctivities. In the finite difference analysis the reluctivity t) is
obtained by calculating the average flux density within the rectangle element,
for instance, 0152 in fig. (3.9). The two triangles formed from each rectangle
can have the same value of reluctivity or individual values. This difference
disappears in regions where the permeability is independent of the flux
density such as in air or copper. In these regions there is no difference at
all in the resultant equations between the triangular and rectangular grids.
A combined grid of triangles and rectangles can therefore be used in the nodal
method as introduced in sec. 3.5.1.
3.5.6 Nodal method and first order finite elements
Comparison between the results of sec. 3.5.3.3 and those of
sec. (A.8) shows that the elements of the S-matrix are the same whether
they are derived from the nodal or the variational method. There is thus
no need to find the variational formulation of the problem in order to get
a solution. In other words, if the field to be studied is energy dissipative
and it is difficult to find the variational formulation of the problem, then
the discrete form of the equations can be obtained from the nodal method
without recourse to the variational formulation. Higher order elements have
been formulated by variational methods and it is likely that the same sort
of results may be obtained with the nodal method.
The main advantage of the nodal method over the first order
finite element analysis is that a contour line is defined by approach B.
of. Fig.(3.8) Nodes on Neumann's boundary.
44
Fig.(3.7) Typical node surrounded with M-triangles.
3
7 3 4
Fig. (3.9) Mesh-block of the finite difference method and the contour of the nodal approximation.
.2
3,J3 )Z , J1
I 0 t I I I
..1 1 ,l' 13.1 1
'..)4 , Jst
Fig.(3.10) Optimal grid of triangles.
a1 '
cc2
contours of triangles T1 and T
2 respectively.
Fig.(3.11) Unsatisfactory contour of integration with an obtuse-angled triangle.
45
The choice of a grid of triangles subdividing the machine region becomes
limited to acute-angle triangles a.3 explained in sec. (3.7). Otherwise
big errors in the resultant field arise. The correct values of the currents
located at the grid nodes are obtained from the nodal method. Wherever an
accurate solution is important tha nodal method is necessary, but in other
regions of low current density approach A may be used without a severe loss
in accuracy and with economy in computation. This point is explained in
more detail in chapter 7.
3.6 NUMBERING SYSTEM
The best numbering system of the grid nodes is that in which the
bandwidth of the resultant S-matrix is minimal. It is clear in the nodal
method that any node is related only to its surrounding nodes. Therefore in
order to get the minimal bandwidth of the S-matrix, the grid numbering must
be done in such a way that the modulus of the maximum difference between a
node and any of its surrounding nodes is a minimum.The grid nodes are best
numbered sequentially in the direction of the minimum width of the grid.
The S-matrix then has a minimum bandwidth limited by the periodicity of
numbering of the grid nodes.
It should be noted that because all the nodes lying on the
Dirichlet boundary are not included in the formulation of the S-matrix, they
are all numbered sequentially at first. All other nodes of the grid.are
numbered afterwards.
3.7 CHOICE OF THE GRID
The choice of a grid of triangles has the flexibility to satisfy
the boundaries with a reasonable and acceptable error. The optimum choice
of the grid occurs (in a region of constant permeability) when a node is
related to its adjacent nodes by equi-dimensional coefficients in the S-matrix.
46
This can be satisfied when the grid has the hexagon as a base with the
nodes lying at their centres as shown in fig. (3.10). This choice is not
always practically possible because of the need for a minimal computational
cost. Such a grid may not satisfy the internal shapes or the external
boundaries readily, nor does it give equal coefficients when the reluctivity
of the material is variable. However, all triangles should have angles less
than 90°.
If an obtuse-angled triangle is used, one of the vertices give
a negative addition to the diagonal element of the S-matrix. It makes the
matrix ill-conditioned and the grid is weak at that node. At the same time
the centre of the triangle T1, a1
shown in fig. (3.11) lies outside the
triangle. This makes the contribution of the current at the node unsatis-
factory because of the awkward contour line shown in fig. (3.11). A part
of the contour line also lies in the adjacent triangle of different reluc-
tivity.
47
CHAPTER 4
DISCRETE FORMULATION OF TRANSIENT. FIELD EQUATIONS
4.1 INTRODUCTION
The methods available for obtaining numerical form of the static
magnetic field in a machine were discussed in Chapter 3. Here, the work is
extended to obtain discrete equations of fields under transient disturbances.
There the values of A were sought at discrete points, here values are sought
at the same points but also at discrete instants in time. Now there is an
additional "boundary" condition, the initial condition, which must be satisfied.
Once the initial condition, has been taken into account the field has a unique
solution at each instant, being a physical realisable system.
Methods of adapting the machine equations (Chapter 2) for simple
transient disturbances, so that discrete equations are obtained,are discussed
first. The nonlinear diffusion equation is thus considered. It is found that
any of the following approaches may be used:
a) Variational calculus followed by Ritz's method of finite element
formulation;
b) Gurtin's variational formulation followed with the finite
element method;
c) Vector analysis;
d) Galerkin's method;
e) The nodal method.
The first four methods lead to the same set of equations when the
same degree of numerical approximation is used. Although the last one is
simpler than all the others it gives a superior field distribution.
In all the above methods, because the field is expressible. at
discrete instants in time, the variation of the field in the time domain is
separated from its variation in the x—y plane. Moreover, the field, being
48
approximated over a region divided into elements, the potential at any point
within an element can be written as
S A(x,y,t) = Ni(x,y).
i=1 (t) (4.1)
whereNi(x,y)aretheshapefunctionsandA.e (t) are the potentials at the
nodes of the element at a given instant. Thus the field at any instant in time
can be expressed in terms of the potentials at the nodes and their derivatives,
when the field at the previous instant is fully defined. The approximation
(4.1) is used, at various stages in the above methods. It can be shown that
the separation of the field variation in time from its spatial variation is
the only way to find a solution of the problem. For instance, if the variation
of the field along one axis with time is required then the whole field prob-
lem has to be evaluated at a set of successive instants.
The nodal method gives a direct formulation of the discrete
equations. Being based on Ampere's Law there is no problem of dissipation
and the discrete equations are directly related to the physical problem. The
many mathematical derivations and arguments aimed at solving the nonlinear
diffusion equation, are thus eliminated. What otherwise is a difficult prob-
lem, is with the nodal method straight forward and more difficult conditions
can be tackled. Thus conductivity matrices are introduced to present the
effect of a winding in a magnetic field. The method clarifies also the
coupling between electric and. magnetic fields in a discrete form.
4.2 DISCRETE FORMULATION OF THE NONLINEAR DIFFUSION EQUATION USING THE METHOD OF SEPARATION OF VARIABLES
Any solution of the nonlinear diffusion equation is faced with
two main problems:
a) The nonlinear dependence of the reluctivity on the resultant
field;
49
b) The spatial variation of the field and its dependence on the
rate of change of the field itself in time.
The solution of the first problem has been sorted out as shown
in Chapter 3 when the field was considered stationary. The inclusion of
time effects is best understood by assuming initially that u is constant.
Later the variation of u can be included.
4.2.1 Separation of the time variable
Considering a magnetic field problem expressed by the partial
differential equation
32A 32A 3A
= Cr 3x2 ay
at (4.2)
over a small area or element. The potential at any point in a defined region
in which the partial differential equation holds can be approximated by a
limited sum of 2m functions.
A (x,y,t) = y ni (x,y) Ei.(t) i=1
where m is large enough to give a reasonable approximation
(x,y) denotes the variation of the function within the spatial
region D of the element e.
(t) the time function which corresponds to fl(x,y).
The continuous function expressing the diffusion equation (4.2) can be
expressed in terms of the functions II., E., i=1, in as
m .,2
ail (t) d 11(x,y) m2ri (x,y)
0( 1 . Ei. CO + X 2 E. (0)- a i n.(x,y) 1 at
i=1 3x2
i=1 ay i=1
(4.4)
Equation (4.4) can be written in the matrix form
U(E(t)) 02n(x'Y) + 32n(x
2
,Y)) = a ( DE(t) {il(x,y)} (4.5) Dt
Dx2 3y
Premultiplying both sides of equation (4.5) by a matrix which has its
1 diagonal elements as 7.-
c(t>
UCUI{121- (x'Y) 3x2
gives
32Ti(x,Y)
3y2
1 K(t) - (t) at
50
(4.6)
where, the matrix [1 3(t) -Z-(t) at
1 is a diagonal and EU1 is the unity matrix.
Equation (4.6) is a matrix form of m separate partial differential
equations. Each is independent of the other or there is a singularity in the
solution. Therefore each of these equations has a solution. The function to
be selected from these equations is the one that satisfies the boundary and
the initial conditions. It is known as the particular solution. Because the
solution is unique, therefore, there is only one function in equation (4.3)
which gives the required solution. Equation (4.3) can be written as
A(x,y,t) = n(x,y) E (t) (4.7)
wherein for simplicity suffix i and the summation sign are omitted.
4.2.2 Restricted variational calculus and discrete equations
Restricted variational calculus may be used to derive the discrete
equation equivalents of the nonlinear diffusion equation, which express the
field at discrete instants in time. Since equation (4.6) is an expression
of the function n which depends on x and y this implies that
1 3 = constant = C1
(4.8) at
therefore, substituting equations (4.7) and (4.8) into equation (4.6)
gives
32n 2n u( — ) = c1 a n 3x2
Dy2
This equation has an integral form using Euler's equation (A.1) as
• u Dn 2 3n 2 1 2 } (n) II (__) —C an dxdy
3y 2 1 De
"De 2 Dx
(4.9)
(4.10)
Similar functionals with different reluctivities can be evaluated over
each element. Hence, the functional to be minimised over the region of
51
elements which satisfies the nonlinear partial differential equation subject
to Dirichlet's and Neumann's boundary conditions. is
01) = (1) e=1 De
1 ff C011)2 -4- 1-1)2] cl o n2} dxdy
e=1 De 2 ax ay 2
The substitution of equation (4.8) into equation (4.11) gives
;c r f { r( ue DA.2 DA 2 1 a 2 (n) 1 53) dxdy
e=1.1.1
De 22 D xi
(4.11)
(4.12)
since2
is a positive quantity, the minimisation of equation (4.12) implies
the minimisation of
sy;(A) f
{uevali,2 (DA .2, 1 DA 1 2 ‘Dxl Dy
) j + --a A — dxdy e=1 De
2 Dt (4.13)
with respect to the spatial variation of A only subject to, the formulation
of equation (4.10).
The discrete solution of the problem is obtained by expressing
the field at nodes. The variation of the potential A expressed by the functions
n and of equation (4.7) can be approximated within each element in terms of
the potentials at the nodes by equation (4.1) repeated here:
S -e -e A(x,y,t) = X Ni (x,y) Ai(t) (4.14)
i=1
where S is the number of nodes within the element. In the first order triangle
element S=3. Since the functional (4.13) is essentially an expression for the
space variation of the function as given by equation (4.11) then Ritz's method
can be used only at discrete instants in time. Therefore by applying the
relavent minimisation condition in sec. A.6 on equation (4.13) yields
where
• 'DA = Es] {A} EN1 9t i{--4
= 0
12, ES] X [Se]
e=1
(4.15)
(4.16a)
52
and
ei. L [N1 e=1
(4.16b)
The components of the matrix ES1 are the same as those expressed in
equation (A.54). The matrix [147] has its elements derived from
Me1 = a ffDe (Ne)
T (N
e) dxdy (4.17)
The above expression , with a linear approximation of the potential A gives a
similar relation to equation
e M1
=
(A.51),
A 2
1
1
1
2
1
1
1
2
(4.18) 12
Boundary conditions are justified in the same manner as explained
in sec. 3.6.6. Additionally values of 2A
on the Dirichlet's boundary con-
dition dition are equal to zero. Initial conditions are verified by evaluating the
initial derivatives of the potential with respect to time from the initial
field distribution. This can be done as follows:
Since the solution of the field at t = -o is given by
[S] {A(o)} {I o} 0
(4.19)
which must be the same as the initial value at t = +o for the solution of
equation (4.15),
[S]{A(o)} Ey.1(°).1 (4.20)
It follows from equations (4.19) and (4.20) that the initial values ofM at
can be determined. This part of the analysis has also been mentioned in
references 18,19,20.
It is applied in chapters 6 and 7 on micro- and turbo-
alternators. It is found that the method of approximating the derivative of
18,19 the potential given in references
18,leads to an oscillation on the
solution. These oscillations are small when the induced eddy currents have
small effect on the machine response. A method for avoiding these oscillations
which are otherwise large when the induced eddy currents have considerable
effect is explained in chapter 7.
53
As a result of the above analysis, it may be seen that although
the field is dissipative, at any given instant in time it arranges itself
to have minimum stored energy. The minimisation condition does not become
the minimum of the stored energy with respect to the potential A as a function
of x,y and t but the minimum of the stored energy at a given instant with
respect to the potential A which is a function of x and y at that instant.
The alternative methods 1-4 listed in sec. 4.1 are explained in
detail in appendix B. The arguments and mathematics used in these methods
are summarised in the next four sections.
4.3 VARIATIONAL FORMULATION OF THE FINITE ELEMENT METHOD
The variational method starts by altering the partial differential
equation into integral form with the condition that the obtained integral
should be minimal. When the field is dissipative Euler's equation cannot be
applied and consequently the condition of minimising the energy stored becomes
unrealistic. Thus it is not possible to obtain a general integral which
expresses the field at any instant independent of the initial conditions.
2221, A number of authors
13,21,22 have used the integral of the static field
problem and after evaluating the discrete equations of the static problem,
or even the integral of the static field, they substituted for the time-
varying field. The mathematical arguments then become loose.
It is not clear that the discrete equations derived for the, static
problem hold under transient conditions because the integral has not been
shown to satisfy Euler's equation under the circumstances that it has been
used.
When current density is replaced by a function of at the the minimi-
sation condition may not be satisfied.
54
4.4 GURTIN'S METHOD
Gurtin23 overcame the difficulty of using Euler's equation and
performed an integral which satisfies specified initial conditions. The
method uses the convolution and a particular formulation is derived which
contains the initial conditions implicitly. Ritz's method is then applied,
where approximation (4.1) is used, to put the field in the form of a set of
discrete equations19.
Once convolution is applied to the diffusion equation it implies
that there should be a separation of the time variation of the field from
its spatial variation. The use of equation (4.1) emphasises the separation
of the time from the space variation but in a later stage in the method.
The method contains some mathematical complications as shown in Appendix B
and could be solved in much simpler ways. Nevertheless, it allows for the
initial conditions in the solution in a way which is not possible in other
methods.
4.5 DISCRETE FORMULATION OF THE NONLINEAR DIFFUSION EQUATION USING VECTOR ANALYSIS
This method does not need the formulation of an integral with which
to replace the nonlinear diffusion equation. It solves the problem using a
self-consistant approximate scheme for the potential at the nodes of the
element 24. The method proves that the field should be stable at any given
instant in time and independent of any assumed field variation about the
stable position. Because the method uses approximation (4.1) it describes the
field at discrete instants in time.
4.6 FORMULATION OF THE DISCRETE EQUATIONS USING GALERKIN'S PRINCIPLE
Galerkin's method is a direct method of formulating the discrete
equations for the given partial differential equation subject to the stated
24 boundary condition . The method is similar to the previous methods and
55
assumes that the field can be approximated by the sum of approximating
functions defined individually over each element. The unknowns in these
functions are their coefficients, which could be the potentials at the nodes.
This method utilizes the linear independence of the approximating
functions from one element to the next. Since the potential in terms of the
approximating functions should satisfy the partial differential equation
with zero residual, then the partial differential equation in terms of these
approximating functions is orthogonal to each approximating function. This
condition leads to a system of equations where the unknowns are the coeffi-
cients of the approximating functions at the grid nodes. The method leads
finally to the same set of equations as the other methods as shown in
Appendix B.
Careful examination of the method shows that constant coefficients
are assumed in the approximating function, but later approximation (4.1) is
used in which the constant coefficients are replaced by the potentials at
nodes expressed as functions in time. This raises doubts as to the validity
of the method. Silvester uses it ref. (26) but calls it a "weak" formulation.
4.7 DISCRETE FORMULATION OF THE NONLINEAR DIFFUSION EQUATION USING THE NODAL METHOD
The transient two-dimensional nonlinear diffusion equation is
3 „ DA D DA _ DA Dx - 3x 4. Dy u Dy aDt
(4.21)
The static problem can be expressed by the nonlinear Poisson equation (3.1),
D , , DA D DA a Dx Dy By' =
(4.22)
The left hand side of the above two equations are the same. The
terms in the right hand sides show that the current density in equation (4.22)
takes the special form in equation (4.21) of
56
J = a (4.23)
The integral over the area of both_ sides of equation (4.22) corresponds to
Ampere's circuital law in the form
= ff J.dxdy (4.24) r s
The contour integral around each node encloses a current which is attributed
to the node. Thus for each node a discrete equation can be derived. Since
Dirichlet's boundary nodes with. A=o are not included in the solution, the
discrete form of equation (4.24) is
[S]. {A}={I} (4.25)
When the field distribution changes with time currents attributed to nodes
are consequently functions of time. Their values can be derived from
equation (4.23).
The left hand side of equation (4.22) corresponds to i H.dk which
has the discrete form [S] {A}. Therefore the same discrete form has to be
assigned to the right hand side of equation (4.21) when the same approximation
over the elements is used. Current density in the right hand side of equation
(4.22) is replaced in the discrete form by the vector {I} at the nodes.
Therefore the discrete form of the right hand side of equation (4.21) can be
derived in the same manner as the vector {I} but its formulation depends on
the time variation of the field at the nodes. The discrete form of equation
(4.21) can be written as
[s]' {A} = -[M1] frd
(4.26)
where the elements of the [S] matrix are exactly the same as derived in
sec. (3.5.3.4). The M1
matrix has its elements derived from the assumed J
distribution within the elements, the contour lines and the conductivity
of each element surrounding the nodes.
Dirichlet's boundary condition is satisfied in the formulation
(4.26), because all nodes on the boundary with A=o are not included in the
57
• derived equations. Subsequently at these node at = o is automatically
satisfied. This is one of the essential conditions that any solution of
the transient problem must fulfill. Neumann's boundary condition is satis-
fied by the image concept explained in sec. (A.6). It is naturally satis-
fied at any time.
4.8 DISCRETE FORMULATION OF THE MACHINE TRANSIENT EQUATIONS
The electromagnetic field theory introduced in chapter 2 leads
to an expression for the field at any point within the machine cross-
section in terms of partial differential equations subject to boundary
conditions. Although the equations obtained are different from one region
to the other, all originate from Ampere's Law. As explained in the previous
section, Ampere's Law has two components; one expresses i H.(12, and the other
is the current I. Application of this Law around the grid nodes alters the
field problem to the discrete form of equation (4.25). When the current
density takes the form of - a DtA- the discrete equations are those of
equation (4.26) where the matrix—MI
appears as a weighting matrix of
conductivities.
The discrete form of the machine equations is rather more compli-
cated than the obtained form of equation (4.26). The method of formulating
a discrete form for such a problem can be clarified by grouping the nodes
into different types: Nodesfed with zero currents, nodes where eddy currents
can be induced, field winding nodes and stator winding nodes. When the node
numbering is applied to the various types of nodes sequentially, the
formulation is simple. The following discrete forms can be derived:
a) Nodes carrying no current
ES1] fA} = (o}
(4.27)
where ES1] is a part of the S matrix for the whole region
58
It is a p1 x n matrix
{A} is a vector of n values
{o} is a vector of p values
that p1
number of nodesA carry no current.
b) Eddy current nodes
[S2] {A} = —Lm 104 1 at (4.28)
where
[S2] has a dimension of p
2 x n. It is a part of
the S matrix for the whole region.
[M1] is a matrix of weighted conductivies related
to the conductivities of the media surrounding
the nodes. Its dimensions are p2
x n
• DA . Is a column vector of n values.
at
p2
is the number of eddy current nodes.
c) Field nodes
The discrete form of the field circuit given by equation (2.11) is
[S3] {k} = + fill - 1 [w] (4.29)
at Rf+2,op
when [S3] is a part of the S matrix with dimensions pf x
r 11 is a vector of p elements where nodes are supplied
from external d.c source.
[W] is a weighted matrix of the field nodes with
dimensions pf x n. It has p
f x p
f filled elements.
pf
number of field nodes.
Equation (4.29) may be enlarged to:
or
Es3IfAl + Es 7{2-@t
162} R
f 3
{I1} - Lr”t4/J , OA} af at (4.30)
[S3] {A} -Ea + 2 at (4.31)
where [a2] is a matrix of equivalent conductivities of the field winding.
59
c) Stator winding nodes.
The discrete form of the stator winding nodes from equation
(2.15) is
ES4]. {A} = 1
Z(p) ({v} + [142 / Idt ] ) (4.32) 514 4
Other simpler and more useful forms for the numerical solution can be
derived with the aid of the relevant analysis of chapter 2.
A simple transient machine problem can be assumed to have the
discrete form
[S] {A} = [Au] {} +} 9t (4.33)
, , where the current vector LI1 J has its components on the currents attributed
to the nodes from the externally applied sources. [Ac] is the equivalent
conductivity matrix where areasaround nodes are included in its formulation.
It is also an assembly of the equivalent conductivity matrices of the
various regions.
The formulation of the conductivity matrix [Aa]has to be made
to suit the prescribed transient problem. A start is made with simple prob-
lems in which the conductivity matrix contains constant values. The detailed
application to a simple condition is presented and more complicated con-
ditions are also considered.
4.8.1 Formulation of the conductivity matrix
The conductivity matrix can be divided into four matrices because
of the four types of nodes. Nodes with zero currents have their corresponding
rows and columns zero. The other type of nodes has elements in the CAa]
matrix. When the grid numbering system is done so that nodes of zero currents
are numbered first followed by eddy current nodes then field nodes and
finally stator winding nodes, then the matrix fda1 takes the fond
60
0 : 0 ! 0 0
0 1 . a1; 0 1 0 1 [
0101a10 1 21 -T !
0 00 :Cr • 3..
(4.34)
This form appears when grid nodes are chosen in such a way so that any node
has only a contribution from one type of current. This choice avoids overlap
between the matrices al, 62, and 63. Here the matrices al, 62 and a3 repre-
sent respectively the iron and/or damper circuit, field circuit and stator
windings. The derivation of the matrices al' a and a3 is discussed below.
4.8.1.1 Evaluation of the eddy current matrix
Accurate modelling of eddy currents depends on a proper choice of
a suitable model. This is simulated in the al - matrix. It has been shown
in sec. (3.5.3.3) that the nodal method leads to a proper distribution of
the currents to the nodes and to the choice of a grid of acute-angled triangles.
However, the calculation of the contours of integration of the nodal method
requires significant computer time. As their function is largely to dictate
the division of elemental currents to nodes, where the current density is
low a reasonable result is obtained by using the finite element model, i.e.,
3 1 -Td of the area to each vertex. It is also a good approximation in acute-
angled triangles. Therefore the finite element current density model, ex-
plained in sec. (3.5.3.2) and (A.7) is used for evaluating the elements of
the eddy current matrix al. A linear approximation of the eddy current dis-
tribution between the nodes of the element can be chosen which satisfies the
analysis of sec.(4.2).
4.8.1.2 Evaluation of the field matrix
The main property of the a2
is its coupling between the field
nodes.Time changes of potential at any field node alter the currents fed
61
to all field nodes. The matrix a2 is a completely filled matrix. By
expressing the field using equation (4.31), it is implicitly assumed that
either the overhang leakage inductance of the field circuit is ignored
(DT2] contains constant elements) or the matrix-la2 contains the effect of
the overhang leakage inductance.Ignoring overhang leakage for the moment
(but see chapter 6), the matrix-a2
can be evaluated by assuming that the
current density distribution is uniform. Simson's rule can be used to find the
average of the induced voltages for a number of field nodes. Knowing the
number of conductors in the field slot and the area in the slot around each
node, weighting coefficients of the field nodes in the a2-matrix can be
determined. This method is explained in chapters 6 and 7 where two diffe-
rent weighting coefficients are employed.
4.8.1.3 Evaluation of the 03-matrix
The matrix a3
cannot alone represent the stator conductivities
because of the rotation. It only becomes valid alone when the machine is at
standstill. Therefore there must be at least another matrix where elements
are derived based on the induced voltages arising from rotation. Therefore
this formulation (4.33) is incomplete with respect to the stator nodes.
Nevertheless formulation (4.33) is a useful statement of the method and the
modifications necessary to suit special transient problems are given in
Part II. The matrix a3 represents only the effect of transformer voltages.
Although it is possible to neglect the effect of this matrix in most tran-
sient problems its consideration is sometimes desirable.
The matrix-a3
is naturally different from the other two matrices
a1 and a2. It is separated from both and it cannot also overlap with either
of them because there is no node in the grid which is fed from both stator
and rotor. The method of evaluating the elements of this matrix depends on
62
the type of model used for the stator winding. The use of the two—axis
theory simplifies the effect of motion but the matrix 63 then becomes skew
symmetric. This makes the numerical solution more difficult.
63
CHAPTER 5
METHODS OF SOLUTION OF THE DISCRETE EQUATIONS OF THE MACHINE
5.1 INTRODUCTION
In the previous two chapters it was shown how the partial diffe-
rential equations of the machine could be altered to a set of discrete
equations.. These equations are either nonlinear equations in the steady
state or nonlinear ordinary differential equations in transient conditions.
In both cases the source of nonlinearity is the elements of the S-matrix,
which are nonlinear functions of the resultant flux density B. The local
value of B depends on the differences between values of A at neighboring
nodes. Small errors in the values of A lead to big variations of B which
may be amplified again by the characteristic of the iron material. This
difficulty cannot be solved except by a powerful iterative procedure. As
the problem starts from the nonlinearity of the S-matrix, it is appropriate
to look for a technique which solves this problem first. Thus the solution
of the static field distribution of the machine may be obtained, leaving
the transient solution for separate consideration.
The techniques available for the solution of nonlinear algebraic
equations are either based on relaxation procedures or on optimisation
techniques. There are several techniques that could be used but the following
have been recommended for the solution of n-dimensional algebraic equations.
A) Relaxation techniques
1. Successive over relaxation of A and under relaxation of U.
B) Methods based on , optimLsation techniques
1. Steepest descent
2. Newton - Raphson
3. Fletcher and Powell
4. Conjugate gradient
5. Modified Newton Raphson
64
The above methods were tested in order to see which. gave the
least computing time and required the least storage. The nature of their
algorithms and an appropriate reference is given for each of them.
The limited corestore of digital computers hinders the use of
conventional methods for the inversion of large matrices. Thus the best
way to solve the nonlinear problem is to look for a technique which takes
advantage of the characteristics of the S-matrix. For instance, the S-matrix
is symmetric and therefore it is appropriate to store only the diagonal
elements and either the upper or lower triangles of the matrix. The analysis
in chapter 3 shows that the matrix is sparse. This advantage allows the use
of sparsity techniques.
The transient problem could be tackled by two types of methods:
1) Runge-Kuttats method
2) Predictor corrector method
The analysis of these techniques is explained briefly in order
to show the various concepts and appropriate references are given.
In the solution of the transient problem in a machine it is
best if the interaction to obtain a solution for the nonlinear equations is
made simultaneously with the step forward in time, for the elements of
the S-matrix are nonlinear and changing. This means that the choice of
the method for solving the transient problem must also fit in with the
technique chosen to solve the static equations.
5.2 THE ITERATIVE PROCEDURES FOR THE SOLUTION OF A SET OF NONLINEAR ALGEBRAIC EQUATIONS
Iterative procedures using successive over relaxation of values
of A and under relaxation for values of U have been recommended in ref.27.
The methods are based essentially on the point Jacobi--Gauss method. Although
the details of each method are very important if convergence and stability
are to be obtained, they are mentioned here briefly for only some of them
have been used.
65
5.2.1. Basic iterative methods for - SuCcessive'relaxation
The set of nonlinear algebraic equations in'the steady state
problem is
[S1 {A} = .{I} 5.1
The physical considerations of the machine problem, which lead
to one and only one derived equation at each node, subject to the boundary
conditions, leads to an expression of the field in n-equations. These are
the only equations required for a unique solution, giving a non-singular
matrix. The diagonal elements of the S-matrix are non zero because of the
choice of a grid in which finite elemental areas are used. Acute-angled
triangles give larger entries to the diagonal elements of the S-matrix.
In iterative analysis using relaxation techniques the following
iterative method may be used28: S, .
1 . A ilk+1 = ( 21) A. I + 1.<1<n, k>o (5.2)
j=1 -i2i k 1,1 hi
An acceleration factor a , known as the relaxation factor can be used as
A.I k+1 + a(A.1 1 - Ail )
k k+1 (5.3)
where Al is the value of A calculated from equation (5.2) during the (k+l)th 1
iteration.
5.2.2 Successive relaxation methods
The analysis in the previous section represents the basic
iterative procedure for the solution of n-equations. Since the values of the
reluctivities u in each element change from one iteration to the next de-
pendant on the resultant solution, it is found necessary to use an additional
acceleration factor for values of u. Earlier work has used these techniques
for nonlinear magnetic problems where rectangular grids are used. However,
it has seldom been applied to a triangular base grid27. The following are
some of the best proposed techniques.
66
1. A technique which starts by taking the distribution of A
everywhere to be zero. The elements of the S—matrix are calculated with U
corresponding with B=o. Accordingly, the vector A is first calculated
and then an over relaxation factor is used to give a better distribution
for values of A. The vector A determines a set of U values in each element.
From the old values of U an under relaxation factor is used for u. The S—
matrix can then be determined after which the iterative procedure is re-
peated. It is ended when a predefined condition for the errors at the nodes
is met.
2. A is calculated for a given S—matrix by using line iteration'.
This means that after completing the computation of values of A on one line
of the grid then all the relevant A and u values are changed. The procedure
is repeated on each line of the grid. It has also been proposed on the
return to start from the end line going back to the first. This procedure
is repeated until a predetermined condition is satisfied.
3. The distribution of A is calculated by a point—by—point
method, also called a chess—board (or quin curex) method29. Instead of
calculating A.+1 immediately after A., values are obtained initially at
alternate nodes. The corrections to values of u are made at the end of
all the changes of potential at all the grid nodes.
4. At the start, values of u can be estimated. Although the
method is similar to the previous methods, it usefully brings the values
of A towards the solution from a good starting point. The method is successful
because small errors in A enormously affect the values of u. The final
solution can be obtained by using any iterative procedure mentioned above.
In all the above methods the over relaxation factor used for
A ranges from 1.3 to 1.9 and the under relaxation factor for u range from
0.2 to.0.9. All have been obtained by trial and error, but change with the
problem considered, the level of saturation, etc.
67
These relaxation techniques are easy to program but the com-
putational time is large compared with other methods. Also, many trials
have to be done to find out the best over and under relaxation factors to
suit the problem. The main disadvantage of these techniques is the possi-
bility of nonconvergence or even oscillation in the solution, which has
appeared in some programs3. In general such techniques give poor convergence,
requiring more than 200 iterations1.
5.3 METHODS BASED ON NONLINEAR OPTIMISATION
Other techniques based on optimisation can be helpful in ob-
taining fast convergence. The field problem is expressed in the discrete
form by a set of nonlinear algebraic equations. Direct solution of these
equations as shown in the previous section is difficult. However, the real
problem is to solve a partial differential equation which has different
forms in each region and is subject to boundary conditions. In Appendix A,
it was concluded that elliptic partial differential equations are related
to the energy functional. Differentiation of the energy functional within
each element gives the element matrices [S]e in which 1.) takes only one value.
By combining the element matrices the equations in the form (5.1) are
obtainable. On the other hand, the energy functional can be expressed in
the discrete form
.54 (A) = y (aelLslae) - ael T fie}
(5.4) e=1
Functional (5.4) has quadratic properties and has to be minimised.
According to the above formulation, the problem can be regarded
as one of minimisation of a functional of several variables without con-
straints. This kind of problem has been studied using the methods of Newton,
Fletcher and Powell, Davidson and others. These are discussed below. Re-
sults of their application to the machine problem are given in chapter 6.
68
5.3'.1. Steepest descent method .
This method is based on the property that a function decreases
rapidly in the. direction of its gradient30. The method is not recommended
for application to problems with. more than two variables. It might appear
simple to implement but has serious disadvantages. The method does not
guarantee convergence after a definite number of iterations even for simple
and well behaved functions31. When the function is complicated and has
local maxima and minima, oscillations occur and convergence may not be
obtained. Nevertheless, the method has been recommended for use at the
start of a solution in conjunction with other techniques to bring the function
near to its minimum rapidly. The difficulty is then that more than one tech-
nique is needed and therefore much computer storage is required. A program
could be written for this approach as explained in ref. 32.
5.3.2. Newton Raphson technique
The Newton—Raphson technique is one of the most widely used
techniques in power system applications. It is also recommended in articles
concerned with the solution of nonlinear magnetic field problems.6. The
method is based on the possibility of evaluating the second order derivative
of the functional and using to advantage the quadratic properties of the
function to be solved. The method is derived from Taylor's expansion.
During the iterative procedure, values of A differ from the
exact values of the field distribution. Thus there must be incremental
changes of A, SA so that the gradient of the function reduces to zero. If
G(A) is the vector of gradient of the functional AA) then
3 Jr (A) . {G(A)} — 3 {A}
(5.5)
• Equation (5.5) can be rewritten as
{G(A)} = Es] {A} — fIl (5.6)
69
The solution of the field problem is reached when all the elements of G
are zero. If G(A) is the value of the derivative at a given iteration and
G(A+6A) is the improved value, then
A) G(A+6A) = G(A) +
G( dA + DA
(5.7)
Neglecting the higher order terms in equation (5.7) and making the approxima-
tion that G(A-1-dA) = o, because the change is made to minimise G, gives
SA = ( 9G(A))-1
G(A)
(5.8)
The above equation can be used to.reformulate the magnetic problem
{dA} = - [H]-1 {G} (5.9)
where
{G(A)} [H] = 9 {A}
(5.10)
Since the matrix [H] is the first order derivative of {G} , it is the
Jacobian matrix of {G). H is also the second order derivative of 54 and it
is known as the Hessian matrix33. The method of evaluating [H] for an
electrical machine magnetic problem is explained in sec.(5.5.1). The Newton-
Raphson technique predicts new values of A given by
{A}k+1
= {A}k
- CH] 1 {G}k (5.11)
It should be noted that the method does not, as stated in the references, imply
an inversion of a matrix, but it does need an inversion multiplication. This
might seem more difficult than inversion only, but it is possible to do
inversion multiplication in one operation without inverting the matrix
separately. The method might not converge from an initial distribution of
A which is far from the correct distribution.
5.3.3. Fletcher and Powell method
The algorithm derived by Fletcher and Powell34 is recommended
in many recent books on optimisation as a practical algorithm. The method
combines some of the more desirable features of steepest descent and the
70
Newton-Raphson method. It is a development of other techniques34. The
method is stable and convergent with quadratic properties.
This technique uses the recurrence relation
{A}k+1
= { }k AEH11k { k (5.12)
where A is a scalar quantity which minimises the functional 5r(A). The
matrix[H1] is updated after every'iteration. Equation (5.12) can be re-
written as
{A} =. {A}k (5.13)
where {U}k
is defined by comparing equation 5.12) and (5.13). The matrix
[H1] is initially the unity matrix. In other words, the method utilizes
initially the properties of the steepest descent. After several iterations
its value reaches the inverse of the H-matrix of equation (5.11). Thus it
finally reaches the result obtained by the Newton-Raphson technique. The
updated formula for H1
in equation (5.12) is
[111kt1 = CH1]k + k
+ ER]k
(5.14)
where {U}
k {U}
T k
[P] - k • T {u}k {v} k
(5.15)
and EH1]kfv1k{V}k[H1]k [R]
k = • {V}
Tk
[Hl
]
k {V}k
wherein
{V}k = {0}1(4-1 - {0}k'
A computer program has been written and is discribed by Ralston35.
(5.16)
(5.17)
5.3.4. Conjugate gradient method
The conjugate gradient method is an alternative approach for
minimising the energy functional. It is based on the computation of the
71
function gradient in two successive iterations to determine the direction
of search s... This algorithm utilizes the conjugate properties of quadratic
functions in an approximate manner. It is known that a set of conjugate
directions ti, =1,...n, with. respect to matrix H is determined by
j t (i)T
Ht =o j (5.18)
' The algorithm is available in references 30'36.
It can be
summarised as floows:
Initially the function value 9(A) and the gradient G(A) must
be found by employing estimated values of the potential A. After the first
iteration, using the method discussed in ref.37, a conjugate gradient pa-
rameterp,.is calculated from the division of the inner product of the 1
gradient of two successive iterations in the form:
P.k4.1 = }k+1 {G}
k k+1 {G}T {G} k with S1=o
The direction of search sk+1
is determined from
fslk+1 = {G}k+l
1310-1 {s}k
(5.19)
(5.20)
The new values of the magnetic potentials can be obtained from the recurrence
relation
{A}10.1 = {A}k + 1k {s}k+1 (5.21)
where 1k is determined by performing a one dimensional minimisation along the
direction of search. The procedure is repeated until the given convergence
criteria are satisfied. This algorithm has been recommended in reference 37
for solving a problem relating to a distributed parameter system. The changes
of the magnetic permeabilities from one point to another in the machine
region and theirdependence on the resultant A values can be regarded also as
a distributed parameter system. This algorithm has been tested on a dynamic
system with a small number of parameters and gave good results. It seems
72
that it is useful in cases where the main object is to solve a transient
problem. However, it has yet to be proved that the method would be efficient
for a static system with a large number of variables. Results of this
study are explained in chapter 6.
Among the several algorithms that are based on the change of
amplitude in the direction of search is the modified Newton-Raphson
technique.
5.3.5. The Modified Newton-Raphson technique
Collatz 14 showed a method of improving the Newton-Raphson
technique. It is based on extending the Taylor expansion in equation (5.7)
to include one additional term. This allows the incremental changes in
the Newton-Raphson technique to take the form
{SA} = - a Eli]-1{G}
(5.22)
where a is a modifier which changes from one iteration to the next. Its value
is always less than unity and is given by:
DIJI] = 1 — (EH]-1 10T )(DI I ] atm.)T (5.23)
Thus additional computations are required compared with Newton's method to
calculate the value of a.
5.4 SPARSITY TECHNIQUES
The analysis in chapter 3 showed that any node is coupled only
to its adjacent nodes. This means that the derived S-matrix is sparse. It
implies also, as indicated in sec.(5.5.1), that the H-matrix is sparse.
Its non zero elements are the same as those of the S-matrix. The main diffi-
culty in applying the previous techniques is the size of the problem to be
solved. The Newton-Raphson technique has been considered impractical mainly
73
because it was thought to need a matrix inversion but as was shown in
sec. (5.3.2) the method does not need inversion but only inversion-multi-
plication, uhich can be done in one. operation. The problem can be regarded
at any iteration as a solution of a set of equations
CHEISAl = - {G} (5.24)
The problem of solving a set of sparse equations is receiving
much attention at the present time, since the importance of developing good
methods is coupled with an enormous number of potential applications in
engineering. Methods of solving equations (5.24) have the objective of
making, the maximum use of the sparsity property of the matrix so that minimum
core storage and computational time can be gained. Among these methods are:
1. Trials to obtain the best ordering of the equations 38 i.e.,
best ordering of the node numbers, and using factorisation techniques.
2. Use of graph theory. The method regards the problem equivalent
to the determination of a minimum essential set of directed graphs39.
3. Finding the "fill-in", i.e., the number of nonzero elements
that change in value, and choosing a technique which reduces them 40,41,42
.
4. Implementation of triangularisation-factorisation techniques43.
The symmetry of the S-matrix in the machine magnetic field
problem reduces some of the difficulties encountered. Jennings 44 overcame
the difficulty of the various types of fill-in of the nonzero elements,
which depend on the ordering system and the type of elimination. The method
stores the elements from the diagonal up to the first nonzero element in a
row as shown in fig. (5.1). The rows are stored sequentially including
some zero elements. Although the method
implies the storage of some zeros it has Fig. (5.1)
the advantage that the maximum size of
the stored elements is known at an early
74
stage. Moreover, the difficulties which arise at some grid nodes, perhaps
due to unfortunate numbering, do not affect the resultant number of stored
elements of the matrix much. The method uses a compact elimination method
or a modified Gaussian elimination technique.
5.5 NEWTON-RAPHSON AND SPARSITY TECHNIQUES
Full use of the Newton-Raphson technique can be obtained when
its disadvantages are eliminated by the use of the sparsity techniques. The
method implies the derivation of the Hessian matrix of the functional ;g(A)
or the Jacobian matrix of its gradient.
5.5.1. Derivation of the Hessian matrix of the static field problem of the machine
The elements of the Hessian matrix can be derived by differen-
tiating the gradient (G} of the functional :j-4(A) with respect to values of
A at the nodes. The gradient {G} represent at the same time the error in
the solution of equation (5.1). Differentiation of the vector G in equation
(5.6) with respect to values of {A} gives
p{0 [H] = a{A}
when values of I are independent of the values of A then,
[II] = [H1]
where
[S] { A }
(5.25)
(5.26)
The matrix H1
can be derived as follows:
Since the matrix S is a combination of element matrices [Se] in
the form of equation (A.55a), then
Q [S]{A} y [Se]{X.e}
e=1 (5.27)
3{111 e=1
JC [S] {A}= X AA--e} [Se] {A—e}
afAel
3U 3ue DB2
aB2 a{} (5.31)
75
Differentiation of the vector [S] {A} with respect to the vector {A} is
equivalent to the differentiation of the element matrices [Sel {Ae} with
respect to{Xe}. The differentiation produces zeros unless i is one of the
triangle vertices. Therefore
(5.28)
From equation (A.53), equation (5.28) becomes
31) afil [S] {A}. — y (A 2e C] {A—e} + Be])
e=1 e a (B ) (5.29)
The reluctivity u within the element e depends in general on the resultant
field. The characteristic of the material of the element e can be expressed
as a function of the flux density squared as
therefore,
U = v (B2) e e
(5.30)
The value o can be obtained by comparing equations (A.25) and (A.41):
3B = 2[].{X.e} awl
Substituting equations (5.31) and (5.32) into equation (5.29) yields
(5.32)
a l [S] {A} = ((Ae)`
{X
afA e} + [Se]) (5.33) e=1 OB
2
The Hessian matrix H1
in equation (5.26) becomes
where
[S] + X [11] e=1
(5.34)
[uT {xe}T {Ae} El (5.35)
76
The. element matrices are combined in the. same manner- as the S-matrix around
each node as explained in sec. (3.5). The matrix Hl has thus the same lo-
cation of nonzero elements as the. S-matrix. Since the H1 matrix is the
second order derivative of the functional 5(&) then any element in the
matrix is given by
32,(A) DA. A. 1 3
(5.36)
Therefore the H1
matrix is symmetric. The element matrices [H2]and conse-
quently the combined matrix H1
tends to [S] when the solution of equation
(5.25) is reached. During the iterative procedure {G} is not zero and
consequently H1 is not the inverse of B1 . When the solution is achieved
then the vector {G} is equal to {0} , and from equation (5.6), (5.24) and
(5.34) the above result follows.
5.6 NUMERICAL TECHNIQUES FOR THE TRANSIENT PROBLEM
In the transient problem the discrete equations of the machine
contain nonlinear algebraic equations and ordinary differential equations.
The solution of this combination can only be obtained by knowing the methods
available for each type. With that knowledge, methods of solving the combined
equations can be developed. Methods of solving the set of nonlinear algebraic
equations were explained in the previous section.. Methods of solving ordinary
differential equations are discussed below.
There are several methods for the numerical solution of a set
of ordinary differential equations. Attention has to be centred here on the
method of evaluating a relation between the potential at a point in terms
of its previous history without reference to the previous history of other
points. This means that the variation of the function in the time domain is
separated from its variation in the x-y plane. The other important point
is that all the potentials at the nodes are related together by the discrete
equations only at discrete instants in time. The reason being that the
77
formulae for discrete equations, derived in the previous chapter, are only
valid under these conditions.
When a solution is required in the time domain, there are two
approaches:
1. Indirect method: The problem is put in a standard form. Time
intervals are taken sufficiently small so that the nonlinear values of the
reluctivities do not change significantly during each time interval. The
solution at the end of the interval has then to be corrected so that the
field at a given instant of time satisfies Poisson's equation rather than
the diffusion equation. The procedure is repeated at each time step.
2. Direct method: The solution has to satisfy both the relation
between the nodal values from one instant to the next and at the same time
the relation between the nodal values at a given instant of time.
It looks obvious that the second method is better than the first.
Nevertheless, the explanation of the second method is assisted by a discrip-
tion of the first and its associated numerical techniques.
5.6.1. Indirect method for the numerical solution of a transient problem
5.6.1.1. Standard formulation of the problem
Solution of the transient problem in the discrete form can be
explained by considering a related problem, in which a solution is required
to
[si {A} = - EAa4 } (5.37)
The S-matrix contains nonlinear coefficients and it is not readily inverted.
The matrix Ao can be expressed in the form
[A a1 (5.38)
11 elements 1
[U] = 11
11
(5.41)
78
where a1
is a nonsingular symmetric matrix and g2
is a singular matrix.
A zero matrix occurs in the top left hand corner of the matrix [Ag].
A solution of the above problem has been suggested by Dr.
J.C. Allwright. The matrix Ao can be decomposed into the matrices L,D and U
satisfying
EAU] = [L] ED1 [u]
(5.39)
where [L]is a lower triangular matrix of the form
[L] =
(5.40)
[D] is a diagonal matrix
and [U] is an upper triangular matrix of the form
The method of decomposition can be referred to in the standard numerical
books (Reference 45). By substituting equation (5.39) into equation (5.37),
[S] {A} = -[L] [D] EUlt}
But from equation (5.40), [L] is nonsingular matrix and [L]-1 exists.
Premultiplying both sides of equation (5.42) by [L]-1 gives
DA [L]
-1 [8] {A} = - [D] [10{57t4
Introducing a vector {y} defined by
{y} = [U] {A}
Differentiating equation (5.44) with respect to time gives
(5.42)
(5.43)
(5.44)
(5.45)
79
By .substituting equations (5.44) and (5.45) into equation (5.43) then
-CL]-1 Es]. Eu]-1 {y} = ED]. {-32(-} Dt
(5.46)
Equation (5.46) brings the problem into the standard form in y rather than
A so that
ay= - ED*] EL]
-1 Es] EU]
-1- I {y} (5.47) at
* i where the matrix D is a diagonal matrix and its elements are defined by
* 1 D.. = 0 if d.. = 0 (5.48a)
11 11 1
= d.. 1.1
if d.. 0 (5.48b)
The subsequent solution for values of A can be obtained from equation (5.44).
5.6.1.2. Numerical methods for solving ordinary differential equations 46
Two methods are widely used, the Runge-Kutta and the predictor-
corrector methods.
The fourth order Runge-Kutta method is self starting and accurate.
It can be expressed by the relations,
where
1 = A
t + (w
0+2w
1 + 2w2 + w
3) A
t+St
w = St ft , t)
- St
w1 = St f (A---t + , t -2--) 2
w2 = St f+ w St
t 2- 1, t + 2- )
w3
= (St f (At + w2, t +St)
(5.49)
(5.50)
The error in this method is proportional to (St)5
and for small values of St,
the error is negligible.
A commonly used predictor-corrector method is the modified Euler.
In this method the corrector is defined by the relation in the vector form
{A}
4. St IDA., .3A1 t = {Alt-St 2 ` 1Dt.f t-6 tj (5.51)
80
and the predictor is given by
DA {A}t --='{A}t_ t at'{
Dt---} t-St
(5.52)
The error starts from the third derivative of the function in equation (5.51)
and from the second derivative of the function in equation (5.52). Thus
the predictor is not as accurate as the corrector. This means that the correc-
tor, most likely, will have to be used more times to arrive at successive
values of {A} at vi-St that are equivalent, than would be necessary if the
predictor and corrector had the same accuracy. Moreover, the fact that most
predictor-corrector methods are not self starting requires particular
attention.
Comparison between the two methods shows that the Runge-Kutta
method requires evaluation of the field four times as often as the other
method if the time step is the same. Furthermore, in a predictor-corrector
DA method employing a sufficiently small step, -5-t- will seldom be evaluated
more than twice. Thus, the Runge-Kutta method is poor in comparison with the
predictor-corrector method. However, the Runge-Kutta method is self starting
and thus is recommended for starting the predictor-corrector method.
The main difficulty in both methods is that the field distri-
bution must be corrected again to satisfy Poisson's equation. This leads to
a change of the field values derived by either method. If we assume that the
time interval is so small that the second correction in the field distribu-
tion does not lead to big changes in the derivatives then these methods
can be used However, relatively large time intervals might require more
computation to satisfy both the time relation and the field distribution
at a given instant. This type of analysis might lead to either extra com-
putation to obtain convergence or it might lead to divergence of the nume-
rical solution.This difficulty can be removed by a direct method of.analysis.
81
5.6.2 Direct method of numerical solution of the transient'problem
The transient field problem has a numerical formulation defined
at each instant in time by equation (5.37), which may be rewritten as
[S] {A}t = - [Aa]fN at t
• 3A The vector { t
can be replaced using the corrector method of equation
(5.51). It is a reasonable approximation because the error starts from the
third order of the function derivative. Therefore equation (5.53) becomes
[S] {A} = -[A0](-{-22} + ({A}t -.{A}t-St)) at (St t-St (5.54)
As values of {A} and'{-9-4-} at the previous instant; t-(St are known,then the
only unknown in equation (5.54) is the vector {A} at time t. The correct
values of A are obtained iteratively as shown in sec. (5.6.3). Once'{A}t
is known the derivatives {3LA-} , at the same instant are readily determined. at
The elements of the S-matrix are also corrected in the iterative solution
at each instant of time.
The solution does not require the use of.the predictor formula.
However, the predictor formula of equation (5.52) can be used in the first
iteration at each instant. This ensures that the errors at successive
instants in time are of the same sign, the approach to {A}t being made from
the convex side of A, and the error in calculating 3A is reduced. 9t
5.6.3. Iterative procedure at each time interval
The solution of the nonlinear equations (5.54) can be obtained
iteratively. In the iterative procedure the error vector {G} appears in the
solution. It is defined by
2 r,„, 2 {G}t = [S]. {Alt + [Aci]
-1§1 {A}t 6t
f114-6t ) (5.55) t-(5t
The solution is obtained when the error {G} becomes less than a lower
(5.53)
82
acceptable value and the incremental changes of the potential 16A1 from one
iteration to the next tend to zero.
Equation (5.55) is nonlinear. The source of nonlinearity is the
dependence of the elements of the S-matrix on the resultant field. Therefore
the various techniques explained for the solution of the steady state prob-
lem could be applied here allowing for the additional term in equation
(5.55) which contains the conductivity matrix [Ac]_
The advantages of the Newton-Raphson method with sparsity techniques
can be extended to solve the transient problem. The Newton-Raphson technique
implies the evaluation of IG1 D{A}
where {G} is given by equation (5.55). Thus
by differentiating equation (5.55)an H-matrix is obtained and is defined at
the kth iteration at time t by
[H] = 1{.91 (5.56a)
t,k 3{A}t'k t,k
B 2 -(A}t,k "] fAlt,k WAG] )
) (5.56b)
The first part of equation (5.56b) is derived in sec. (5.5). Therefore by
substituting equation (5.27) into equation (5.56b) gives,
2 [H]t,k =
St[AG] (5.57)
where [Hot ,k is defined by equation (5.34) and the relevant equations.
The incremental values of {A}t from the k
th to the k+l
th
iteration at a given instant of time are thus
-1 16Alt,k [H]t,k IGlt,k
The modified values of {A} at time t are then
(5.58)
fAlt,k+1 = 1111t,k 16114t,k
This iterative procedure at a given instant of time is repeated in order to
reach an acceptable lower limit. At the end of the time step values of {A}t
(5.59)
83
DA and {--4 replace values of{A}
t-dt-and _p_A1 for the next time at t at t-St
inverval. This procedure is then continued. At the. start of each time
interval values of A are determined using equation (5.52).
5.6.4. Initial value problem
The initial value problem appears in equation (5.54) in that
value of.2A at zero time, when the transient starts, are not known and 3t
there is only one circuit equation for each winding. The solution of this
problem can be illustrated as follows:
In the steady state solution, the vector of the current, {I} ,
supplied to each node and the A-distribution are determined. The current
vector has its nonzero entries from all nodes which lie on current carrying
conductors. Values are determined by the external and internal voltages
(see chapter 2.).
When a transient disturbance starts in a winding, equation (5.1)
still holds but it takes the form of equation (5.37) for that winding.
Initially other windings still continue to supply their steady state currents
but later additional terms appear in their equations resulting from trans-
, atformer action. It is not possible to calculate the initial values of
-5-tE
at each node by comparing equations (5.1) and (5.37) because the matrix [AG]
is singular. However, equation (5.55) can be rewritten as
f01t+dt = ES] {A}t4.6t + {I}t + Tit CAo] {A}t+St -
St [Aa] {A}
t (5.60)
in which the vector {I} is equal to {I o} when t=o and 2A o is not required 3t
explicitly. Thus the calculation commences from the initial steady state
values with the minimum of ancilliary calculation.
This method is applied to a micro alternator as shown in
84
chapter 6. However, when it is applied to a turbo alternator oscillations
in the solution occur. A method for avoiding this difficulty is given
in chapter 7.
Methods of solving equations (5.37) have been explained in
this chapter. More complicated machine transients could be solved following
the same procedure. Applications of the method are explained in the next
part of the thesis.
85
PART II
PRACTICAL APPLICATION OF THE METHOD ON MICRO- AND TURBO-ALTERNATORS
The theoretical results developed in part I are used to solve
some of the magnetic field problems in micro- and turbo-alternators. So-
lutions of the steady state and transient conditions on a micro-alternator
are discussed in chapter 6. The method is also applied to a solid rotor
turbo-alternator in chapter 7. In these two chapters a half pole pitch of
the machine is sufficient to analyse the problem in both steady and tran-
sient conditions. Loading of the turbo-alternator distorts the syuuuetry of
the flux lines about a pole centre line and full pitch analysis is necessary
as is explained in chapter 8. In this chapter machine transient conditions
are considered when armature currents and motion exists.
86
.CHAPTER 6
STEADY AND TRANSIENT APPLICATIONS ON MICRO ALTERNATOR
6.1 INTRODUCTION
The micro-alternator is a 3-ph synchronous machine which possesses
the same time constants and per unit reactances as those normally found in
large altelmators in modern power stations47. It is an electrical scale
model of machines of up to 1000MW rating and is rated at 3RVA, approximate-
ly one millionth of the rating of the simulated machine. It is used to study
the performance of large electric generators. It can demonstrate transient
performance as part of model network.
A micro-alternator is employed here because its geometry was
known and it was available for tests. In that it has a damper winding and
a fully laminated magnetic circuit it was a good machine to use for an
initial study of transient behaviour.
The two-dimensional steady state field problem of a machine has
been formulated in chapter 2, and the partial differential equations changed
into discrete form as shown in chapter 3. The equations are nonlinear alge-
braic equations and numerical methods for their solution are explained in
chapter 5. The method of application for steady state conditions is similar
to other previous workers in the subject. However, small differences are
explained in chapter 3. A comparison between the possibility of using
various iterative procedures is given. A general flow chart of a complete
program and a method of numerical modelling the iron characteristic are
provided. The method is used to solve the 2-dimensional field problem of
the micro-alternator. It evaluates the open circuit characteristic of the
machine which is compared with test results.
The transient problem is formulated using partial differential
equations in chapter 2. Methods of altering the partial differential
87
equations to the discrete form and their iterative solution are explained
in chapters 4 and 5. respectively. There are several transient applications
of interest and the simplest of them is the field decrement test when the
armature windings are open circuited. The simplicity in this application
lies in the use of the symmetrical properties of the field distribution
around the pole centre line, during the transient period. This reduces
the number of nodes necessary for the solution because consideration over
half a pole-pitch is sufficient to analyse the problem. Additionally the
number of entries in the Acs matrix are reduced by absence of entries in
the stator, and rotational effects are not involved. Results from the micro-
alternator show the interplay between the distributions of induced eddy
current densities and the flux lines. During the transient period the flux
lines change their position and their "motion" is clear. Eddy current den-
sities are also changing their distribution and magnitudes. Comparison
between computed and test results are checked at the terminals of the machine.
6.2 DISCRIPTION OF THE MICRO-ALTERNATOR
The nominal rating of the micro-alternator used is 3KVA,
220/127 volts, 3-phase, 4 wire, star connected 50 c/s, 1500 r.p.m. (54
stator slots). Constructional details are shown in Fig. (6.1).
6.2.1. Stator
The stator is a strip wound in the form of 4-pole, two tier,
double layer, pull through winding,where 3-phase, 4-wire connections can
be made. The following data are provided in Table (6.1).
88
nibs
Fig, (6.1) Constructional details in a 2-dimensional
corss sectional area of the micro-alternator.
89
Table (6.1) Details of the stator
....... •
phase voltage 127 volts
Full load current 7.9 A
Number of moles 4
Frequency 50 c/s
Number of slots 54.
Number of conductors per slot 12
Number of turns per phase 108
Conductor size (copper) 2 (0.4" x 0.045")
Type of winding 2-tier, double layer
Coil pitch 1-12
Winding factor 0.94
Length of mean turn 40' (102 cm)
Phase resistance 15°C 0.080 ohms
Slot skew 1 slot pitch
Axial length, gross 5.5" (14 cm)
Axial length, effective 5.2" (13.2 cm)
Slot dimension see fig. (6.2)
6.2.2. Rotor
The rotor is of cylindrical construction with four groups of
equally spacial slots forming a four pole construction. The field winding
comprises a 4-pole double layer winding and an identical auxilliary shadow
winding occupying about 10% of the available winding space. The rotor slot
openings are wedge shaped into which detachable damper bars• are
attatched by screws at each end of the damper bars to two end rings to form
a complete cage damper winding.
91
The rotor has 24 slots, the double layer winding is pitched
1-7 slots of the divided type having two halves whose axes are displaced
by approximately 67.3 degrees from each other. This is fitted into the
bottom portions of the rotor slots, with separate "shadow"windings incorporated
from each half of the field winding. A 24 bar squirrel cage winding is fitted
into the top portions of the rotor slots. The following data are provided
in table (6.2).
Table (6.2) Details of the rotor
Excitation voltage one section full load 23 V
Excitation current one section full load 4.5 A
Typical Air gap length 0.028" (.7 mm )
D-axis excitation winding
Type 100 V double layer
divided winding
Number of conductors per slot 100 excitation.
Number of turns per section per pole (effective) 285
Conductor size (copper) 3 x 0.032 dia
0.018" dia (shadow)
Coil pitch 1-7
Winding factor 0.95
D-axis damper winding
Type 28-bar squirrel cage
Number of conductors per slot 1
Conductor size - bar 1.27 x 1.27 cm
Copper End ring 2.54 x 1.27 cm
Slot dimensions are shown in fig. (6.3a)
A
198.11
103.34
Fig (6.3a) Rotor slot of the micro alternator. Fig (6.3b) Approximations and dimensions of the rotor slot for the numerical solution.
93
6.2.2.1 Rotor winding connection
The rotor windings are connected so as to produce the resul-
tant Ampere turns, and consequently the flux in the direction of the pole
axis. This connection is used to compare computed values with test results
in both steady and transient conditions.
6.3 APPROXIMATIONS FOR THE NUMERICAL SOLUTION
The model used is different from the actual machine in the
following ways:
a) Fringing of the flux lines at the machine ends.
The effect of the fringing of the flux lines is taken into
account by using the effective length of the machine instead of the physic,ii
length.
b) Extra spacing in the field slots.
_,The field exciting and shadow windings and slot insulation are
replaced by an equivalent field winding occupying the same area with a
reduced conductivity.
c) Simplification of the machine shape.
The round bolts in th stator teeth holding the laminations
together are replaced by square bolts of the same area. They are assumed
to be insulated and nonmagnetic.
d) Positioning.
The machine stator is considered to remain in one position
relative to the rotor lying symmetrically about the direct axis in this
instant in the position of maximum radial flux density.
e) The effect of skewing is ignored in the 2—dimensional analysis
but its effect on the results could be included in the conventional way.
• 94
6.4 CHOICE, OF THE GRID
Two grids of triangles, are used, the first to evaluate the no-
load flux distribution at different field currents and the second for
computation of the field transients, (fig.(6.4) and (6.5)) The second grid
has elements for the nibs and the space which separates the field and the
damper windings. The grid of fig. (6.4) has 192 nodes, 26 of which are of
fixed potential. The second grid of fig.(6.5) contains 210 nodes with the
same number of fixed nodes. In both meshes the number of triangles around
any free node, excluding the Neumann's nodes varies between 4 and 7.
In the stator yoke, one set of triangles is used between the
back of the stator slots and the machine outer diameter. Each stator slot
is divided into six triangles to make the grid compatible with the two
triangles in the narrow slot mouth. The round bolts replaced by square ones
are approximated by two triangles. In the stator teeth 15 triangles are
used for each tooth.
The air gap is very small. Nevertheless 42 triangles are used to
allow for the variation of the flux density between•stator and rotor. It
allows at the same tine for the correct drop in Ampere-turns over the air gap.
The part of the field slots containing the winding are divided
using four triangles in each slot. They are the same in both grids. Allowance
for the nibs makes the number of triangles in the damper region different
from grid of fig. (6.4) to that of fig. (6.5). The rotor slot is approxi-
mated as shown in fig. (6.3b). Heavy lines are used for the approximation
and dotted lines for the original slot configuration.
A number of triangles are used in the pole and the body of the
rotor of the machine to allow for the variation of the flux density in
these regions.
95
Fig (6.4) Grid of triangles used to divide
a half pole pitch of the micro-
alternator for steady state flux
calculations.
96
Fig (6.5) Grid used to divide a half pole pitch of the micro alternator including the nibs.
97
6.5 GRID NUMRRRING SYSTEM
The grid is numbered in the order which minimizes the computer
core storage and the computing time14. It is found that the greatest speed
is obtained when the nodes on the boundary have fixed zero value of poten-
tial (in contrast with the approach of Ref. 4) and are numbered consecutively.
These nodes require no computational effort and their function is to limit
the values in the region considered and specify the boundaries. All other
nodes are numbered so .that the maximum difference between any node and its
surroundings is minimal. This method of numbering gives an H-matrix of
equation (5.9) with a diagonal strip which requires minimal storage space
and it is inverted using the modified-Gaussian elimination technique.
The grid of fig. (6.5) is used for the transient calculations. The
pattern has to be modified in regions where currents are induced in conducting
paths. The numbering does not affect the static field solution but increases
the required core storage.
6.6 MODELLING OF THE IRON CHARACTERISTIC
6.6.1. B-H curve of the iron material
The magnetic circuit is made of laminated sheet steel, Transil
92, 0.355 mm thick. The B-H curve of this type of material is shown in fig.
(6.6) being the same as for laminations 0.5 mm.
6.6.2. Methods of modelling the iron characteristic
Various workers have sought simple approximations for the mag-
netic characteristic of iron. Since all iterative methods require repeated
evaluation of reluctivities and when the Newton-Raphson technique is used
the derivatives of the reluctivities with respect to B2
are also required,
a computationally cheap method is obviously needed. Therefore the choice
of a suitable method of modelling the iron characteristic is important4.
MAGNETISING FORCE - AMPERE TURNS PER CM.
1 /5
fs 9 10 I1 /2
inehrtioN fogvist
2.2 —
71 —
13 14
Induct ion 4‘10(1,ics per SquAft
.10 —
Ig —
18 —
17 —
IC —
IS —
14 —
13
1.2—
11 —
JO —
— 130
— 110
—110
—100
— 90
—so
— 70
—60
Fig(6.6) TRANS/I. 92 TYPICAL. CURVE
DC MAGNETISATION THICkNES5: .0zo" (0.50 rritn) TEST HEFW OD: D.C. PERIV:AMETER
—
8 —
7 —
6 —
5
—
3 —
a
—
— 4u
—50
— 20
— 10
5 10
MAGNETISING FORCE - OERSTED
98
99
A substantial portion of the previous work seems to concentrate
on finding models for 131-1 characteristic.. The models used are either
analytic where an approximate function is used over the entire region of the
B-11 curve or a numerical model. Silvester4, showed that modelling of the
iron material using the B-H curve is inefficient from the computational point
of view. The alternative model is the use of u-B2
characteristic. This charac-
teristic is only computed once at the start of the program, for a given iron
material. In the iterative procedure, which solves the magnetic field prob-
lem, values of B2
are simply determined. Values of u can then be determined
from the u-B2 characteristic.
6.6.3. Methods of modelling the u-B2
characteristic
The choice of the method of modelling the u-B2
curve depends
on the method used for solving the magnetic field problem. For instance if
an analytical solution is used then the modelled can be considered well-
approximated by an analytical expression. However, this model becomes appro-
ximate when accurate numerical methods are employed in the solution. There-
fore the use of a numerical model becomes preferable when accuracy is re-
quired in the solution of the magnetic field problem.
Silvester et el 4showed a method using the spline-fit approxi-
mation. Here in this thesis two different numerical models are used. One is
applied to micro-alternator magnetic field problems and the other on turbo-
alternator magnetic field problems.
6.6.4. Modelling of u-B2
characteristic of the iron material for micro-alternator applications
A piece-wise linear model of the characteristic of the iron
material is used. It is found necessary not only to model u-B2
curve but 2
also 311 - B2 and 30 - B2 curves. bE2 3B4
100
Fast convergence of the iterative process can be achieved by
adjusting the above curves so that they are positive continuous. The non-
linearities in the low flux density region of the B-H curve are consequently
neglected. The numerical procedure can be explained using figures (6.7a,b,c).
A finite number of points are selected on the B2-axis. On this axis the
distance between B2' (saturated value of the flux density squared) and the
end of the initial linear portion is divided into equal intervals. It is
assumed that the curve is linear between successive points. The slope of the
U-B2
curve between two successive points i and i + 1 is
U. -U. 31) I 1+1 3_
aB2
i 5 (B2)
This slope is correct at B.2 1 + -5B.
2 and for programming it is stored 2
at the point i.
(6.1)
The reluctivity U can be computed for a given B2
by finding
the appropriate section j, given by the first integer below
B2 - B
2 m + 1 for B2 2 Bm
2 6132
2 If B
2 B
m then j = 1
Then
2 2 Du = (B - B.) I + U. -1 3B j
Similarly, the slope of the au B2 curve between two successive
3B2
2 points i and i + 1 on the B axis is
D2
( T I. DU 2 .) 6132
a(B2)2
aB i+1 aB
the above value is assumed correct within the region from
2 1 Bi+1 2 - (SB2 to B2 + 2 SB2 .
(6.2)
(6.3)
(6.4)
(0)
0.0 • 82 11-1 i i+1 Bz B
2
(a) -B characteristic
2. n0 B BS
(b) as
i-1 t i +1 2. Bm
characteristic
1 (b)
(c)
• .2 B characteristic
Fig (5.7) Modelling of the iron characteristic
0 0 .
(c) Bm
101
1. -1 1. 1 +1 2. 5s
102
6..7 ITERATIVE PROCEDURES AND RESULTS OF THEIR APPLICATION ON STEADY STATE MAGNETIC FIELD PROBLEMS
6.7.1. Relaxation techniques
The following programs were written:
1. Over relaxation for values of A and under relaxation for
values of II-A constant value for the over relaxation of 1.9 and under
relaxation of 0.2. (Values of 1.9 and 0.1 are recommended by Anderson 27).
It is found that the number of iterations is excessive and the computational
time is unacceptable.
2. Corrections of values of A directly after their evaluation
leads to a divergent solution. This technique is replaced by correcting
values of A after calculating all values of A. The program corrects values
of A in the reverse order but it is found that this technique leads also to
a divergent solution.
6.7.2. Optimisation techniques
1. Steepest descent: It is not recommended and no program
was written.
2. Fletcher and Powell method: A program explained by Ralston35,
is available in the scientific manual of the IBM computer. It is found that
the size of the matrix to be stored is equal to the lower triangle matrix
plus the diagonal. It is also found that the computational time is greater
than three minutes on CDC 6400 computer for each value of field current.
These results indicate that it is not advisable to use this technique for
the solution of the magnetic field problem.
3. Conjugate gradient method: This program is also available
in the scientific manual of IBM Computers. It is found that the core
needed is much less than that needed by Fletcher and Powell method. However,
the computational time is still of the same order. Therefore its use is
103
not recommended.
4. A program was written using the. Newton-Raphson technique.
The inversion of the Hessian matrix is obtained by inverting the diagonal
elements only. This gave a divergent solution.
5. Newton-Raphson technique and sparsity techniques: Good re-
sults are obtained as shown in sec. (6.10).
6. Modified Newton-Raphson technique and sparsity technique:
This program is convergent. However, it is found that the calculation of
the coefficient a in equation (5.27) increases the time per iteration above
that of the previous program.
The Newton-Raphson technique and sparsity are found the best
for nodes up to 700.
6.8 COMPUTER PROGRAMS
The computer programs to solve the steady state field problem
have three parts.
1. A program which calculates the coordinates of each node
for given dimensions, constructional details of a machine and a chosen grid.
2. A program which calculates the dimensional entries to the
S- and H- matrices.
The output of the above two programs are stored in files. They
become input to the third program.
3. A program which calculates the essential numerical method of
solving the magnetic field problem.
The first two programs are so simple so that their method of
programming is not given. The flow chart of the third program is shown in
fig. (6.8). It is explained in the next section.
104
6.8.1. Flow chart of the main program
The flow chart of the main program shown in fig. (6.8) consists
of the following operations:
Box number 1 : Start of the main program.
Box number 2 : Reads the input data calculated from programs 1 and 2.
It reads also other data from the input such as the
coefficients of the iron material(s) used and data about
special nodes, ... etc.
Box number 3 : Preliminary distribution of the currents at the various
nodes; Field nodes only in the case of open circuit
characteristic.
Box number 4 : For a given field current, the current located to each node
from a specified distribution is determined.
box number 5 : The program feeds the information to the essential subroutine
of the numerical solution. It is fed mainly through common
statements.
Box number 6 : Calls the routine of the numerical solution,and also gives
the results.
Box number 7 : The results are written in the required format or stored for
further usage.
The process from box 3 to box 6 has to be repeated for computa-
tions of the flux distributions for different values of field current.
5.8.2. Flow chart of the routine of the numerical calculations
The flow chart in fig. (6.9) describes the operation of the
routine. It consists of the following operations:
Box number 1 : The subroutine starts with a given value of the field
current as well as the distribution of currents at the
grid nodes.
105
Start the Main Program (1)
Read data and from
from files
input (calcu-
(2)
lated in other programs
(3) Preliminary specification of currents at grid nodes
-1.10
Calculate the currents at grid nodes (4)
(6) (5)
Call thG subroutine of the numerical solution
Write the result
Ston End
Subroutine of the Numerical solution
(flow chart fi.6.9)
(7)
Fig.(6.8) Flow chart of the main program
106
Box number 2 : Vectors used in the iterative procedure are all set to
zero except those of the currents fed to nodes.
Box number 3 : The start of the iterative procedure for all the grid nodes.
Box number 4 : Precalculated data in the files related to the S- and H-
matrices are read.
Box number 5 : Calculation of the flux density in each triangle around
the grid nodes.
Box number 6 : A selection of the characteristics of the material of the
triangle which it is made. It calls the suitable routine
corresponding to the material of the element.
Box number 7 : The routine calculates the required properties for a given
element from predefined characteristics of the material. It
is repeated for all elements around a given node.
Box number 8 : The elements of the S- and H-matrices from each triangle
are calculated.
Box number 9 : The boundaries are checked. If the node lies on the boundary
(Neumann's boundary) box number 10 is called, if not,
number 12.
Box number 10: Call the subroutines which augment the elements of the
S- and H- matrices on the boundaries (Neumann's boundary).
Box number 11: The routine which calculates the S and H elements on the
boundaries.
Box number 12: Augments the elements of the S and H matrices in the correct
order.
Box number 13: Calculate the error resulting from a given field distri-
bution at each. node.
Box number 14: Store the elements of the H-matrix using
a storing routine.
Subroutine of (7a) Material 1
Subroutine of material 2
(7b)
107
boundary
no
the ele:,,en ts (12
(8) Calculations of the entries in the S- and. H- matrices for each element
:111T,r1t- ne
arrahLemen
Store the elements of the H-matrix Call storinr; routine
Inversion of (16)
the H-matrix
(10 Storing Subroutine
Inversion libroutine
(15)
(17)
Subroutine start with input data (1)
Do k=1,M
Preliminary preparations Of the vectors (2)
J=1 ,N((3)
Read the constants of the vectors from the files
1 ,L
Calculations of the flux density (5)
(6) Select the type of the iron material used or air. call the suitable subroutines
Check the bouncari7 (9) (lc)
Subroutine o u ndary
(13) calculation of the error in th
Correct the values of A at the end of this
(18)
iteration
ieturn
Fig.(5.9) Flow chart of the numerical subroutine
108
Box number 15: The subroutine which stores the elements in the way which
suits the sparsity techniques.
The process from box 3 to box 14 has to be repeated for all
the grid nodes.
Box number 16: The file which contains some of the data must be rewound
so that the iterative procedure can be repeated. The H-
matrix has to be inverted.
Box number 17: Inversion of a sparse matrix, H-matrix, is done by the
inversion routine for a given error at the nodes. This
determines the incremental changes in potential at each node.
Box number 18: Values of potential are corrected from the previous itera-
tion using the incremental changes in potential.
The iterative procedure has to be repeated a number of times so
that the error at the nodes and their incremental changes from one itera-
tion to the next is less than acceptable lower limites. The results of this
routine are returned to the original program for further processing .
6.9 NO-LOAD FLUX PATTERN
The no-load flux distributions of the micro-alternator are shown
in fig. (6.10) to fig. (6.17), values of field current ranging from 0.1 to
lA at intervals of 0.1A. The number of lines in each diagram are the same.
They divide between the maximum potential and zero to a fixed number of equal
divisions. The difference in potential between any two successive lines is
thus not the same in different diagrams. Lines terminating on the interpolar
axis have been omitted. The flux distributions in fig. (6.10) to (6.17)
are obtained when the grid in fig. (6.4) is used. From the obtained flux
plots it is clear that the flux lines have almost the same distribution for
all the values of field currents considered. The reason is that the flux
density is low and the iron is working in the linear region of the B-H curve.
It may be seen that:
109
a) The flux line which_ crosses the field slot is approximately
perpendicular to the iron wall which satisfies the air-iron boundary con-
ditions.
b) The flux lines go around the bolts holding the stator lami-
nations together because they are considered non-magnetic.
There is a discritisation error which occurs in the flux pattern
because of the grid used. This error affects the flux pattern but not the
values of the field at the nodes. The distortion occurs because the diffe-
rence between the current needed to obtain curved flux lines instead of
straight ones is very small. Also, the permeabilities in these regions are
very small.
Potential and flux density distributions on the rotor surface
are plotted at 0.8 and 3A field current. They are shown in figures (6.18)
and (6.19) respectively. These plots show the effect of the stator and rotor
slotting.
The use of the grid shown in fig. (6.5) gives a small difference
in the flux distribution from those obtained from the grid of fig. (6.4).
The flux distribution is plotted as shown in fig. (6.22) at field current
of 0.8A.
6.9.1. Open circuit characteristic
The results of the flux plots are checked by comparing the
computed values of the open circuit voltage at different field currents
with test results.
The computer program has been used to find the radial induction
at the armature surface and along a line at the centre of the stator slots.
The no-load voltage E is calculated from the fundamental component of the
flux linking the stator winding.
Fig (6.10) Flux distribution at no load for 0.1A excitation current of a 3KVA micro alternator
Fig (6.11) Flux distribution at no load for 0.2A excitation current of a 3KVA micro alternator
Fig.(6.12) Flux distribution at no load for 0.3A excitation current of a 3 KVA micro-alternator •
Fig (6.13) Flux distribution at no load for 0.4A- excitation current of a 3 KVA micro-alternator
Fig (6.14) Flux distribution at no load for 0,5A excitation current of a 3 KVA micro-alternator
Fig (6.15) Flux distribution at no load for 0.5A e.\'citation current
of a 3KVA micro-alternator
rig (6.16) Flux distribution at no load for 0.7A excitation current of a 3KVA micro alternator
Fig (6.17) Flux distribution at no load for 0.8A excitation current of a 3 KVA micro alternator
114
Flu
x den
sity
B
0.8
0.7
1,0
L
Q,
0.9
5. 0.6
0b
0.5—
0.4—
0,3
0.2—
0.1
00
(a) Potential (b) Flux density
(b)
rotor surface (lo scale) /7/,-///,////// -v.,././/////,/ N..(/// //////./ Ne,// ///
Fig (6.18) Flux density and potential distributions along the rotor surface of the micro alternator at 0.8A field current
0.4 — Po
ten
tial in
We
ber
/m.
,
028
Flu x
de
nsity B
0.24
(a) potential ( b) flux density
0.16—
0.12—
0.08—
(to scale)
y////// /// \///// // ////// "V77 rotor surface
(b)
020—
0.36—
0,32—
115
Fig,(6.19) Flux density and potential distributions along the rotor surface of the micro alternator at 3.0A field current
116
6.10 TYPICAL RESULTS
The methods of solution and programming applied to the two
grids gave the following res.ults:
Grid number 1 in fig. (6.4)
Number of nodes 192 with 26 fixed nodes'
Size of the H--matrix 3709 words
File required to load the program using FUN compiler =
23300 Oct. no.
File required to run the program in the same compiler =
53500 Oct. no.
Average time per iteration on CDC 6400 computer = 0.77 s
Number of iterations for each field current = 10
Maximum error in the solution at the nodes = 0.6 x 10-4 A
Maximum incremental change (5A) in the last iteration =
0.5 x 10-5 Web/m.
Grid number 2. in fig. (6.5)
This grid is essentially constructed for the solution of the
transient field problem.Nevertheless it is used to study the no-load flux
distributions.
Number of nodes 210 with 26 fixed nodes
Size of the H-matrix 3879 words
Average time per iteration on CDC 6600 computer = 1.1s.
Number of iterations = 15
Maximum error in the solution at the node = 2x10-3A
Maximum incremental change {6A} in the last iteration =
1.5 x 10-6
Web/m.
The first grid is used to calculate the open circuit characte-
ristic of the micro-alternator. Test results and computed values are shown
0.0 00
117
co 160
cc 140 —
V)
c)
120
100
80 —
60
40
20 — EXPERIMENTAL RESULTS COMPUTED VALUES
1 I 0.2 0,4 0.6 0.8 1.0 1.2
FIELD CURRENT IN A.
Fig.(6.20) Open circuit characteristic or the micro alternator
118
in fig. (6.20) but as the difference is small values are compared in
Table (6.3).
Table 6.3: Comparison between points on the open-curcuit curve obtained by mesurement and calculation.
Field current A
Computed values V
Test results V
0.1
14.5
15.8
0.2
29.
30
0.3
43.5
45.
0.4
58.
60
0.5
72.5
74.6
0.6
86.97
88.8
0.7
101.3
102.4
0.8
115.45
115.4
0.9
129
127
1.
141.6
137.8
3.
230
224
(largest error is 3%)
6.11 FIELD DECREMENT TEST OF THE MICRO-ALTERNATOR
6.11.1. Discription of the test
With the stator on open circuit and the machine rotating, the
field supply is suddenly short-circuited. Thus there is an initial distri-
bution of flux in the machine which then decays accompanied by induced eddy
currents wherever a closed conducting pass exists in the rotor. In the micro-
alternator, the closed conducting paths were the damper bars, connected by
the end rings.
6.11.2. Grid of triangles for the transient solution
The grid of triangles shown in fig. (6.5) is used to study the
characteristic of the flux lines during the decay period of the field current.
It includes the nibs that separate the field and the damper windings.
Although the separation between damper nodes and field nodes is small, it is
important for the numerical solution. When the grid contains the nibs it
leads to a separation between the field and damper conductivity matrices
as explained in sec. (4.8.1). Therefore there is no problem at the initial
conditions. The effect of end leakage flux is included in the solution
which is different from that in solid rotor machines as explained in the
next chapter.
119
6.11.3. Determination of the [AG]-matrix
Only those nodes which_ are the vertices of triangles carrying
current have nonzero entries in the [LT] matrix, as explained in sec. (4.8.1).
The remainder are zero. Since the armature windings are open circuited,
therefore the stator nodes have zero entries. The position of the nonzero
entries depends on the node numbering system. Here, the grid is numbered so
that the AG-matrix takes the following shape:
[AG] = O 0 0
O a1 ' 0
O 0 a2.
(6.5)
This shape is obtained by using the grid of fig. (6.5). There
is no node supplied from both damper and field,and consequently there is no
overlap between the matrices a1 and a2.
The [a1] matrix is generally symmetric. Here, by assuming a
uniform distribution of current density within the part of each triangle
enclosed by the contour of integration, al becomes a diagonal matrix. Its
elements depend on the areas of triangles in the damper.Assumingone third
the area of the damper triangle to each vertex then,
T — L a 1i,i 3
j=1 Aj eff
where Geffis
the effective conductivity of the jth
triangle of area Ai
from the m damper triangles surrounding node i. The value of Geff
is explained
in sec. (6.11.3.1)
In the G2-matrix (representing the field winding) the finite
element model is used and the entries of each row are all considered equal.
Their values are.
L,k
1
3 cieq A k = 1,Nf (6.7)
(6.6)
120
where laeq
is the equivalent conductivity,obtained in sec. (6.11.3.2)
q number of triangles carrying field current and surrounding node i.
Nf is the total number of field nodes.
6.11.3.1. Equivalent conductivities of the damper circuit
During the transient period, the induced e.m.f.s. inside the
damper bars vary over the cross-section. By considering a triangle in the
damper region, a network model could be constructed for it either in the
direction of the flow of the eddy currents or in a plane perpendicular to
it or in both. A network model in both directions, when the end ring re-
sistance is not included, is given in ref. (16). Here the model is considered
only in the direction of the eddy currents but includes the end ring resis-
tance and that of the bars beyond the slot portion.
The network model of a damper triangular element is shown in
Fig. (6.21). The voltages induced in the
slot portion el, e2 and e3 each face a
resistance r determined by the slot length
l' one-third of the area of the triangle
and the resistivity of the material. The
resistance R represents the equivalent re-
sistance of the triangle in the bar beyond Fig.(6.21) Network model
the slot and the end ring. The currents at of a triangular element in a damper circuit.
the triangle vertices can be expressed as
1.
it
i2
3
r + 2R. -R -R
-R r + 2R -R
R -R r + 2R
1 2 r +3rR
el
e2
e3
(6.8)
The resistance R representing the equivalent length 2,2 (outside the slot
is given by
r 3Q 1
(6.9) R 2,
2
n (average) = 2 -1- .1- a e
turn at i =1
DA. 1 (6.12)
121
The three current densities acting in the triangle and attri-
butable to the vertices, Ji, J2 and J3 can be expressed in terms of the
induced voltages per unit length_El, E2 and E3 as
2 1 1 k 4. Q2 T
2,2
2,2 1 3 2
1 -
-5'- k
2 k 2, k 1+ 3 2 2
5 2,
2 3 4'2. k 4- 1 k
2 1 3 1 1 ,
Jl
J2
J3
El
E2
E3
(6.10)
If the circulation of the eddy currents inside the triangle is neglected,
the effective conductivity matrix[aeff
] becomes
2,1
aeff
= a CU] (6.11) Q
where [U] is the unity matrix.
6.11.3.2 Equivalent conductivity of the field circuit
The average induced e.m.f. in any turn can be expressed as
where k is the average effective length, n is the number of field nodes
and a- are weighting coefficients dependent on the induced e.m.f. distri-
bution within the slot. Assuming ai = 1 for this laminated machine, the N
series turns/pole give an induced e.m.f. of
kN n A. • E.M.F. = 2
n at 1=1
This voltage produces a field current If in the field winding so that
nCC
A. If =
kN nRfs i=1
at
where Rfs
is the resistance of the field winding per pole.
Since there are Zs conductors in each slot of area As
, the
average current density J is therefore
(6.13)
(6.14)
122
Z If J -
As
and the equivalent field conductivity can therefore be defined as
Zs a = 2 —
eq As
Rfs
a n 3A. j = eq y Jf
1L=1 3t
and
(6.15)
(6.16)
(6.17)
6.11.3.3 Approximations for the damper circuit inductances and Resistances
The inductance of the damper circuit can be divided into three
portions. The mutual inductance which is the main inductance, the slot
leakage and the overhang leakage inductance. The mutual and the slot leakage
are automatically taken into account during the evaluation of the flux dis-
tribution. The value of the overhang leakage inductance per damper bar can
be calculated as given in reference 48. The ratio between the damper slot
leakage and the overhang leakage changes the initial period of the transient
current decay by a factor Ktwhich changes with time as
Kt = 1 + (K - 1) e-t/T (6.18)
where K and T are explained in Appendix C.
6.11.4 Unsymmetry of the conductivity matrix and the iterative procedure
It has been shown in sec. (5.3.2) that the use of Newton-Raphson
technique implies the evaluation of the H-matrix, of equation (5.25). The
matrix Hl is symmetric but the , addition of the matrix 2 [Aa] to it destroys
the symmetry of the H-matrix. The matrix 62
in the Acs-matrix expressed by
equation (6.7) causes this asymmetry. The H-matrix can be portioned as
Q
P
R
(6.19)
T
where Q is symmetrical and is the largest matrix of the four when the
123
stator windings are open circuited. T is a square unsymmetrical matrix.
R and P are rectangular matrices.
At any given instant of time during the transient period, the
incremental changes of the potential A during the iterative process is found
from equation (5.58). The numerical solution is obtained by partitioning the
vectors {SA} and {G} to the vectors {SA1'
{SA2} {G1} and {G
2} so that the
algebraic manipulation can be achieved. Using the partioned form (6.19) of
the H-matrix equation
from which
{6A2} =
and
OA1 1 =
G2
4[T]
- EQ]
(5.58)
- EPHO-1
-1 (G1}
becomes
R
{SA2}
SA1 SA2
2}- [P]C0-1 {G1}
(6.20)
(6.21a)
(6.21b)
P
- CQ]
T
CR] -1]{G
-1CR]
The iterative process is continued until convergence is reached.
The solution is considered satisfactory where the largest element of the
gradient vector {G} approaches an acceptable lower limit of .001 per unit
current and the incremental changes in the values of {A} are very small
(1.5 x 10-6 Weber/m).
6.12 FLUX DISTRIBUTIONS DURING THE TRANSIENTS
The flux lines are plotted at successive instants in time as
shown in fig. (6.22) to fig. (6.31). In all these diagrams the potential
difference between any two successive lines is kept constant. The initial
flux distribution shown in fig. (6.22) is that which gives the rated voltage
of the machine. It is clear in these diagrams that there is a "motion"
in the flux lines, i.e., lines of fixed potential are changing their position.
For instance, the flux line crossing the rotor slot adjacent to the centre
124
Fig (6.22) Flux distribution at no load for 0.8A excitation current of a 4-pole 3KVA micro alternator;
125
Fig (6.23) Flux distribution 0.002s after the field winding is short circuited.
126
Fig (6.24) Flux distribution 0.004 s after the field winding is short circuited
127
Fig (6.25) Flux distribution 0.006 s
after the field winding is short circuited.
128
Fig.(6 .26) Flux distribution 0.008s.
after the field winding is short circuited.
129
Fig.(6,27) Flux distribution 0.01s.
after the field winding is short circuited.
130
Fig. (6,28) Flux distribution 0.015s after • the field winding is short circuited.
131
Fig. Flux distribution after 0.02s. after the field winding is short circuited.
132
Fig. (6.30) Flux distribution 0.03s. after the field winding is short circuited.
133
Fig. (6.31) Flux distribution 0.13 s.
after the field winding is short circuited.
134
polar axis has different positions in the various diagrams.. Its motion is
accompanied by the appearance of a flux line which_ crosses the stator slot
adjacent to the inter polar axis shown in fig. (6.26). This line rapidly
changes its position in the stator followed also by a change in position of
another flux line around the fourth nonmagnetic bolt from the centre polar
axis which. is shown in fig. (6.27). In fig. (6.28) the flux line crossing the
rotor slot adjacent to the centre polar axis has changed its position to be
completely in the tooth while the flux line crossing the stator slot becomes
crossing it in the narrow part. Therefore a new flux line in the stator tooth
adjacent to-the centre polar axis is found. This means that there is a re-
duction in the total flux lines in the machine region. There is a small change
in position of the flux line crossing the stator slot between fig. (6.28) and
(6.29). In fig. (6.30) there is a reduction in the flux lines which makes
the flux line crossing the stator slot of fig. (6.29) to come directly from
the rotor iron. This line disappears in fig. (6.30). Therefore the flux lines
are reduced by an amount equal to the difference in potential between two
successive lines times the length of the machine, during the considered
period. The decay of the number of flux lines and their redistribution con-
tinues indefinitely approaching zero.
Reduction of the flux lines is affected by the induced eddy
currents in the damper bars. Their values and distributions are different
from one bar to the other and from one instant to the next.
6.13 EDDY CURRENT DENSITY DISTRIBUTIONS DURING THE TRANSIENT PERIOD
During the transient period eddy currents are induced in the
damper bars. Their distributions in these bars at successive instants are
shown in fig. (6.32) to fig. (6.35). Induced eddy current densities are
plotted in the form of contour lines. Each contour line represents a constant
value of the induced eddy current density. A number of contour lines are
, 1 \
/ \ /
r \
r
0.1015
0117
0,104
POLE
0.104
LINES OF CONSTANT CURRENT 0.1197 DENSITY AT INTERVALS OF 0.0025A/mm 2
— - - - - -- CURRENT DENSITY OF 0.115 A/mm 2
Fig.(6.32) Eddy current density distribution in the damper bars 0.006s
after the field winding is short circuited.
0,1315
POLE E -Kt
0.139 ,
0.138
0.1387
LINES OF CONSTANT CURRENT
DENSITY AT INTERVALS OF 0.0025 A/mm 2
CURRENT DENSITY OF 0.141 A/mm2
Fig (6.33) Eddy current density distribution in the damper bars 0.01s after the field winding is short circuited.
0.1557 0.170
POLE •
LINES OF CONSTANT CURRENT DENSITY AT INTERVALS OF 0.0025A/m m2
- ----- CURRENT DENSITY OF 0.173 A/mm 2
0.1 72
Fig (6 -34) Eddy current density distribution in the damper bars 0.02s after the field winding is short circuited.
POLE
LINES OF CONSTANT CURRENT DENSITY AT INTERVALS OF 0.0025A/mm2
0.1854
0.1894
Fig. (6.35) Eddy current density distribution In the damper bars 0.03s after the field winding is short circuited.
139
plotted to show the concentration of the eddy currents.
Eddy current plots show that maximum induced eddy current
densities in the dampers are those nearest to the source of disturbance.
Here the source of transients is the field winding and consequently, at any
instant, the maximum eddy current densities in the damper bars are those
closest to the field. During the rise time of the induced eddy currents, the
nibs projecting into the rotor slots reduce the fast changes in the flux
pattern. The result.is that there is a reduction in the density of the induced
eddy currents in the "shadow" of these nibs. This is clear in the far end
triangles in each damper bar as shown in fig. (6.32) to fig. (6.34). As time
increases and the induced eddy currents start to fall, the intervening nibs
also act to prevent fast variations in the flux and consequently in the in-
duced eddy currents. The result becomes then the inverse of that previously
obtained. Eddy currents behind the nibs in the same triangles of the previous
case cannot decrease very fast from their last values. This results in the
distribution of the induced eddy currents shown in fig. (6.35).
Although a uniform conductivity is assumed for the damper bars,
the time constants of the induced eddy current densities are different. Some
of them such as those adjacent to the magnetic nibs are affected by the adja-
cent extra flux in the iron and are greater.
Eddy current density distributions are plotted against time for
three points on the damper bars. In fig. (6.36) the time variation of the
eddy current densities at three points a, b and c in the damper bar adjacent
to the centre polar axis are shown. At a similar point in the middle damper
bar eddy current densities are plotted against time as shown in fig. (6.37).
Fig. (6.38) shows the eddy current densities as a function of time for the
damper bar adjacent to the interpolar axis.
At a given instant in time, values of eddy current densities are
different at the similar points in the three dampers. 'The flux linking the
140
dathper bars adjacent to the interpolar axis is greater than the flux linking
the damper bar adjacent to the centre polar axis. Thus changes in the flux
induces eddy currents with.different values in the damper bars. Eddy current.
densitiesin the damper bars in the q-axis are greater than those nearest to
the d-axis.when the flux lines have their main direction in the d-axis. These
results are consistent with the overall effect of the flux lines and the
induced eddy currents on the field current.
6.14 CHARACTERISTIC OF THE FIELD CURRENT
Sudden short circuiting the field terminals lead to a decay of
the field current. This decay depends on the induced eddy currents in the
damper bars and the nonlinear characteristic of the iron material The response
of the field current is similar to a disturbance which occurs in one of two
mutually coupled circuits, when one of them is short circuited. The first
part of the field decay depends mainly on the response of the damper circuit.
Therefore the overhang leakage inductance of the damper and the field cir-
cuits have to be taken into account in the calculations, otherwise errors can
Occur.
A comparison between test results and computed values are plotted
in fig. (6.39). Computed values agree with test results within 5% . This
shows that the method of analysis gives the correct overall picture and its
application to other transient conditions likely to be useful.
6.15 TYPICAL RESULTS
The grid of fig. (7.5) is used and the following results corres-
pond with the data given:
Number of grid nodes = 210 with 26 fixed nodes
Size of the H-matrix = 3879 words
Two other matrices are also used of size 166 x 19 and 18 x 18.
File used to load the program using FUN cOmpiler = 33100 Oct.
0.10—
0,08
0,06
0.04
0.02
0.00 0.00 0.002 0.004 0.006 0.008 0.01 0.012
Time in
141
Fg. (6. 3 6) Variation of the induced eddy curr en t s in the damper bar at points A,B &C in the slot nearest to the pole axis during the growth of eddy currents.
E
field winding
. 0.002 0.004 0.1006 0_ 1008 0.101 0012 Time in s
142
Fig.(6.37) Variation of the induced eddy currents in the damper bar at points A,B &C in the middle slot during the growth of eddy currents.
Edd
y c
urr
en
t
143
0 00 0,002 0.004 0.005 0.008 0.01 0.012 Time in
Fig.(5'.38) Variation of the induced eddy currents in the damper bar at points A B&C in the slot nearest to the in
axis during the growth of eddy currents.
00.1 0.18 TIME IN S. 01202 00.04 00.06 00.08 0.22 0.15 0.0 0.12 0.14
EXPERIMENTAL CURVE • • COMPUTED VALUES
CUR
RE
NT
IN
Fi g (6.39) Field decrement test on the micro alternator
145
Average time per iteration on CDC 6400 computer = 7.35 s.
Number of iterations used per time step = 14
The maximum error in the solution = 0.1 x 10-4
A.
Total time to run the program on a CDC 6400 computer and to
obtain the plotted results = 1024 s.
146
.CHAPTER 7
STEADY AND TRANSTFINT APPLICATIONS ON A TURBO-ALTERNATOR WITH_ SYMMETRICAL FLUX DISTRIBUTION
7.1 INTRODUCTION
In the previous chapter the analysis of Part I was applied to
a micro-alternator in which eddy current., occurred only in the damper bars.
Here in a solid rotor machine, eddy currents occur in the rotor body and
provision must be. made for it. In particular the choice of grid must allow
for elements representing the rotor iron to be not adjacent to those repre-
senting the field windings, otherwise there is a discontinuity in the numeri-
cal solution.
Eddy current distributions and the field current are obtained
for a field decrement test. This is similar to that performed on the micro-
alternator except that the field circuit is suddenly shorted through a dis-
charge resistor. Again, only half a pole-pitch of the machine is considered.
The initial flux pattern is also obtained for a range of field
currents and the machine open circuit curve is calculated and is compared
with measured values.
7.2 DISCRIPTION OF THE TURBO-ALTERNATOR
. The nominal rating of the turbo-alternator used is 275 MW,
0.85 power factor, 18KV, 10377A, 50 cis, 4 pole, 1500 r.p.m., Hydrogen cooled.
7.2.1. Stator
The stator is constructed from laminated sheet steel arranged
in packets between which ventilation ducts are provided. Relevant data is
given in table 7.1.
147
Table 7.1 Details of the stator
Phase voltage 18/V KV
Full load current 10377 A
Number of poles 4
Frequency 50 c/s
Number of slots 72
Number of parallel circuits 2
Conductors per slot 2
7.2.2. Rotor
The rotor is a single forging, the magnetic characteristics of
which are given by characteristics of other samples of the same composition.
The shape of the rotor slots is chosen so as to maximise the cooling. The
normal concentric rotor winding is employed together with nonmagnetic end-
winding retaining rings. The following numerical data is provided:
voltage 330 v
Current 1800 A
Number of slots 4
7.3 APPROXIMATIONS FOR THE NUMERICAL SOLUTION
The model differs from the actual machine in the following ways:
a) Fringing of the flux lines at the machine ends:
The fringing of the flux lines is taken into account by using
the effective length of the machine instead of the physical length.
b) Stator ducts
The effect of the radial ducts in the stator is taken into account
by altering the characteristic of the stator iron material as shown in
sec. (7.4.2.).
148
c) Extra spacing in the field slots:
1) In the steady state, the equivalent field winding is
assumed to fill the space occupied by the insulation and the cooling ducts,
and is given a reduced conductivity to maintain constant resistance.
2) In the transient state allowance is made for the slot leakage
in both radial and tangential directions as explained in sec. (7.5.1).
d) Contact resistance between the rotor and conducting wedges.
The contact resistance between the wedges and the rotor iron is
assumed zero as the centrifugal forces on the rotor ensure intimate contact.
e) Positioning
One relative position of the stator and rotor is used.
f) Return path of eddy currents
Eddy currents are assumed to flow in the retaining rings from
one pole to the next. This does not prevent the internal circulation of the
induced eddy currents. The resistance of the end paths is neglected.
7.4 STEADY STATE NO-LOAD CHARACTERISTICS
The steady state analysis explained in Part I in this thesis
was applied on the turbo-alternator. No-load flux distributions are obtained
at different values of the field current. The computed values of the flux
distributions are checked with the experimental results by comparing the
induced voltage at various values of field current.
7.4.1 Choice of a grid for steady state computations
The nodal method introduced in chapter 3 lead to the possibility
of choosing a combined grid of triangles and rectangles. In the two-dimen-
sional cross section of a half pole pitch in the machine, the grid chosen is
as shown in fig. (7.1). A grid of rectangles is used in the stator and rotor
slots where the permeability is independent of the resultant field. The
.149
Fig. (7.1) Grid used to divide a half pole pitch of the turbo alternator for steady state flux calculations.
150
rectangles are indicated by St. Andrew's crosses in the rectangles.
Triangles are used in the remainder of the cross-section. Many
small elements are introduced where flux density gradients are large, and
provision is made for flux to be able to change direction. Wherever possible
large triangles are used so as to minimise the number of nodes.
7.4.2 Modification of the characteristic of the iron material
The characteristics of the stator iron are modified so as to
allow for the presence of the radial cooling ducts.
Thus the reluctivity used corresponding to a given
flux density B1 is that corresponding with a higher
density B2, fig. (7.2), where
B2 gross B1
2,effective
Fig. (7.2)
7.4.3 Multi-function modelling of the iron characteristic
A new method of modelling the characteristic of the iron material
is used for solving turbo-alternator field problems. For computational
purposes4 as explained in chapter 7, the iron material is modelled in terms
of the u-B2 characteristic. In the previous methods
4,49, including that
explained in the previous chapter, the same method of approximation is used
all over the curve. In other words, the curve is divided into parts and the
same approximation is used in each. It is found that the use of one approxi-
mating function with different coefficients for an equally divided B2 scale
needs much preliminary work. This can be avoided by using a multi-function
model.
In the multi-function model of the iron characteristic, different
analytic functions are employed over a non-uniformly divided B2-axis. There
are to conditions to be satisfied:
151
a) The continuity of the U-B2 characteristic at the points
dividing the B2 axis, i.e., points of intersections of the functions, has
to be ensured.
b) The continuity of the function derivatives must be satisfied
at points of functions intersections. These conditions can be written
mathematically as
limt f.(B2 2 limt f.+1 (B
2) .
• 1
B2
L
fa.. B +a.
1 i_
(7.1)
for all u.=1,...,q
and
limt
B2 + a.
3f.1(B
2)
DB2
= limt
B2 a.
Df1. -,(B
2) +1
DB2
where f.(B2) and
f.1+1 (B2) are two different adjacent analytic U functions
of B2. They are valid in two successive regions of B
2 with a common point
at B2 = a.
The functions employed in the divisions are straight lines,
exponential functions and fourth erder polynomials. The first step is to
determine the regions where straight lines are adequate followed by those
that can be given by exponentials. Fourth order polynomials are then used
for the remainder.
Regions where straight line portions may be used, may be ob-
tained from the U-B2
curve directly. Exponential functions are obtained
from the log(u) -B2
curve and fourth order polynomials are used for the
remainder.
Straight lines and exponentials satisfactorily represent most
of the U-B2
curve and the polynomial is chosen to have the right values
and gradients at the points where it meets the other functions. It has the
form
= a + alB2 + a + a
3(B2)3 + a
4(B2)4
(7.2)
152
2 At the boundarias.1) B2, and Q
V R2
the differential of the
function,
au2 -
a + 2a2
R2 + 3 a
3(B2)2 + 4 a
4 (B2)3
DB 1
is made to match the value given by the other functions.
(7.3)
Thus the value of the coefficients may be obtained by solving
1 (B2 ) (132 )2 (B
2)3
1 1 1
(B2) (B
2)2
(B22)3 (B2)4
2 2 2
a
a1
a2 (7.4)
a3
a4
1 (B23) (B
2)2
(B2)3
(B23)4
3 3
0 1 2(B2) 3(B
2)2
4(B2)3
1 1 1
2 2 0 1 2(B
2) 3(B) 4(B
2)3 2
vl
2
03
D01
DB2
Du2
DB2
2 where U3, B
3 is a point on the curve in the centre of the region represented
by the polynomial.
7.4.4 Flux pattern of the unloaded machine
The flux distributions in the unloaded turbo-alternator at
several values of field current are shown in fig. (7.3) to fig. (7.11).
In each of these diagrams the number of flux lines plotted is the same.
However, the flux plots differ because of the change in the levels of satu-
ration in the various regions.
The stator yoke shows little variation in flux pattern, below
600A, the region is unsaturated and the pattern reflects this. The flux
in the teeth. near the q--axis becomes relatively larger as the pole saturates
at higher field currents (cf. 7.6 & 7.7, the second tootH frOm the
interpolar centre line).
153
The rotor teeth show transverse effects superimposed on the radial pattern.
Towards the bottom of the teeth, the flux is passing towards the pole, and
nearer the air gap normal slot leakage occurs in the reverse direction.
The teeth on the q-axis are dominated by leakage flux as might be expected.
The effect of a magnetic wedge is shown in fig. (7.12), con-
ditions which are otherwise comparable with those in fig. (7.3). At this low
field current the high leakage flux in the magnetic wedge virtually reverses
the flux in the tooth adjacent to the pole. At higher excitations this
effect tends to disappear (fig. 7.13 & fig. 7.4) because the wedge saturates.
Potential and flux density distributions along circumferential
paths are plotted in fig. (7.14) to fig. (7.22). They correspond to the flux
distributions of fig. (7.3) to fig. (7.11).
7.4.5 Typical results
The following data and computational results are obtained.
Number of nodes 292, 37 of which are fixed nodes
Size of the H-matrix = 3774 words
Average time per iteration = 0.421s on CDC 7600 computer
Number of iterations used for each field current = 15
Maximum error in the solution at the nodes = 0.5x10-10A
Maximum error in the potentials at the last iteration =
0.2 x 1012
Total time of the program to give the open circuit characteristic
shown in fig. (7.23) = 58.5 s
The open circuit characteristic and flux plots are also obtained
for the grid of fig. ((7.24) which is used to start the transient solution.
The following data and results are obtained:
FIG(7.3). FLUX DISTRIBUTION AT NO-LOAD
FOR 200 AMPS EXCITATION CURRENT
OF A 4-POLE.333MVA TURBOALTERNATOR.
FIG(7.4). FLUX DISTRIBUTION PT NO-LORD
FOR 400 AMPS EXCITATION CURRENT
OF V4-POLE.333MVP TURBORLTERNATOR
FIG(7.5). FLUX DISTRIBUTION AT NO-LORD
FOR 600 AMPS EXCITATION CURRENT
OF A 4-POLE.333MVA TURBOALTERNRTOR
FIG(7.6). FLUX DISTRIBUTION AT NO-LORD
FOR 800 AMPS EXCITATION CURRENT
OF R 4-POLE.333MVR TURBORLTERNATOR ta,
FIG(7.71. FLUX DISTRIBUTION AT NO-LOAD
FOR 1000 AMPS EXCITATION CURRENT
OF A 4-POLE,333MVR TURBOALTERNATOR
FIG(7.8 ) . FLUX DISTRIBUTION RT NO-LORD
FOR 1200 AMPS EXCITATION CURRENT
OF .R 4-POLE,333MVR TURBOALTERNATOR
FIG(7.9 ). FLUX DISTRIBUTION AT NO-LORD FOR 1400 AMPS EXCITATION CURRENT
OF A 4-POLE .333MVA TURBOALTERNATOR
FIG(7./0). FLUX DISTRIBUTION AT NO-LOAD
FOR 1600 AMPS EXCITATION CURRENT
OF R 4-POLE.333M\/R TURBOALTERNAJOR
FIG(7.771. FLUX DISTRIBUTION AT NO-LORD
FOR 1800 AMPS EXCITATION CURRENT
OF A 4-POLE.333MVA TURBORLTERNATOR
FIG(7.72). FLUX DISTRIBUTION RT NO-LOAD
FOR 200 AMPS EXCITATION CURRENT
OF A 4-POLE.333MVA TURBOALTERNATOR 1--. (7, (...)
FIG(7.13). FLUX DISTRIBUTION AT NO-LORD
FOR 400 AMPS EXCITATION CURRENT
OF R 4-POLE.333MVR TURBORLTERNRTOR
of
165
(Stator surface)
( PO
TEN
T I AL
S )-
->
a-
O
tD
O
CO
A
( PO
TEN
T I A
LS
) ->
B
(Middle of the air gap)
A
CD
Li
B O f
co cr co _-
-----;-<--</i-9--./:-"---/"." 7 FA
(Rotor surfaL)
Fig (7. 14) Fl ux density and potential distributions for field current of 200A•
A
166
N ( PO
TENT
I AL
S )-
--).
UHLINHHU (Stator surface)
O to 0
•Kr
A' C\J
B O O
(Middle of the air gap)
(Rotor surface) Fig (7.15) Flux density and potential distributions for
fi.e!d current of 400A
O
O O
R4
( PO
TENT 1 AL
S )
CD
CNI
O
O
167
( PO
TEN
T I R
LS )
-->
a_
A
B
(Stator surface)
(Middle of the air gap)
O
A
B
O O t_i I
(Rotor surfaceY Fig (7.16) Flux density and potential distributions for
field current of 600A
OD
( PO
TEN
T I A
LS
)->
0
0
0
( PO
TEN
TIA
LS
)-->
168
B
'OULIUUNHH (Stator surface)
A
I (Middle of the air gap)
( PO
TEN
TIA
LS
)--->
B
(Rotor surface) Fig (7.77) Flux density and potential distributions for tr800A
(PO
TEN
TIA
LS
l-->
c•
0
O
O O
a
CO
( PO
TEN
TIA
LS
) )-)
•cr
(Stator surface)
169
(Middle of the air gap)
(PO
TEN
TIA
LS
)---
>
A
1 Li
a_
ct-
s(R°tfar urce)
Fig (7.18) Flux density and potential distributions for fi'ld current of 1000A
O
0-0
z 1J ■--
a-
a_
'ouHmunuu B
O
O
O
C7
O
I 00
(Middle of the air gap)
(f) _J CL
z
O a_
a_
O
I (Rotor surface)
170 A
(Stator surface)
Fig (7.19) Flux density and potential distributions for field current of 1200A
( PO
TEN
T IA
LS
) ---
>
O O
1
B
171
to CO
`7 Cl_
•
( PO
TEN
T IA
LS
)-->
IOUHUULIULI
(Middle of the air gap)
A
B f
(Stator surface)
Flux density and potential distributions for field cu;',,-sent of 1400A
Fig(7.20)
(Rotor surface).
( PO
TEN
T I A
LS
)-->
0.,
a72
A
co
z LU
rL CD
a_
CO
O
O ftr
UUUU (Stator surface)
( PO
TEN
T I
ALS
)-->
O co
A
O
(Middle of the air gap)
(Rotor surface) Fig (7.21) Flux density and potential distributions for
field current of 1600A
173 T
( PO
TENT I ALS ) --->
a_
T
U)
co uJ
a.
a_ > ----------
e r----. I o_.,______,---- 9 1
... (Stator surface)
O C
O CO
O
A
I B .
ti
(Middle of the air gap)
O
O OD
( PO
TEN
T I A
LS )
a_
(Rotor surface)
Fig (7.22) Flux density and potential distributions for field current of 1800A
174
Number of nodes 322, 37 of which are fixed nodes
Size of the H-matrix <5000 words
File required to load the program using FUN compiler =
34400 Oct. no.
File required to run the program using the same compiler =
41600 Oct. no.
Average time per iteration on CDC 6400 computer = 4.8 s
Number of iterations used for each value of the field current = 10
Maximum error in the solution at the nodes = 0.1x10-6A
Maximum change in potential at the nodes = 0.2x109 Web/m
The open circuit curve of the machine was calculated and compares
well with that measured (fig.7.23).
7.5 FLUX AND EDDY CURRENT DENSITY DISTRIBUTIONS DURING THE FIELD DECREMENT TEST
Switching the terminals of the field circuit on to a discharge
resistance induces eddy currents in the wedges and the rotor iron, and the
flux pattern alters progressively. Test and computed values for field
current are compared.
7.5.1 Choice of the grid
Numerical solutions of transient problems require a grid which
is similar, in principle, to that for the steady state, as explained in
sec. (7.4.1). However, the grid has to be modified in the field slots to
overcome the difficulties that arise with solid rotor machines. It is
important to have an accurate model for the slot leakage and to overcome
the discontinuity of the numerical solution at the initial time step.
Term
inal v
olta
ge
in K
V 28
26
24
22
20
28
16
14
12
10
8
6
4 -
2
0.0
0.0 1
-r-t i I I 1 I [ I
2 00 400 600 800 1000 1200 7400 1600 18 00 2 000 Field current in A --
Fig (7.23) Open circuit characteristic of a 333MVA turbo alternator
Experimental results X Computed values
176
Fig (7.24) Grid used to divide a half pole pitch of the turbo alternator for transient analysis (field decrement test) .
177
7.5.1.1 Initial value problem with solid iron and a field winding
At the start of a transient condition, say a field decrement
test, the external voltage supplied to the field winding is suddenly
removed. However, the field current and flux distribution do not change
abruptly and induced voltage is produced at t = o+ by an initial value of
rate of change of flux linkage. Thus at each node which draws current from
the cross-section of the field winding values of at
appear. Dt
It has already been stated that nodes in the field winding must
be separated from those in solid iron. If this distinction is not maintained,
at t = o+ values of at occur in the solid material and a spurious current
is produced. This gives rise to a discontinuity in the numerical solution
for not only do eddy currents suddenly appear but the field current suddenly
drops. When the nodes are separated the value of at
can change suddenly 3t
without producing currents in the iron, and at i — n the slot can have the
Dt
square-wave distribution of fig. (7.25) at t = o+. After a very short time
(St
the distribution spreads into the iron as shown.
7.5.1.2 Slot leakage flux and distribution of field nodes
The distribution of field nodes does not have a significant
effect on the steady state distribution of the flux pattern. however, in the
study of transient performance an improper choice of field nodes leads to
a distortion of thecuirents calculated. Therefore there is a need for a
model to present the characteristic of the field winding faithfully. During
the transient period, the induced field voltage is the sum of the induced
voltages in the field conductors. Throughout any slot, these induced vol-
tages differ from one point to another in both radial and tangential direc-
tions. They depend on the flux in the teeth. and the induced eddy currents.
The effect of the induced eddy currents in the solid iron or damper circuits
is to oppose any variation of the flux lines. On the other hand, the conti-
(d)
178
aA It
dA Fig,(7,25) — distribution in a rotor slot dt at t =0 &
kAt
slot i --aA at
(6)
Illustrative diagram of a cross section in the rotor slot.
cir• (7 • 25)• ci-A distribution in a rotor slot J
179
nuity of the field current is only maintained by the rate of change of the
flux linking the winding. These contradictory effects lead to a rise of the
slot leakage and a change in the flux distribution in the field slot. Nor-
mally maximum changes in the flux (and maximum induced e.m.f.) occur in the
middle of the winding with minimum changes at the edges. It is possible to
estimate the induced voltage curve in both radial and tangential directions
by comparison with the slot leakage flux of an induction motor. Here, in
the field decrement test, induced eddy currents completely surround the
field winding. The expected e.m.f. curve has the shape shown in fig. (7.26a).
These curves are not symmetrical about the slot centre lines and they vary
from one slot to another and from one instant to the next. Typical computed
values of the induced e.m.f. in the first slot (nearest to the pole axis),
after 0.005s is shown in fig. (7.26d and c) for a slot shown in fig. (7.26b)
Fig. (7.26c) shows the induced e.m.f. distribution in the circumferential
direction at a-a & b--b. The distributions of the e.m.f. at the edge of the
slot and at a radial section along the slot are shown in fig. (7.26d).
Thus accurate representation of the field winding needs sufficient
nodes so that these distributions may be represented. It is found that six
nodes, three on each side of the field winding as shown in fig. (7.26b) are
adequate. These nodes are positioned so that the integral of the induced
e.m.f. over the slot area occupied by the field winding is equal to the sum of
the induced voltages at the nodes evaluated from Simson's rule.
7.5.2 Conductivity matrix of the field decrement test
The conductivity matrix is occupied by two matrices. One is
related to the field nodes and the other for the rotor iron and damper nodes.
All other nodes have zero rows and columns in the conductivity matrix.
180
7.5.2.1 Field conductivity matrix
The induced e.m.f.s. at the nodes of the field winding vary
from one node_ to another as explained in the previous section.The field
current resulting from the induced voltages depends on the function dis-
cribing the distribution of induced voltages. In other words it depends on
the triangular grid considered. The turbo-alternator rotor has parallel
sided slots and assuming a linear distribution of the induced voltages
between the adjacent field nodes, Simpson's rule can be applied to evaluate
the average induced voltage in the slot. The currents supplied to the field
nodes depend on the position of the contour lines and the assumed current
density distribution in the triangle as explained in chapter 3.
As the current density is constant in the field winding and if
nodes are equally spaced in the field winding cross-section, the conductivity
matrix can be arranged to be symmetric. The weighting coefficients of the
induced voltages at the field nodes can be determined. The field conductivity
as a part of the whole conductivity matrix is
2 =
eq (7.5)
where aeq
depends on the characteristic of the field circuit resistance and
Ampereconductors as explained in chapter 6. All the matrices a11'
a12'
are identical coefficient matrices whose number is equal to the number of
the field slots in a half pole pitch squared; n2 . Each matrix a has its
elements for the 6 nodes in the field slot in the form:
181
Eal = i6
0.25 0.5 0.25 0.25 0.5 0.25
0.5 1 0,5 0.5 1 0.5
0.25 0.5 0.25 0.25 0.5 0.25
0.25 0.5 0.25 0.25 0.5 0.25
0.5 1 0.5 0.5 1 0.5
0.25 0.5 0.25 0.25 0.5 0.25
(7.6)
Alternatively, the matrix [a] can be expressed as a multiplication of two
vectors as
[a] (f3)T (f3)
(7.7)
where the vector is given by
0 1) = (0.5 1 0.5 0.5 1 0.5) (7.8)
,,, The column vector (0)
T and the vector 0) are respectively the ratios of
voltages in Simpson's rule and the ratios of the field areas around the
field nodes.
7.5.2.2 Iron conductivity matrix
The iron conductivity matrix is based on the assumption of
linear distribution of the current density in triangles. This gives a
conductivity matrix for each element dependent on the contour line of
the nodal method. however, for the sake of simplicity, the finite element
model was used, one-third of each triangle being allocated to each node.
7.5.3 Initial value problem and oscillation in the solution
The second order predictor corrector technique is difficult to
apply at the initial time interval. Although in small problems, such as the
micro-alternator, eddy currents have not a powerful effect on the behaviour
of the solution, it leads to severe oscillations in big problems such as
in a turbo-alternator. The problem arises from the variation of the second
182
derivative of the potential with time. In other words, the eddy current
denSity distribution starts the values from zero but with. nonzero initial
rates of change. When the initial rate of change is assumed zero this means
that the derivative at every other time interval takes approximately double
its value and approximately zero in between. The solution of this problem
could be obtained by starting with the Runge-Kutta method as explained in
chapter 5. However, the main disadvantages of applying the Runge-Kutta
method are the very large computational time, the core storage and the
switch. in programming required.
Oscillation has been avoided by using the first order approxi-
mation of the potential initially with a very small time step. At subsequent
instants uses the second-order approximation. Results are shown in sec.
(7.5.9).
7.5.4. Modelling of the switching operation
In the field decrement test measurements are available, the
field terminals were disconnected from the exciter and connected to a
discharge resistance. The test circuit is shown in fig. (7.27a). Records
were obtained for:
a) Stator induced voltages
b) Current and voltage across the discharge resistance
and c) The decay of the field current.
It is found that in the test circuit shown in fig. (7.27a)
Switches s1
and s2
did not operate at the same instant. Switch s2 was
closed first for a small time interval t1 before switch shwas opened. This
gives a value and a direction to a current i1 in the discharge resistance.
At the interruption of the circuit by the swi.tch.sl, current i3 in the
field circuit passes through the discharge resistance. Since the discharge
resistor is not a pure resistance, a time delay has to be taken so that
183
the current may change. This means that during a time interval t2
the current
i3 is not equal to i
2 The difference is supplied from the main circuit.
In other words there must be an arc through the switch s1 which carries a
current of -i.2+13 . This arc disappears at the end of the time interval t2
when the currents 2 and i3
are equal. The variation of the current in the
discharge resistance is shown in fig. (7.27b). Its initial characteristic
depends on the time constant of the discharge resistance. The current res-
ponse in the discharge resistance forms a voltage drop across it. This
voltage is nearly the same as that across the field terminals.
(a) (b) (c)
Fig. (7.27) Switching circuit of the field winding.
Modelling the characteristic of the switching circuit could be
done by knowing the characteristic of the arc. The discharge resistance
and the time response of the switches. Since the characteristic of the arc
is not available, the switching circuit is modelled by replacing the discharge
resistance by an equivalent resistance, and by assuming that the same current
has to pass through the field winding and the equivalent discharge resistance
without any current from the exciter circuit. This implies that the equiva-
lent resistance has a time response. The characteristic of the equivalent
resistance is similar to the response of the current in the discharge
resistance, as shown in fig. (7.27c). Since the time interval t1 is small,
the following approximation can be made:
R = -R1 + (R
d +R1 ) (1-)
(7.9)
where
(7.10)
184
and t1
is_the time difference in operation between switches s1 and s
2.
Since the time t1
is small it is assumed that R is constant and equal to R1
during the time interval tl. This model alters the computed values of the
field current and the results are explained in sec. (7.5.9).
7.5.5. Overhang leakage reactance
The overhang leakage reactance of the field winding could be
calculated either using 3—dimensional analysis of the potential as given
in ref. 50 or in an analytic way similar to that of ref. 51, but it, is
simplified and calculated according to Kilgords formula52. It is taken into
account by altering the initial period by a factor kt which depends on the
slot and overhang leakage in a similar way to that explained in chapter 6.
7.5.6. Flux distribution during the transient period
The flux lines are plotted during the rise of eddy currents
as shown from fig. (7.28) to fig. (7.37). This period is small and the flux
is maintained by both field current and the induced eddy currents. There is
a small reduction in the total flux which is indicated by the induced voltage
in the stator windings. This is shown in the test results in fig. (7.38).
In the stator region there is a small change in the flux lines.
It appears in the diagrams by the change in position of the flux line that
crosses the stator slot the nearest to the interpolar axis. There are also
changes on the other lines but they are not significant.
In the air gap region the flux lines have &small variation.
It may be seen by looking at the position of the flux lines reaching the
stator slots from the air gap between the diagram of fig. (7.78) and that
after 0.075 s. in fig. (7.37).
In the rotor, the only apparent variation in the flux lines is
in the region of the teeth and slots. In the rotor teeth, the flux lines
FIG(7.28) • FLUX DISTRIBUTION AT NO-LOAD
FOR 600 AMPS EXCITATION CURRENT
CF. A 4-POLE.333MVA TURBOALTERNATOR
FIG(7.29). FLUX DISTRIBUTION AFTER 0.005 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.30). FLUX DISTRIBUTION AFTER 0.01 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.3/). FLUX DISTRIBUTION AFTER 0.015 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED Co co
FIG(7.32). FLUX DISTRIBUTION AFTER 0.025 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.33). FLUX DISTRIBUTION AFTER 0.035 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.34). FLUX DISTRIBUTION AFTER 0.045 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.35). FLUX DISTRIBUTION AFTER 0.055 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.36). FLUX DISTRIBUTION AFTER 0.065 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
FIG(7.37). FLUX DISTRIBUTION AFTER 0.075 SEC.
FROM THE SWITCHING OPERATION IN THE FIELD CIRCUIT
WITH THE STATOR WINDINGS OPEN CIRCUITED
1 11 i I HI I illiiii
HI
1 i
Peak value of generator stator voltage 9480 v/cm
Generc t or rotor current 191.5 A/cm. •
Discharge resistance voltage 57.9 v/cm
- .7s,
, _ resistance current ›. Discharge • time
Fig,(7.38) Generator on open circuit at rated voltage and speed field circuit breaker tripped with discharge resistance
99'0 8020 0E20 U20 sI20 9020 zocd-
1 \ .
08'0 09'0 01,0 OZ'O oa.d' a-a-
41ISN30 xrru ONd (g1U1INI10d1 '•=1'A•w—e
OZ"I 00'1 08'0 09'0 Or') (2'0 00'0
<-111SN30 xnid ONd (S1H101310d) 'd'A'W —4
Fig.(7.39) No-load flux density and potential distributions along (a) Rotor surface, (b) Middle of the air gap, (c) Stator surface and (d) Middle of the stator slots
Q
196
OS '210 01r20 0E20 OZ 200I20
kilsNm Xflld ONU IS1el(iN310d1
oo-a
xn- A ONE ES-It:JUN310d1 •d•A'W —4
cf.) I
76'
oz.'t 00 21 00'0 0920 oz''o 00'0'
4--A1IgN30 Xflld GNU (S1811N310d1 •d'A•W —4
OE'0 02'0 01'0 00'0
—AtisN30 xrilA GNU (S1H11N31081 -E—A1ISN30 xcliJ GNU (S1U11N310J1
Fig.(7.40) Flux density and potential dist,^ibutions along
(a) Rotor surface, (b) Middle of the air gap,
(c) Stator surface and (d) Middle of the stator slots
0.75s. after the switching operation in the field winding.
LI)
00'0 ow0 or'o oz 2o owe oz.o-
AitsN30 xn1J GNU (S11:111N310d) •J'A'W —4
BE*0 OE •0 ZZ-0 Y1 20 90 20 201 e1- 1
9■'0
197
198
alter their position because of the slot leakage which plays an important
role during the. transient period. Its effect can be seen in the flux lines
in the rotor tooth. nearest to the interpolar axis and also in the flux lines
crossing the rotor slot nearest to the polar axis, where a magnetic wedge
is employed. Although the magnetic wedge is saturated, the effect of the
slot leakage flux is still clear. Flux lines in teeth become closer to the
slot walls than in the initial distribution and they cross the tooth approxi-
mately in the tangential direction. The changes in the flux lines become
clear when the eddy current diagrams are studied. Potential and flux density
distributions are plotted at four constant radii at the start and after .075s
from the switching as shown in fig. (7.39) and fig. (7.40).
7.5.7. Eddy current density distributions
The distribution of the induced eddy currents in the solid iron
and wedges at various instants in time are illustrated by contour lines.
Each contour line represents an equi-eddy current density. Eddy current
density distributions are plotted in two different groups. The first
group shows eddy currents in a fixed number of divisions between the maximum
and minimum induced eddy current densities at each instant (A). Thus the
contour lines represent eddy current densities of different values from
one instant to the next. Fig. (7.41) to fig, (7.49) show the distributions
of the induced eddy current densities at successive instants in time. The
second group of eddy current distributions (B) uses contour lines having
the same values in all the distributions at various instants in time. They
are shown in fig. (7.50) to fig. (7.58).
Values of induced eddy current densities vary with time and are
illustrated by plotting their values through radial and tangential sections
in the rotor iron at successive instants. Eddy current densities along
radial directions at angles indicated in fig. (7.59) are shown in fig. (7.60)
199
Fig (7.41) Eddy current density - distribution 0.0025 s after the transient operation in the field winding
(Contour lines type A)
200
N
N \
Fig (7.42) Eddy current density distribution 0.0075s after the transient operation in the field winding
(Contour lines type A)
201
Fig (7.43) Eddy current density distribution 0,0125s after the transient operation in the field winding
(Contour lines type A)
202
Fig (1 44) Eddy current density distribution 0.02 s cfter the transient operation in the field winding
(Contour' lines type A)
203
N
Fig (7,45) Eddy current density distribution 0.03s after the transient operation in the field winding
(Contour' lines type A)
N
204
Fig (7.46) Eddy current density distribution 0.04s \-\\ after the transient operation in the field winding
(Contour lines type A)
205
Fig (7.47) Eddy current density distribution 0.05s after the transient operation in the field winding
(Contour lines type Al
Fig (7.48) Eddy current density distribution 0.06 s after the transient operation in the field winding
206
(Contour lines type A)
Fig(7.49) Eddy current density distribution 0.07s after the transient operation in the field winding
207°
• (Contour lines type A)
208
Fig (7.50) Eddy currentdensity distribution 0.0025s after the transient operation in the field winding
(Contour lines type B)
Fig (7.51) Eddy current density distribution 0 0075 s after the transient operation in the field winding
209
(Contour lines type B)
Fig (7. 52) Eddy current density distribution 0.0125 s after the transient • operation in the field winding
210
(Contour lines type B)
Fig (7.53) Eddy current density distribution 0.02s after the transient operation in the field winding
211
(Contour lines type B)
Fig.(754) Eddy current density distribution 0.03s after the transient operation in the field winding
212
(Contour lines type B)
Fig.(7.55) Eddy current density distribution 0.04 s after the transient operation in the field winding
.213
(Contour lines type 8)
Fig (7.56) Eddy current density distribution 0.05s after the transient operation in the field winding
214
(Contour lines type B)
Fig (7.57) Eddy current density distribution 0.06 s after the transient operation in the field winding
215
(Contour lines type B)
Fig (7.58) Eddy current density distribution 0.07s after the transient operation in the field winding
216
(Contour lines type B)
217
<a
Fig(7.59) Illustrative diagram of the positions of radial and tangential lines along which eddy current densities are plotted at successive instants.
0. RADIAL DISTANCE -
I
Fig.(.7.60) Eddy current density distribution along the radial direction as in fig(7.59).
01
c!.
A 004s t
002s
006s
I
- RADIAL DI STANCE
Rg.(7.61) Eddy current density distribution along the radial direction bb in figi159).
t
4-- RADIRL- DiSrANcE
Fig.(7.62) Eddy current density distribution along the radial direction cc of fig.(7.59).
RADIAL DISTANCE
I
Fig,(7.63) Eddy. current density distribution along the I' a dial direction - dd in fig.(7.59).
RADIAL. DISTANCE
Fig(7.64) Eddy current density distribution along the radial direction ee in fig(7.59).
0.04s O
RADIAL DI Sr INCE
ViR
EATION OF INF EDDY
Fig.(7.65) Eddy current density distribution along the radial direction ff in fig.(159)
rotor surface
10-- TANGENT I At DISTANCE
Fig.(7.66) Eddy current density distributions along the tangential direction AA in fig,(7.59)
10--E
DD
Y C
URR
ENT
DENS
ITY
IN AM
PS
/MN .
1D'-
-►
N.) Ln Fig.(7.67) Eddy current density distributions along the tangential direction BB in fig.(7.59).
.- TANGENT I AL DISTANCE
*—EDDY CURRENT DENSITY IN AMPS/Iirt--). -0.04 -0.02 0.02 0.04 0.06 0.08 0.10 0-12 1' '06 0.14 0.16 0.11 0.2
Cb
l ue,iJ
n ,
cpp3
4-4444■scs606-,,--.-'rfr
suoi
,znq
s.ip
0
tQ
dire
ction DD
in fig,(7.5
9).
9zz
►
- E
DDY
CU
RREN
T D
EN
SIT
Y IN
IIM
PS/M
M--1
.
pole
Fig.(7.69) Eddy current density distributions along the tangential direction EE in fig.(7.59g
o-- TANGENTIAL DISTANCE
Fig.(7.70) Eddy current density distributions along the tangential direction FF in fig.(7.59).
229
to fig. (7.65). Eddy current densities are also plotted in the tangential
direction at radii shown. in fig. (7.59) are shown in fig. (7.66) to fig.
(7.70). Where wedges of different materials are used, the plotted values
of the eddy currents need to be modified in the wedge. The modification
depends on the ratios between the conductivities of the wedge and that of
iron. Fig. (7.67) shows the induced eddy currents at a radius equal to that
at the bottom of the wedges. The flat tops of approximately zero slope are
in the slots and do not represent eddy.currents.,They. are only plotted to
clarify the relation between the induced eddy currents at each instant and
the slot positions (following diagrams have the same feature).
7.5.8. Discussion of the eddy current plots
It is found that eddy currents have big values near to the
source of disturbance (field circuit). They have the same direction as the
field current. The results show induced eddy currents in the opposite di-
rection to the field current in the middle of the teeth. These negative
eddy currents might be thought to raise doubts as to the validity of the
results and consequently some justification is sought.
7.5.8.1. Negative eddy currents and the ideal analytical solution
The distribution of eddy currents can be explained by considering
a one dimensional linear problem by assuming that the hetht and length of the
tooth both tend to infinity. The currents in the two adjacent slots are
represented by currents at the boundaries. This problem can be expressed
by the partial differential equation
2A DA • = Pa at
Dx2
(7.11)
subject to the boundary conditions
CO
X P.2m = Paa
m=1 (7.14)
DA at
= _a eat X A cosf3 x m at 111 m=1
CO
(7.15)
230
1. J = Jo
att=o,x= a (7.12a)
2. J = Jo
at t = o, x= -a (7.12b)
3. A(x, ) = f (x) at t = o, -a < x< a (7.12c)
as illustrated in fig. (7.71a).
The solution of the problem is the A-function which satisfies
the partial differential equation and its boundary conditions. Assuming
that the solution has the form
co A = eat y A cosfx
m=1 m irt
(7.13)
substituting equation (7.13) into equation (7.11) gives
Equation (7.14) is valid because p, a and a are all positive quantities.
Since the value of is at
therefore, the current density J becomes
j = _a DA at (7116a)
co
= eaeat
A conmx (7.16b)
m=1 m The two boundary conditions at t=o, (7.12a) and (7.12b) satisfy the equation
CO
= aa X. A cos + p, a m=1 m
m
(7..17)
Since the source of the transient is on the edges therefore the maximum
value of J occurs also on the edges at x = a and x = -a. The value of (3m
in equation (7.17) becomes
= mir a
m = 1,3,... (7.18)
231
The constants (31-ri and ()tare determined from equations (7.14) and (7.18).
The only unknowns in equation (7.13) are the values of Am. They can be de-.
termined from the initial distribution of the potential A. Thus equation
(7.13) is the solution of the problem.
From equation (7.17) and (7.18): CO
Jo = -aa X A
m (7.19)
m=1 The effect of the sudden change which gives a current density Jo at the
boundaries induces current density J1 at t = o on the middle'of the tooth.
The current density J1 (t=o, x=o) can be determined from equation (7.16b) as
CO
J1 = aa : A (7.20)
m=1
substituting equation (7.19) into equation (7.20) yields
1 = (7.21)
This shows that the exdstance of the negative eddy currents at t = o.
The same problem can be regarded in a different way. Since
B = curl A and the current density J = -68t , then the partial differential
equation (7.11) can be written as
( a ) = 9x "-y/ -J (7.22)
The flow of two equal values of current in the same direction at the edges
of the iron produces a net flux in the iron equal to zero. This does not
mean that the flux density distribution in the iron in zero but its integral
over the cross sectional area vanishes. The flux density distribution through-
out the iron cross sectional area can be assumed at t = o as shown in fig.
-DA (7.71 b). The unit step functions to replace J by -0- implies the Dt
differentiation of with_respect to x as given by equation (7.22). The
differentiation is shown in fig. (7.71 c). This means that there are negative
values of the induced eddy currents. Thus according to the variation of the
x
(a) (b)
Fig.(7.71)
Fig.( 7.72)
232
2L
(c)
(a) 11
233
•
flux density in the iron, the induced eddy currents also vary.
A similar problem has been studied by Rudenberg53. He studied
the comutation of the flux in rotating machinery. In that problem the per-
formance with time of the total current of every armature conductor has a
curve shown in fig. (7.72c). The steady state current in the conductor
embedded in a slot produces a flux density distribution as shown in fig.
(7.72b). In his solution of the problem, it is assumed that the current
changes its value from 21 to 0 instead of from +I to -I. The analysis shows
that with an initial flux density distribution of triangular shape shown
in fig. (7.72d), the current density distribution becomes as shown in fig.
(7.72e). It also alters the flux density at the successive instants in
the sequence 1, 2, 3 ... etc. He wrote:
" It is interesting that for t = o the series solution does
not converge, yielding an infinite current density at the top edge of the
bar. However, this lasts only for an infinitely short time, after which the
current density rapidly decreases. Finally, the current decays in cosinu-
soidal distribution over the bar. The upper parts of the conductor carry
"negative" current which spreads into the depth, gradually extinguishing
the positive current in the lower part of the conductor. In both figures
(7.72d) and (7.72e) dashed contour lines are shown indicating the zero value
of induction and current when actually commutating from +I to -I rather than
+2 I to 0".
The slot leakage reactance could be represented by the model
of fig. (7.71a), by assuming that the response of the eddy current on the
boundaries is delayed from that in the field winding. Therefore the current
density on the iron boundaries of fig. (7.7a) may take the form
J = Jo (1-e t)
(7.23)
234
where Jo
is the maximum value, of the eddy current density to be reached
on the iron boundary and y expresses the.delay in the response.
7.5.8.2. Physical explanation of the negative eddy currents
The induced negative eddy currents depend on the leakage flux.
In any tooth there are radial and tangential components of the flux intensity.
The tangential component shows the effect of the leakage. Therefore, the
study of the leakage flux and its effect during the transient period can only
be clarified in regions where there is nearly no radial flux. For instance
a tooth in the q-axis does not contain flux lines in the radial directions
when the excitation of the field winding produces a flux in the d-axis. The
only flux lines are those due to leakage. These flux lines cross the inter-
polar axis normally at the top and the bottom of the tooth. This does not
prevent leakage flux flowing in opposite directions in the two sides of the
slot produced by the currents in the two adjacent slots as illustrated in
0 (a)
0 0 0 (b)
0
Fig. (7,73)
fig. (7.37a). When a sudden disturbance occurs in the source maintaining
the flux lines, they try to keep the same pattern - unaltered. This implies
that in the right hand side of the.tooth, fig. (7.73b), a current has to
be induced in the opposite. direction as shown in fig. (7.73b). The same
occurs in the left hand side as in fig. (7.73 a). Both_ have to be maintained
by a current in the middle of the tooth. as shown in fig. (7.73b). This may
also be seen in an alternative and more immediate way. In order to maintain
the flux lines in two opposite directions as shown in fig. (7.73b) a current
235
has to pass in the negative direction in the middle of the tooth.
Tha results can also be proved using Faraday's Law rather than
Ampere's Law. The reduction of the field current leads to a reduction in the
leakage flux. The reduction of these lines induces e.m.f. which is additive
in all directions around the flux. Therefore the e.m.f. has the direction
in the middle of the tooth which induces negative eddy currents.
7.5.8.3. Eddy currents in the core and pole
The eddy current modes explained in reference 54 appear in the
machine core and pole. The modes up to the third appear in the core which
depends on the grid division. The eddy current modes up to the sixth are
consequently in the pole. When the number of grid divisions in both radial
and tangential directions increases more eddy current modes are added. They
improve the shape of the eddy current curves. Nevertheless, similar to the
flux lines, the eddy current curves contain numerical errors but they are
close to those pre-estimated. The effect of saturation and the constructional
details of the machine on the induced eddy currents and their variation in
the core and pole is clear in the results obtained.
7.5.8.4. Eddy currents in the wedges
The induced eddy currents in the nonmagnetic wedges are plotted
separately as shown in the diagrams of the eddy current density distributions
(fig. (7.41 - 7.49)). The induced eddy currents in the magnetic wedge
are plotted in the same diagram as the rotor cross-section. From the diagrams
the induced eddy currents have maximum values in the parts of the wedges
nearest to the field winding. Their values reduce towards the outer surface
of the rotor. This confirms the view that the eddy current progresses from
the source of disturbance.
236
7.5.9. Typical results
The grid of fig. C7.24) is used to solve the transient problem.
Its data and the results of programming are:
Number of nodes 322 with_ 37 fixed
Size of the H—matrix < 8000 words
File required to load the program using FUN compiler = 37400 Oct.
File required to run the program using the same compiler =
54100 Oct.
Average time per iteration was not recorded
Number of iterations per time interval = 10
The maximum error in the gradient = 0.1x108
The maximum incremental change in potential at the last iteration
= 0.5 x 109 Weber/m
Total time to run the program from the start to finish on
CDC 6400 computer for .075 s of real time = 1137.262 s.
The results have been checked by calculating the field current
and comparing it with the experimental results as shown in fig. (7.74). Good
agreement between computed and test results is obtained.
7.5.10. Effect of the time step on the results
The program for calculating the field current was repeated six
times with different time intervals. The first time interval was kept con-
stant at 0.005s. The other time intervals were 0.005 s, 0.008 s, 0.01 s,
0.012 s, 0.015 s, and 0.025 s are observed. In these programs the time steps
are constant in each, while the number of iterations is variable but is
restricted to be less than 10 iterations,. The control on the number of
iterations at any time step is the maximum error which is specified at
0.00001A. It is found that in order to reach errors less than the above
specified value, the required number of iterations ranges between 2 and 5.
1 0.28 0 32 0.36 0.40 0.44 Time In s
0,0
0.0 0 04 0.08 0.12 0.15 0 20 0.24
Fiel
d c
urr
ent
Nc 300
200
100
A experimental curve
B computed curve as if the machine is laminated
C&C' starting computed curves of solid rotor iron machine with two different time steps
d as C and C' but oscillations are removed
e as d but with the effect of changing the grid dimensions in the field slots
f. as e but with switching operation included
g as f but with overhang leakage reactance taken into account
500
400-
Fig (7.74) Field decrement test of a 333 MVA turbo alternator
c
700
600
0.08 0,12 0.0 0.04 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.44
Experimental curve
Computed values at different time steps ,
(a)
Time in s
X A X'-X 4( -
Fig.(7.75) Effect of time intervals on : (a) computed values of field current ( b) diviation from experimental results
0.02 Time step 0,01 Time step 0,005 Time step
(b)
Time in s 1
0,12 0.08 0,04 0.16 0.2 0.24 0.28 032 0.36 0,4 0.44 00
609
500 L L
0.) U-
300
250
• 20.—
1 0— •
L L Li 0.0
239
As the time interval increases the number of iterations for the first period
becomes larger. This number reduces as the time increases. In the small
time interval program of 0.005 s it is found that the number of iterations
per time step is fixed at 3 over the entire period. Errors from the experi-
mental valuasinthe results obtained are shown in fig. (7.75b) when diffe-
rent time step programs are used. The best results are obtained with small
time steps but the errors with the biggest time step of 0.025 are not
acceptable. The deviations of the computed values from the experimental
curve are plotted to a large scale for three programs as shown in fig.
(7.75b). It is also interesting to note the variation of the error from one
iteration to the next in the Newton-Raphson procedure and predictor correc-
tor method. In general, for the time intervals shown, the error starts with
a value of the order 101. In the next iteration the error goes down to the
order of 104 to 106. In the third iteration it reduces to 10
5 to 107. This
shows that the number of iterations can be automatically limited by the
maximum number of iterations. The error reduces more rapidly in the first
few iterations than in others. For 15 time steps in the six programs, the
total time needed to run these programs on a CDC 6400 computer ranged
between 482 s and 528 s. This shows a very big reduction in the computational
time (approximately half) when computed with the results in sec. (7.5.9)
where the number of iterations is fixed for each time step.
240
CHAPTER 8
STEADY STATE AND TRANSIENTS OF A LOADED TURBO ALTERNATOR
8.1 INTRODUCTION
The magnetic field problems of machines and their numerical
solution are explained in part I. They are used, as shown in the previous
two chapters to evaluate the steady and transient performances of micro-
and turbo-alternators. Good agreement was obtained with test results.
However, the applications were limited to conditions in which the flux
distribution was symmetrical about the pole axis. When the machine is
loaded and armature currents flow this symmetry is lost in general and
a full pole-pitch analysis is required.
This chapter describes'the application of full pole-pitch
analysis to solve steady state and transient problems of a turbo-alternator
when the armature currents and motion are included. The method of analysis
is essentially the same in symmetric and unsymmetric conditions. When the
method is applied to a full pole-pitch of a turbo-alternator, in the presence
of armature currents, the number of entries in the conductivity matrices
shown in chapter 4 increases greatly, stator nodes fed from armature currents
being added. The periodicity condition of the flux lines one pole pitch
apart has to be used instead of Dirichlet's and Neumann's conditions at
some nodes and there are therefore minor changes in the program. However,
although test results are not available for comparison with the computed
values under transient conditions it appears likely that the results are
correct.
The full pole-pitch analysis is applied first to an unloaded
turbo-alternator to check the results with those obtained from half a pole- .
pitch. The flux plots are also obtained when the machine is delivering full
load at unity power factor and connected to an infinite bus bar. The flux
241
plots obtained are not based on a similar model of the stator currents
given by Erdelyi et ell. In their model, they located currents at stator
nodes from phase values of the currents assuming distributed three phase
windings. As Alger55
points out, the method presented by Erdelyi and Fuchs
for the armature currents gives an instantaneous flux distribution and not
an average over the power cycle. Their method is thus only suitable for
steady state analysis. The method used here is suitable for both steady
state and transient operation.
Sudden short circuiting of the field winding of a loaded solid
rotor turbo-alternator connected to an infinite bus bar is studied. Con-
sequently transient effects appear in a number of ways. Field current decays,
eddy currents are induced in the solid iron and the wedges, transients
appear in the armature currents and the machine starts to slip. These re-
sults, during the transient period are explained in this chapter.
8.2 STEADY STATE CHARACTERISTICS
Steady state full pole-pitch analysis of turbo-alternators is
obtained here using a current sheet in the stator slots. The flux distri-
bution is different from that which would be obtained in the method given
by Erdelyi and Fuchs. It allows for the use of the two-axis theory which
simplifies the computations. Moreover, it reduces the initial value prob-
lem as two axis currents are required under balanced loading conditions
instead of three. There is no problem of overlap between the conductivity
matrices which occur when three windings are used. Thus the two axis theory
has advantages which give a considerable reduction in the computational.
effort.
The use of the current sheet does not reduce the computtions
required for a steady condition: its advantages only appear in the transient
state.
242
8.3 CHOICE. OF THE GRID
The grid of triangles used for full pole pitch of the turbo-alter-
nator is shown in fig. C8.1). One row of extra triangles beyond those
covering the full pole-pitch is located at one radial boundary. They are
exactly similar to those triangles neighbouring the other pole axis and
are used to inforce the periodicity condition. The grid of triangles
dividing the stator and the air gap is otherwise exactly twice that over a
half pole pitch as in fig. (7.2).
The grid that divides the rotor into triangles is different in
some regions from that of a half pole pitch. The grid in the iron is shown
to give a model which is accurate enough and suitable for eddy current
calculations. During the transient period, eddy currents are induced from
the field current reduction and the slip of the rotor. A grid which allows
for the accurate modelling of the field current reduction is chosen in a
similar manner to that for a half pole-pitch. The grid allows for skin
effect and induced eddy currents from the slip frequency by using small
triangles near to the surface. It has been shown by Carpenter16
that the grid
must have at least one set of triangles covering the depth of penetration,
otherwise there would be a big error. It can also be seen from the results
• obtained by Chari56 that the choice of a grid over a depth of penetration
could be sufficient to allow for induced eddy currents at the relevant
frequency. Below the depth of penetration it seems also from the same re-
sults of ref. (56) that it is unnecessary to have a complete grid in which
every triangle is equal to or less than the depth of penetration. The grid
of triangles which takes into consideration these requirements is shown in
fig. C8.1). The field winding is represented in the grid in a way similar
to that of the half pole-pitch analysis.
. 243. :
Fig.(8.1) Grid of triangles dividing a full pole pitch
of a turbo-alternator
244
8.4 PERIODICITY CONDITION IN FULL POLE PITCH CALCULATIONS
Loading of a turbo-alternator destroys the'symmetry of the
flux'lines around the pole axis. A firmly established boundary condition
over a half pole-pitch cannot be given4 . Nevertheless, it is possible to
state that since the flux wave form repeats itself with negative sign
every full pole pitch, then the potentials of nodes one pole-pitch apart
have opposite signs. During the transient period the potential derivatives
with respect to time, one pole-pitch apart must be equal but with opposite
signs. In the evaluation of the error at nodes on the boundaries it is
simple to multiply the elements of the S-matrix by the negative potentials
of nodes one pole-pitch apart. The negative sign may be associated with the
elements of the S-matrix rather than values of A. Convergence is only possible
when A values are defined positive and therefore the negative signs are
required to appear in the elements of the S-matrix. Consequently, it also
has some negative entries in the corresponding elements in the Hessian
matrix.
8.5 NO-LOAD FLUX DISTRIBUTION
The periodicity condition is checked by open circuiting the
armature windings and obtaining the flux distribution for a given field
excitation. The no-load flux distribution for a 600A field current in a
full pole-pitch is shown in fig. (8.2). It may be seen that symmetry is
regained. The following results are obtained:
Number of nodes = 740 with 666 free nodes and 10 nodes of
alternative potentials.
Size of the H-matrix is less than 17.K words
File required to load the program using FUN compiler = 37000 Oct.
File required to run the program using the same compiler =
110100 Oct.
245
Fig.(8.2) No-load flux distribution in a full pole pitch
of a turbo alternator for field current of 500A
246
Average time per iteration on CDC 6400 computer = 81 s.
Number of iterations used = 8.
Maximum error in the gradient = 0.5x108 A.
Maximum incremental change of the potential at the last
' - iteration = 0.1 x 10
10 Web/m.
It is found that the flux lines are slightly affected by the
grid specially in regions around the slot nearest to the pole axis. The
reason is that the number of triangles used near to the magnetic wedge is
less than when half a pole pitch was considered. This does not prevent the
flux distribution in the rotor teeth region from being approximately the
same as that obtained before, but it does show that there are some numerical
errors resulting from altering the continuous problem to a discrete one.
Errors could be reduced by a suitable prediction of the variation of the
reductivities in the various domains of the rotor iron. In the distribution
obtained the errors are small and they do not affect the transient solution.
8.6 CURRENT SHEET FOR A LOADED MACHINE
A current sheet is used to replace the stator effective
magneto motive force. It is known that the m.m.f. distribution of a 3-phase
balanced load is variable from one instant to the next over 1 th
of a cycle.
Despite the different wave shapes of the m.m.f. and current distributions,
a Fourier analysis of each successive shape yields precisely the same mag-
nitudes of the various harmonics55 . The fundamental component of the m.m.f..
gives mainly the fundamental component of the flux wave form and consequently
the voltage wave form.
In the current sheet model, the number of turns are assumed to
have a sinusoidal distribution. The maximum number of turns in both the
d- and q-axes gives 1.35 times the effective Ampereturns per phase. The
distributed equivalent conductors are located at the middle of the stator
slots. It is assumed that at the middle of the stator slot, the slot area
247
is equal to the tooth. area. This assumption associates equal tangential
intervals to the nodes and avoids problems when integration is required.
It is found that the use of the current sheet has double advantages. It
can be treated as a continuous function or discrete whenever, respectively,
the analysis or computations are required. From the analytical point of
view the sinusoidal distribution has its own advantages. From the computa-
- tional point of view, the current sheet is located at the stator slots so
that there is a separation between copper and iron regions. Slot leakage
and the nonlinearity of the iron material are automatically taken into
account.
8.7 FLUX DISTRIBUTION OF A LOADED TURBO-ALTERNATOR
Values of direct and quadrature axis currents necessary to give
a specified loading condition can be approximately estimated using a phasor
diagram57 using machine reactances. Thus the initial two axis currents and
the field current are known. Subsequently, currents fed to each node are
evaluated from:
Field nodes: a weighted distribution of the total Ampere-
conductors in the field slot to its nodes.
Armature nodes: knowing the electrical angle of the node with
respect to the pole axis, the current at a stator
node becomes the average effective equivalent
Ampereconductors over the incremental angle enclo-
sing the node.
The distribution of the currents at the grid nodes gives a
complete formulation of the current vector I. The problem becomes then the
solution of a set of nonlinear algebraic equations. It was found that the
use of the Newton-Raphson technique to find the field distribution of a full-
loaded turbo-alternator with zero magnetic field initially does not give a
248
convergent solution. A solution was obtainedby initially calculating the
• 1 field of
3 of the. currents and then using these flux values for the start
2 of a calculation for Tof the current, etc. Small corrections to give any
precise operating condition may be calculated subsequently.
The flux distribution of a fully loaded turbo-alternator at
5° phase lead is shown in fig. (8.3). The flux distribution is also plotted
for a fully loaded turbo-alternator at unity power factor as shown in fig.
(8.4). Loading of the machine transfers the radial line of zero potential
to an intermediate region in the stator and the rotor. It is found in the
flux distribution obtained, fig. (8.3) and (8.4), that there are two dis-
tinct characteristics of the flux lines in the region of the stator slots
and teeth. They are separated by the stator current sheet. Flux lines near
to the air gap are affected by the stator slot leakage while near to the
stator yoke slot leakage is very small because the current has been located
at the centre of the slots. In the air gap region there is clearly a tan-
gential element in the flux lines and it indicates electric torque.
In the rotor, the flux lines have a distribution which is
completely different from that in unloaded conditions. Stator currents tend
to prevent flux lines from the rotor from linking the stator windings. This
is reflected in the distribution of the flux lines in the region near to
the rotor surface. The number of the flux lines crossing the rotor slots
is very large corresponding with the high field current. Magnetic wedges
become saturated and there is approximately no difference between the
characteristic of the flux lines approaching them and non-magnetic wedges.
The results also show a considerable reduction in the number of the flux
lines crossing the rotor slots adjacent to the. core.
249
Fig.(8.3) Flux distribution of a loaded turbo alternator
at 5° phase lead.
250
Fig.(8.4) Flux distribution of a loaded turbo alternator
at unity power factor.
251
8.8 TYPICAL RESULTS FOR A LOADED TURBO ALTERNATOR
The grid of triangles shown in fig. (8.1), which is used
for the no—load flux distribution, is also used for the loaded machine.
The two flux plots of fig. (8.3) and (8.4) have the following data:
. Size of the H—matrix is less than 20 K words
File required to load the program using MNF compiler = 40500 Oct.
File required to run the program using the same compiler =
110200 Oct.
Average time per iteration on CDC 6400 computer = 92 s.
The field distribution has been obtained in three stages (see
sec. 8.7).
Number of iterations used in the last stage = 10.
Maximum error in the solution = 0.1 x 10-7 A.
Maximum incremental change on values of A = 0.1x1010 Web/m.
8.9 FIELD TRANSIENTS OF A LOADED MACHINE CONNECTED TO AN INFINITE BUSBAR
Regrettably it has not been possible to produce results for a
transient condition on a loaded machine because sufficient computing time
was not available. However, the condition following loss of excitation of
a loaded machine has been modelled and a numerical solution obtained for
two time steps. This demonstrates that there is no reason (other than that
of computational cost) why general transient performance cannot be obtained.
In the condition studied, some simplification was possible but for armature
transient conditions a fuller consideration is necessary. It is however
still possible.
Two—axis theory is used to represent the m.m.f. of the windings
and voltage generation effects. Loss of excitation does not produce large
58 0 (or transformer) voltages but these have been written into the
252
formulation. They are. neglected in the numerical computation.
8.10 NUMERICAL SOLUTION USING THE TWO AXIS THEORY
Transient problems can be handled in the discrete form when
the elements of the conductivity matrix [AaJ shown in chapter 4 are fully
determined. This matrix contains three matrices a, a2 and a3. The first
two matrices correspond respectively to nodes in solid iron or in a damper,
and in the field winding. The elements are determined in the same way as
those derived in the previous two chapters. The third matrix, a3 has its
entries, as previously mentioned, dependent on the method of modelling the
stator magneto motive force. Here the model used for the stator windings is
a sinusoidally distributed current sheet in the middle of the stator slots.
The induced e.m.f.s. in the direct and quadrature axis coils depend on the
total flux linking the windings in these axes. Therefore, direct and quadra-
ture axis currents depend on the potential distribution at the nodes of the
stator current sheet. The matrix a3
becomes a completely filled matrix
when the p terms are included. Its number of elements is equal to the
number of nodes on the current sheet squared.
8.10.1. Stator numerical formulation including plp terms
The conductors in the current sheet have the distributions from
equation (2.16 a and b) as
Zd
Zm
sin 0 (8.1a)
Zm
cos 0 (8.1b)
Assuming that id and i have the same positive direction as the field, the
flux linking the d— and q— axis coils are
,1112
= 2.2th j A sine de 4. ko
id
(8.2a)
-7112
aA at s1
aA s2 at
+ w • •
aA at n
A 1
As 2
(8.5) e q
A
253
k,Zin A cos 0 d0 +o i
o
but from the two axis theory
ed = - rid - 11)00
.rici + dw
therefore cff/2 3A
ed = 2'Zm j sin() dp + 2,o
aid -1T/2 at at - r i
d
(8.2h)
(8.3a)
(8.3b)
kZm 0
A cos() d0 -o
wiq (8.4a)
and 71-
f aA i e = kZ co s0 dO + k =--g- -"
' + o
q m at o at q
rTr/2 kZ w A sin° d0 + k wi
-Tr/2 o d (8.4h)
Equations (8.4a) and (8.4b) have the matrix form
aid at
Diq
at
0
id
Q 0
= 60kZm
sin() s sin®
s2 sin®s
1 n
-2, w 0
q cosi° cos()
s2 s cos()
Si
... -cos®sn
-cosy -cos()() -coss 2
sings sinssin()s2
sin()sn
where the suffixes sl' s2'
srl refer to nodes 1 to n on the stator
current sheet and 60 is the angle between two successive nodes and is
254
given by
Equation
[Z]. {1}
where
60 = n
(8.5) can be rewritten in the compact form
a{i} afAsl - SOU ([01 4-6) CO ] {A}) - iel
(8.6.)
(8.7)
(8.8a)
(8.8b)
(8.8c)
ko at
} =
[0]
[021
d
q
sin()
cost)
-cos()
sine
s m at 2
sine sin() ssn
1 s2
cos() cos()
sl s2 sn'
sl -cos() . . .
s2 -cos()
s
sinesin() () s
sine 2
On the other hand the magnetic potentials at the stator nodes are related
to the magnetic field by the partial differential equation
if ( aA ax
u_ ax 4.
ay u ay ) dxdy = -I (8.9)
It has the discrete form for the set of nodes on the stator current sheet
as
= ES41 {A} k
1(60) ZIT
sin® cos° s1
s 1
sin() () s
cosO 2
d
(8.10)
i. q
sin®s
cos()sn.
where {Gs} is the part of the vector {G} corresponds to the nodes on the
stator current sheet and the factor k1 is simply obtained from equating
the m.m.f. of the 3-phase with that of the current sheet. Equation (8.10)
can be rewritten in the compact form with the aid of equations (8.8a and b) as
255
10 s 1 = CS4 {A} + k1
SO Zm 001I {i} (8.11)
Beside equations (8.5) and (8.11) the torque equation of the machine which
is given by equation (7.18) can be rewritten using equations (8.2a) and
(8.2b) as 7 7/2
w-i Tmm
k f A sin0d0 +d kZm kl f A cos0d0= M
J 9T- (8.12)
q-712
This equation has the numerical form
Tm
-(SOZZm k1
Cidq
] -cos() .... -cos®
s 21 .n
sin() .... sin® sl
sn
(J3 (8.13) = —
J at
A Si
A s2
A
or in a matrix form
Tm -S02Zmkl (i) CO2] {A
s} (8.14)
The set of equations (8.7), (8.11) and (8.14) gives an expression of the
conductivity matrix 03 as well as other terms are included.
The above method is applicable when the piP terms and the
armature resistance are taken into account.
8.10.2 Stator numerical analysis ignoring armature resistance and the plp terms
Simplifications in the formulation and the method of solution
can be achieved when the 0 terms and the armature resistance are ignored.
The method and the simplifications are applicable to the loss of excitation
of a turbo-alternator connected to an infinite bus bar. The voltages e
and e of equations (8.4a) and (8.4b) are then
7 ed = m. f A cos 0 d 0 - k
o iq w
• o
7/2
= wk Zm . f A sin 0 d 0 + ko id w -7/2
(8.15a)
(8.15b)
e q
-ed - k — Z (60) 0 m
q
sine sin()s2
▪
. . sine s1
sn
cos() cos0s
. . . cosO 1 2
s s
or
.{i} = 60 k Zm [0] {Ad. +
o 0
where
Asl 1 As
•
wk
2 0
• As n
(8.17)
(8.18)
256
The currents id and i are
.T/2. i1 (e wAZ
m f A sin 0 d. 0 )
d = a q -1T/2
i w22. f A cos 0 d 0 ) q. wk 0
d m
(8.16a)
(8.16b)
For the numerical solution, the above two equations can be written in a
matrix form over the set of the stator nodes to give,
{6} = q -ed
substituting equation (8.18) into equation (8.11) gives
. {Gs} = CS I{A s } + k1
k (60) Z
mCOiT ((L52Fe) kZ CO] IA 1 + a))
4 o m
s 1 wk o
Here the value of in the stator voltages has been ignored so a3 may be
considered a null matrix. However, stator voltages are still dependent on
i A values and a matrix a3 is used to allow for them,
2, al3 = kZ
2 (60)2'
R ' EMT CO]
0
(8.20)
The convergence of the iterative procedure is obtained by the Newton-
Raphson method. The part of the matrix H in equation (5.57) corresponding
to the stator nodes becomes
(8.19)
b{GAs} 2 2 k
T
-C - EH114- k1
Zm
(60)
O
[01 [0]. at
(8.21)
257
In the. above equations 013 is a symmetric matrix with elements given by
2, 0 = k1 Z2
2 t (SO) --- cos ((j-i) 60) m
(8.22)
Thus the matrix to be inverted 11, is symmetric.
The calculation is simpli-fied if w can be considered to have
the value at the previous time step and the loss of accuracy is small if
the time step is small. At the end of the calculations at a given time
at step value of -5-i- can be determined from
aw edid e
qiq = ( T ) w I2H at 3 m phase rated VA t j (8.23)
where H. is the inertia constant10. The angular speed at the end of the
time step becomes
dw wt+6t -wt St
The change in speed alters the power angle y so that
lt+St = Yt (wt+St -wt) (St
As a result of which values of vd and v are altered• to q
vd = V sin y
t+6t
q t = V cos y
t+St
when Vt is the terminal voltage with y
o is determined from the vector
diagram.
(8.24)
(8.25)
8.10.3 Vector diagram and numerical solution
The above analysis can be explained with the aid of the vector
diagram. A diagramatic sketch of the stator and rotor is shown in fig.
(8.5a). The distribution of the conductors in the stator and the rotor in
the model is shown in fig. (8.5b). Currents in the field and the armature
are of opposite directions as shown in fig. (8.5c), and produce the m.m.f.
1
qv
E du
V, •
258 stator
rotor
Fig.(8.5) Illustrative diagram of the distribution of • the various components along a full pole pitch-
Mf
Fig.(8.6) Vector diagram of the turbo alternator,
259
distributions shown in fig. C8.5d), which gives rise to the fluxwave
form. The fundamental component of the flux can be analysed in d- and q-axes.
The values of (pc, and (1) depend on fA sin 0 d 0 and. fA cos 0 d.O.
These components produce voltages -Ed and E respectively as
q
shown in the vector diagram of fig. (8.6). The components of the terminal
voltage in the d- and q- axis, Vtd
and Vtq
are
Vtd
= Ed w (8.27a)
Vtq
= Eq -id k
o w
(8.27b)
Since the machine is working as a generator, field and armature currents
have opposite directions. Therefore there is no difference between equations
(8.17) and (8.27a and b) where as stated previously in sec. (8.10.1)
posivite currents in the armature were assumed to have the same positive
direction as the field.
8.11 FLUX PATTERNS DURING TRANSIENT CONDITIONS
At the start of the transient, in the field circuit, the flux
distribution is shown in fig. (8.7). After times of 0.005 s and 0.015 s
from the start of the transient, the flux distributions are shown in fig.
(8.8) and.fig. (8.9) respectively.
The difference between the flux distribution after the first
time interval from that at the start of the transient appears in the flux
distribution in the sixth and eighth stator slots. There is also a slight
variation in the flux line adjacent to the top of the first rotor tooth
adjacent to the pole on the right hand side,
The change in the flux lines from 0.005 s to 0.015 s appears
in the second stator slot adjacent to the pole on the right hand side. This
change occurs around the stator current sheet which is at the middle of
260
Fig.(8.7) Flux distribution of a loaded turbo-alternator at the instant of switching the field circuit.
261
Fig.(8.8) Flux distribution of a loaded turbo alternator 0.005s after the switching operation in the
field circuit.
262
Fig.( 8.9) Flux distribution of a loaded turbo-alternator 0.015s after the switching operation in the
ld circuit.
263
the slots. There isalso a change in the flux lines near to the top,of
the rotor tooth adjacent to the pole on the. right hand.side.
8.12 EDDY CURRENT DENSITY DISTRIBUTION
Eddy current density distributions at the middle of the time
intervals are shown in figures (8.10) and (8.11). Eddy current densities
are plotted as contour lines. Each contour line represents equal value of
eddy current density and the difference between two successive contour
lines is kept constant. In these diagrams the skin effect is very clear
and it is indicated by the heavy black lines. These lines indicate a
sharp gradient in the induced eddy current density. The effect of the
wedges on the eddy current density distribution in the teeth may be seen
in the v—shaped contours. The magnetic wedges have special eddy current
density distributions and they also affect the distribution of the induced
eddy currents in the adjacent tooth and pole. As a result of loading the
machine, the maximum changes in potential occur in the second tooth from
the pole on the right hand side. Away from these teeth the values of the
induced eddy currents are lower.
From the two eddy current diagrams shown in fig. (8.10) and
(8.11), it is clear that the rotor teeth carry most of the induced eddy
current and there is little in the rotor body during this initial period.
8.13 TYPICAL RESULTS OF A LOADED MACHINE ON TRANSIENT
The grid of triangles shown in fig. (8.1), which is used for
unloaded and loaded turbo—alternators is also used when the turbo—alternator
is loaded and subject to afield transient. The following results are ob-
tained:
266
Size of the 11 matrix = 42441 words
File required to load the program using MNF compiler =
40020 Oct.
File required to run the program using the same compiler =
165107 Oct. (slightly less than 60 K words)
• The number of iterations varies between 1 and 3. One iteration
to improve the steady state and 3 in each time step.
Time per iteration for steady state on CDC 6600 computer =
92.8 s
Average time per iteration for the transient = 131 s.
Maximum error in the solution of the equations at the nodes =
0.4 x 102 A
Maximum change in values of A at the last iteration = —5
0.1 x 10 Web/m.
267
CHAPTER 9
CONCLUSION
9.1 METHODS
The use of Ampere's Law to form discrete equations for an
approximation to the magnetic field of a machine is a very desirable develop-
ment. It removes the need for the previous methods which were highly mathe-
matical and not easy to justify. It gives a better current distribution than
the finite element method and gives some guidance to the shapes of triangles
that give the most accurate representation.
The use of conductivity matrices to represent closed circuits
in a field is vital to this analysis. These enable the action of a winding
in that it has voltage induced, which with other effects gives rise to
current to be matched with the magnetic field. Maxwell's equations relate
together the values of the electro-magnetic quantities in a medium and this
work may be considered to supplement them, in that the matrices represent
the behaviour of circuits embedded in the region considered.
The method of forming distinct conductivity matrices for each
conducting region by ensuring that no node receives current from more than
one region or winding aids the solution and prevents the generation of false
currents and numerical errors.
The Newton-Raphson method together with the sparsity techniques
of Jennings gives convergence in few iterations with a minimum of computation.
Its combination with the predictor-corrector method of moving forward in
time greatly reduces the computation time, the iteration of reluctivity
and the step in time being made concurrently. The avoidance of oscillation
in the results by the initial use of the first order (predictor) method
avoids the oscillations found by other workers.
268
These methods combine a simplicity of approach in using
Ampere's Law with a full consideration of magnetic conditions in a machine
with its awkward geometry, varying reluctivity and distributed circuits.
The large number of sparse equations resulting are then solved with maximum
economy. Both the basic considerations and the economy of solution are
essential.
9.2 APPLICATIONS
The solution of steady-state field distributions has been ob-
tained with greater economy than previous methods by the elimination of the
Dirichlet boundaries from the problem. Steady-state behaviour is however
not really of very great importance here, because the traditional methods
of calculation supplemented by relaxation and other techniques have already
provided solutions.
It is the solution of transient problems in such detail that is
thought to be particularly significant. It has been shown that it is now
possible to solve any machine transient problem where hysteresis is not
involved. Although for the most part results have only been obtained for the
simplest of transients, the field decrement test, it has been shown how a
general loaded machine may be tackled.
However, the methods devised, although practicable, are very
expensive in computing time. Before their use was attempted in any
commercial situation a very careful appraisal of the degree of detail
necessary, and thus the number of nodes should be made. This would then
enable the cost of calculation to be estimated.
The flux and eddy current plots of the turbo alternator have
revealed two effects which do not appear to have received attention previously.
At low levels of excitation the slot with a magnetic wedge has very high
leakage flux causing a reversal in the flux of the first tooth.
269
Secondly during the field decrement test negative eddy currents
are indicated in the centres of the teeth and in the pole body. Although
it has been shown that similar results have been expected in other circum-
stances, the results do conflict with the normal picture of eddy currents
moving into a solid body from the surface.
9.3 FUTURE WORK
Ampere's Law could probably be used to formulate equations for
higher-order elements which could be used to represent regions in which
reluctivity could be allowed to vary. The use of such elements may lead
to economies in computation, which are very desirable, but it is by no
means certain that this is so. It might be that the extra complication of
these elements would give a nett increase in computation although the number
of nodes might be less.
Hysteresis effects at present ignored might be included with an
appropriate model.
Until computers become even larger, faster and cheaper it is
very likely that this work will only be used to determine machine charac-
teristics which will then be represented in other ways. It is as in investi-
gation and design that its application lies. In power system analysis more
conventional models will continue to be used. However, a worthy, long-term
aim might be that full representation of the power system and the machine
in it under transient conditions.
270
REFERENCES
1. Fuchs, E.F. and Erdelyi, E.A., "Determination of water wheel alter-nator steady state. reactances", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-91, 1972, pp. 2510-27.
2. Erdelyi, E.A., Ahmed, S.V. and Hopkins, R.E., "Nonlinear theory of synchronous machines", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-85, 1966, pp. 792-801.
3. Erdelyi, E.A., Sarma, M.S. and Coleman, S.S., "Magnetic fields in nonlinear salient pole alternators", I.E.E.E. Trans. on Power App. and Syst., PAS-87, 1968. pp. 1848-56.
4. Silvester, P., Cabayan, H.S. and Browne, B.T., "Efficient technique for finite element analysis of electrical machines", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-92, 1973, pp.1274-81.
5. Silvester, P., and Chari, M.V.K., "Finite element solution of saturable magnetic field problems, I.E.E.E., Trans. on Power App. and Syst., Vol. PAS-89, 1970, pp. 1642-51.
6. Chari, M.V.K. and Silvester, P., "Analysis of turbo-alternator magnetic field by finite elements", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-90, 1971, pp. 454-64.
7. Fuchs, E.F. and Erdelyi, E.A., "Determination of water wheel alternator transient reactances from flux plots", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-91, 1972, pp.1795-1802.
8. Fuchs, E.F. and Erdelyi, E.A., "Nonlinear salient pole alternator subtransient reactances and damper winding currents", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-93, 1974, pp.1871-92.
9. Hannalla, A.Y., and Macdonald, D.C., "A nodal method for the numerical solution of transient field problems in electrical machines", Inter-mag Conference, April 1975, paper 17-7. Paper to be published in I.E.E.E. Trans. on magnetics in Sept. 1975.
10. Adkins, B., "The general theory of electrical machines", Book Chapman & Hall Ltd., London 1964.
11. Forsythe, G.E. and Wasow, W.R., "Finite difference methods for partial differential equations". Book pub. Wiley, T., New York, 1964.
12. Ortega, J.M. and Rheinboldt, W.C., "Iterative solution of nonlinear equations in several variables", Book, pub. Academic press, New York, 1970.
13. Norrie, D.H.. and De Vries, G., "The finite element method fundamentals and applications, Book Academic Press, New York, 1973.
14. Collatz, L. "Functional analysis and numerical mathematics", Book, Academic Press, New York, 1966.
271
15. Windslow, A.M., "Numerical solution of the quasi linear Poisson's equation in non-uniform triangular mesh", Journ. Computational physics 1 pp. 149-172, 1967.
16. Carpenter, C.J. "Finite element network models and their application to eddy current problems", Proc. I.E..E., Vol. 122, No. 4, April 1975, pp. 455-462.
17. Stoll, R.L., "The analysis of eddy currents", Book, Clarendon Press, Oxford, 1974.
18. Flatab cg,N., "Transient heat conduction problems in power cables solved by the finite element method", I.E.E.E. Trans. on Power App. and Syst., Paper T 72-508-0, PAS summer meeting July 1972.
19. Wilson, E.L. and Nickell, R.E., "Application of the finite element method to heat conduction.analysis", Nuclear Eng. and Design 4, 1966, pp. 276-286.
20. Donea, J., Giuliani, S. and Philippe A., "Finite elements in the solution of electromagnetic induction problems", Inter. Journal for numerical methods in Eng., Vol. 8, 1974, pp. 359-67.
2]. Zienkiewicz, 0.C., "The finite element method in engineering science" Book 2nd edition, Pub. McGraw Hill, London 1971.
22. Desai, C.S., and Abel, J.F., "Introduction to the finite element method", Book, Pub. van Nostrand Reinhold Co., New York, 1972.
23. Curtin, M.E., "Variational principles for linear initial value problems", Quart. Appl. Math. Vol. 22, 1964, pp. 252-56.
24. Aguirre-Ramirez, G., and Oden, J.T., "Finite element technique applied to heat conduction in solids with temperature dependent on thermal conductivity", International Journal for numerical methods in Eng., Vol. 7, pp. 345-355, 1973.
25. Kantorovich, L.V. and Krylov, V.J., "Approximate methods of higher analysis", Book pub. Noordhoff Ltd., P., 1958, translated from Russian.
26. Foggia, A., Sabonnadiere, J.C. and Silvester, P., "Finite element solution of saturated travelling magnetic field problems, I.E.E.E. Power Engineering Society, 1975 winter meeting, T75 030-2.
27. Anderson, 0.W., "Iterative solution of finite element equations in magnetic field problems", I.E.E.E. Power Eng. Society, paper C72425-7, July 1972.
28. Verga, R.S., "Matrix iterative analysis", Book pub. Prentice-Hall, New Jersey, 1962.
29. Mastero, S., "Empirical convergence algorithm for successive over relaxation", Proc. I-E.E-, May 1974,. Vol. 121, No. 5, pp. 369-373.
272
30. Dixon, L.C.W., "Nonlinear optimisation", Book, puh. English Univer-sities Press, 1972.
31. Ramarathnam, R., Desat, B.G., and Sabha Rao, V., " A comparative study of minimisation techniques for optimisation of induction motor design", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-92, No. 5, 1973, pp. 1448-54.
32. Klerer, M. and Granino, K., "Digital computer user's handbook", pub. McGraw Hill, 1967.
33. Bandler, J.W., "Optimisation methods for computer-aided design, I.E.E.E. Trans. on microwave theory and techniques, Vol. MTT-17, 1969, pp. 533-52.
34. Fletcher, R., and Powell, M:J.D.,"A rapidly convergent descent method for minimisation", The Computer Journal, Vol. 6, p. 163.
35. Ralston, A. and Wilf, H.S., "Mathematical methods for digital computers", book, pub. John Wiley, U.S.A., 1962.
36. Fletcher, R., and Reeves, 0.M., "Function minimisation by conjugate gradients", The Computer Journal, Vol. 7, April 1964 to Jan.1965,pp.149-54.
37. Michel, A.N. and Cornick, D.E., "Numerical optimisation of distributed parameter systems", 1971 Joint automatic control conf. of the American automatic control council, paper No. 3-C7, pp. 183-91.
38. Dembart, B., and Erisman, A.M., "Hybrid sparse matrix methods", I.E.E.E., Trans.on circuit theory, Nov. 1973, Vol. CT 20, pp. 641-649.
39. Cheung, L.K. and Kuh, E.S., "The bordered triangular matrix and minimum essential sets of a diagraph", I.E.E.E. Trans. on circuits and syst., Sept. 1974, Vol. CAS-21, pp. 633-39.
40. Nakhla, M., Singhal, K. and Vlach, J., "An optimal pivoting order for the solution of sparse systems of equations". I.E.E.E. Trans. on cir-cuits and syst., Vol. CAS-21, 1974, pp. 222-25.
41. Hsieh, H.Y., "Pivoting order computation methods for large random sparse systems", I.E.E.E., Trans. on circuits and syst., Vol. CAS-21, 1974, pp. 225-230.
42. Hsieh, H.Y., "Fill in comparisons between Gauss-Jordan and Gaussian eliminations", I.E.E.E., Trans. on circuits and Syst., Vol. CAS-21, 1974, Pp. 230-233.
43. Nabona, N. and Freris, L.L., "New programming approach to the Newton-Raphson load flow", I.E.E.E., Trans. on Power App. and Syst., Vol. PAS-92, 1973, pp. 857.
44. Jennings, A., "A compact storage scheme for the solution of symmetric linear simultaneous equations", The Computer Journal, Vol. 9, 1966, pp. 281-85.
45. Householder, A.S., "The theory of matrices in numerical analysis" Book, Blaisdell pub. co., New York, 1965.
273
46. Balabanian, N. and Bickart, T.A., "Electrical network theory", Rook, John Wiley, New York, 1969.
47. Hammons, T.J. and Parsons, A.J., "Design of micro-alternator for power system stability investigations", I.E.E.E. procedings, Vol. 118, 1971, pp. 1421-41.
48. Liwschitz-Carik, M. and Whipple, C.C.,"Alternating current machines", Book, 2nd Ed., Van Nostrand, 1961.
49. Trutt, F.C., Erdelyi, E.A., and Hopkins, K.E., "Representation of the magnetisation characteristic.of d.c. machines for computer use", I.E.E.E. Trans. on Power App. and Syst., PAS-87, 1968, pp.665-669.
50. Serma, M.S., Wilson, J.C. Laurenson, P.J. and Jokp, A.C., "End winding leakage of high speed alternators by 3-dimensional field determination", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-90, 1971, pp.65-77.
51. Reace, A.B.J., and Pramanik, A., "Calculation of end region field of a.c. machines", Proc. I.E.E., Vol. 112, No. 7, 1965, pp. 1355-68.
52. Kilgore, L.A., "Calculation of synchronous machine constants", A.I.E.E., Trans., Vol. 50, 1931, pp. 1201-1213.
53. Rudenberg, R., "Transient performance of electric power systems", Book, Pub. the M.I.T. press, Cambridge, 1969.
54. Silvester, P., "Eddy current modes in linear solid-iron bars", Proc. I.E.E., Vol. 112, No. 8, 1965, pp. 1589-94.
55. Alger, P.L., "Induction machines", Book, 2nd ed., Pub. Gordon and Breach, New York, 1970.
56. Chari, M.V.K., "Finite element solution of the eddy current problem in magnetic structures", I.E.E.E., Trans. on Power App. and Syst., Vol. PAS-93, 1974, pp.62-72.
57. I.E.E.E. Committee Report, "Recommended Phasor diagram for synchro-nous machines", I.E.E.E. Trans. on Power App. and Syst., Vol. PAS-88, 1969, pp.1593-1610.
58. Rao, K.V.N., Macdonald, D.C., and Adkins, B., " Peak inverse voltages in the rectifier excitation systems of synchronous machines", I.E.E.E., Trans.on Power App. and Syst., Vol. PAS-93, 1974, pp.1685-1692.
59. Gelfand, I.M. and Fomin, S.V., "Calculus of variations", Book, Pub. Prentice-Hall Inc., New Jersey, 1964, translated from Russian language.
60. Courant, R. and Hilbart, D., "Methods of mathematical physics", Book, Vol. 1, Pub. Interscience publishers, New York, 1953.
61. Beteman, H., "Partial differential equations of mathematical physics", Book, Cambridge University Press, 1959.
274
62. Brocci, R.A., "Analysis of axisymmetric linear heat conduction prob-lems by finite. element method", ASME Winter meeting 1969, Los Angeles, Paper 69—WAMT-37,
63. Richardson, R.A. and Shum, Y.M., "Use of finite element methods in solution of transient heat conduction problems", ASNE winter meeting, 1969, Los Angeles, paper 69—WAMT-36.
64. Woodson, H.N. and Melcher, J.R., "Electromechanical dynamics", Book, Part 1., pub. John Wiley, New York, 1968.
65. Zienkiewicz, O.C. and Parekh, C.J. "Transient field problems: two dimensional and three dimensional analyses by isoparametric finite elements", International Journal for numerical methods in Eng., Vol. 2, 1970, pp. 61-71.
275
APPENDIX A
STEADY STATE FINITE ELEMENT ANALYSIS USING RITZ METHOD
A 1. Equivalent variational Problem
The technique of using the variational calculus for solving
the partial differential equations is the inverse of what is normally done
with the variational calculus59.
The basic approach of the variational calculus is to find the
extremum of a given integral by reducing the problem to one involving a par-
tial differential equation60
. The resultant differential equation represents a
necessary but not sufficient condition to furnish the extremum of the in-
tegral61. Euler's equation is one of the methods which can be used for
sufficiently differentiable functionals when an extremum of a given integral
has to be obtained. The rule for finding the minimal of an integral in the
form:
71(A) = ff L (x,y,A,A x ,A y) dxdy
(A.1)
is to find L which satisfies Euler's equation:
3L D (DL _ _aA ax DA
xI 3y ‘DA
Y
0 (A.2)
DA together with the boundary conditions. Here A
x = -
5-X- and Ay =
DA
The use of the variational calculus to solve partial differen-
tial equations satisfying given boundary conditions is based on the following
idea59: If it can be shown that a given differential equation is the Euler
equation of some functional, and if it has been proved that this functional
has an extremum for a sufficiently smooth admissible function, then this
proves that the differential equation has a solution satisfying the boundary
conditions corresponding to the given variational problem. Moreover, the
variational method can be used not only to prove the existence of a solution
of the original differential equation, but also to calculate a solution of
any desired accuracy.
276
The variational method can he used for solving the magnetic
field prohlem with an alternative- integral form of equation (3.3)
B2
(A) = ff 2 f U (B2) dB2--JA} dxdy ± f odL1
fodL2
(A.3) D o L
1 L2
where U is the reluctivity of the material which_Can be expressed in terms
2 of the squared value of the flux density B . This integral may be shown to
give the nonlinear partial differential equation (3.1) as follows:
(A.4)
3A DA where the reluctivity u depends on B
2 and B
2 is a function of
W and — or
Dy
u is independent of A.
The first component in Euler's equation (A.2) can be evaluated
from:
DL D f 1 fB
J (B2) dB
2)
DAx
= DAx `
2 1 2 (-2- f
B2 u(B
2) dB
2)
3B
3B aAx 0
but the differentiation of an integral of a function in the form
x
3x f f (x) d x = f (x) 0
then 2
(A.5 )
f 12--.1)(B2)dB= 1u(B2) = 2
1 2
(A .6) DB o
DB2 The value of
aA can be calculated from:
DB2 3 t( DA \2 f DA 12\ DA DA `" ax A ` By A A x x
= aA DX
Substituting equations (A.6) and (A.7) into equation (A.5) yields
(A.7)
2
277
DL _ DA 0Ax _
u-ax
The. first component in Euler's equation (k.2) then becomes
'DL = D DA ax DA ax Dx
(A.8)
(A9a)
Similarly the second component of Euler's equation can be derived for the y
coordinate which gives
D DL = D u DA (A.9b)
ay DA By By
Substituting equations (A.4), (A.9a and b) into equation (A.2) gives the
nonlinear partial differential equation (3.1).
The magnetic field is physical and stationary. The total stored
energy within the region of the physical problem has to be keptninimum. Moreover,
the field can be expressed by a continuous function, smooth where there is no
sudden change in the reluctivity. When the discrete analysis is applied, the
region of the field problem has to be subdivided and the energy is distributed
all over the elements.
13 A.2 THE SUBDIVISION OF THE REGION
The machine region R is subdivided into discrete subregions or
elements. They have the same general form. Each element encloses a part of
the region where there is no sudden change in the properties of the enclosed
region. The boundaries of each element being plane or curvilinear faces, and
with the adjacent boundaries of any pair of elements being coincident. The
last condition means that where two faces of a pair of elements are in contact,
the edges around these faces and the vertices of these faces are, respectivly,
coincident. The commonly used elements also used here are triangles. At similar
positions in each element a number of points are identified as "nodes" or
"nodal points". For a first order triangular element these are the triangle
278
vertices. The subdivision of a two-dimensional region into triangular ele-
ments with_ nodes at the vertices is illustrated in fig. (A.1).
y
Agt*
element AVAI kl PrA X
Fig. (A.1). Subdivision of the region into finite elements.
The value of A determined at a node with these approximations is the nodal
value of A, denoted by X , where p is the node number. A is assumed to
correspond exactly with the true solution A at the node p.
Let the region R contain the domain D surrounded by the
boundaries L1
and L2
so that R=D+L1+L2 where L
1 and L
2 are Dirichlet and
Neumann boundaries. Let the number of elements into which R is subdivided be
2, and the total number of nodes in R enclosing the boundaries be n, with n1
and n2 nodes on L
1 and L
2 respectively. The nodal values of the elements
will be identified as Ai, 2 Xe' 3 Xe' these being the node identifiers. They can
be listed as a column vector
{Ae}
Te
Xe 2
X e 3
(A.10)
The n nodal values of A in the region R can be written as the column vector
Al
A2
• (A.11)
A n
279
where the subscripts are the node numbers,
The node identifier at a node should be carefully distinguished
from the node number for that node, For instance, in the first order triangle
the vertices are numbered internally from 1 to 3. These vertices are at the
same time nodes of the main grid where they are numbered relative to it.
13,21 A.3 ELEMENT SHAPE FUNCTION AND FIRST ORDER FINITE ELEMENT RELATIONS
The solution A of the magnetic field can be described element by
element across the region, i.e., can be defined piecewise over the region.
Within an element, the simplest form of approximating the field shape
function is linear in the nodal values ae) . The field can be expressed within
the triangular element where nodes are defined at its vertices by the relation:
or in matrix form,
3 A = L r Ne
A e .
i=1
A = (Ne) {Ae}
(A.12)
(A.13)
where (Ne) are known as the shape function, restricted to being functions -
of position.
Since the true solution A is continuous and has continuous derivatives
(up to some order) across the region, the piecewise representation of equation
(A.13) should have these same properties. The shape functions have to satis-
fy the condition that when the coordinates of a node are put in, the value of
A at that node is obtained, that is its coefficient is unity and zero coeffi-
cients from other nodes. Such properties are possessed by pyramid functions.
When the plane is broken up into triangles the surface of the
function becomes approximated by triangular planes. With linear shape
functions, the representation of A over the element e becomes the equation of
a plane i.e.,
A= +a2x+ci, y.
1 (A.14)
280
where al'
a2
and a3
are constants whose values are determined by the nodal
values of A, and hence,
(Al'-e -e
1 A ) = 1,2,3
, 2' 3 k. (A.15)
where the nodal points of the element are identified as 1, 2, and 3 in a
counterclockwise manner as shown in fig. (A.2).
From equation (A.14) the nodal values A e A3
(A.16) y
where (x1,3)); (x2,y2) ; (x3,y3) are
the coordinates, respectively of the Fig. (A.2). Typical triangular
element e and its projection on nodes 1, 2, and 3. The system of
x,y plane.
equations (A.16) yields a unique solution for each of the coefficients al, a2
and a3
provided the determinant given by
24= 2 (area of the triangle 1 2 3 ) (A.17)
is not zero. Since the triangle area is always finite, A 0 and equations
(A.16) yield a unique solution. Introducing the notation
a/ = x2y3 x3y2 , b1 --= y2 y3, = x3 -x2 (A.18)
and solving for the coefficients al'
a2 and a3
yield the representation of A
in the form
A-=
Ual+bix + cly1 +b2x+c2y)
(A.I9)
e Al'
A2
and A3
are given by
-e Al
= al + a
2x1 + a
3y1
e A2 = al + a2x2 + a3y2
e A3 = a, + a2x3 + a3y3
(a3+b3x + c3 y) y) ]
281
where b
•b3' , c3 are obtained by cyclic permutation of the indices in
equation (A,18). Equation (A,19.) can he written in a compact form. 3 1
A = Ca + b.X + c y ) 2A. .i=1
(A.20)
Equation (A.37) is known as the first order approximation of the potential
within the triangle. Equation (A.19) can also be written in a matrix form,
ZA –e e e
A = C(a) {A 1 + (b) {A 1 x + (c) {A } (A.21)
where the row matrices (a), (b) and (c) are given by
(a) = (ai a2 a3) > (b) =(b1 b2 b3), (A.22)
(c) = (c1 c2 c3)
and equation (A.10) gives {Ae}.
Alternatively, equation (A.20) can be written as a matrix equation
in the form of equation (A.13) as
A =1
N2
N3) (A.23)
where the shape functions N1, N, and N3 are given by
1 Ni = 72-6, (a1 + bix ciy) ; i = 1,2,3 (A.24)
A.4 FLUX DENSITY RELATIONSHIP
The flux density is related to the potential A in terms of the
cartizian coordinates as
B2 BA 2 DA (-3x) + (-3y)
With a linear approximation of the potential A within the element the
A.25)
components of the flux density in the x and y direction can be obtained by
differentiating partially equation (A.23) as
DA 2N1
DN2 DNT3
DX `ax 3x Dx ) {Ka}
which_ becomes from equation CA.22) as
-DA 1 Dx 2L (b) {r} (A . 26)
282
and
DA 11-1 DN2 DN 3
ay ay ay ) {A )
and from equation (A.22) it becomes
DA 1 r. ay = 26. (C) f } (A. 27)
It is clear from equations (A.26) and (A.27) that with the linear approxi-
mation of the potential within the element, the components of the flux
density are independent on the position i.e uniformly distributed.
The value of the flux density squared, B2, can be obtained by
substituting equations (A.26) and (A.27) into equation (A.25) to give,
- B2 -
(2A)2 ael ((b)T(b) (c)
T (c)){ A
e } (A.28)
A.5 ELEMENT VARIATIONAL FORMULATION
The subdivision of the region R into elements leads to a sub-
division of the energy functional into parts associated with each element.
The minimization of the total energy within the field (A) can be expressed
in terms of the energy within each element (A) as Q
(A) = y .X (A) (A.29) e=1 e
The linear approximation of the potential within each element leads,
as explained in sec. (A.4), to a constant distribution of the flux density
within each element. Since the reluctivity and permeability are functions of
the flux density, this implies in turn a constant reluctivity within each
triangular element. The energy functional within each triangular element
then becomes from equations (A.3) and (A.25)
= 11 { 1 • DA 2 U (ax)
2 DA 2 .) JA} dxdy
"Dy -De (A.30)
283
A.6 THE MINIMISATION CONDITION
The substitution of equations (A.3) and (A.13) into equation
(h.29) yields ;4(h) as a function of all the nodal values of A, i.e., as
a function of {A} . According to Ritzs method25 the condition for the functional
to be stationary is that the first derivative of ;(A) with respect to
each nodal value separately, equals zero12. This leads to the system of
equations (3.4) which can be rewritten as
G. (A) E 1- 0 DA (A) = 0 i i = 1, ...n (A.31)
To perform the minimisation of the energy functional, it is
convenient to express the surface integral in a way that permits the
evaluation of the functional over one triangle at a time. This can be done
by substituting equation (A.29) into equation (A.31) to give13 ,
Q 0 X (A) = X DA. 5(A) = 0 DAi e=1 i e
or in the matrix form, as
i = 1,...,n (A.32)
k k
L (A) = y (A) = 10)e Da-} e=1 e=1 9{A} e
(A.33)
A.7 THE ELEMENT MATRIX EQUATION
• It is clear from the characteristic of the shape function in sec.
(A.3) that the energy functional within the element depends only on the
element node potentials. The energy within the triangular element can be
expressed mathematically as
37; (A) = f (A1, 2'
A3e
(A.34)
Accordingly, the differentiation of the element energy functional
(A)
. in equation(A-33)withrespecttoA.ives zero unless is one of the
triangle vertices. It is convenient in the consideration of an element to
284
invert the system (node number i) to its corresponding node identifier k.
Thus for an element e, a term such. as --- of equation (A.32) can be
aye DAi
written as =---. aAk
The element matrices can be derived as follows: The first part of
the integration in equation (A.30) can be expressed in terms of the node
potentials using equation (A.12) to give
ff A,2 DA 2 3
De 2 L (9x) dxdy = if 1)2 i4x N? -Z)2 +
L 1 1 De 1=
3 y Z-
BY)2 ]dxdy
i=1 1 1
or with the differentiation under the summation to be
(A.35)
3 N. 3 DN! 1 -e 2 ff (DA) dxdy = ff [( X 1 7e ,2 -- A.) dxdy
2 -gTc ' i=1 BY j De De
(A.36)
Equation (A.36) can be written in the matrix form as
T e T
ff U DNe -e jj (DA)2 dxdy = ff 1)T: {-Ael ) {A } + 2 a x De De
T T ,me tr) ero ael ) dxdy (A.37)
The differentiation of equation (A.37) with respect to the nodal potentials
-e IA I can be obtained from the standard differentiation of matrices
ar
T lYI N1 IYI = 2 EQ1 {Y} (A.38a)
and
T ° aTFT {Y
} EQ1 = EQ1 (A. 38b)
The differentiatin of equation (A.37) with respect to the nodal potentials{Ae}
using relation (A.38a) gives.
285
If (DA) dxdy U aNe T aie
T e _ 4.(-.7;7 ) 1V{Ae} dxdy
D D (A.39).'
The column vector IAe
iS constant with respect to the integration, and
DNe 3 e consequently can be taken outside the integral. The value of (-
57c) and(--
N —)
Dy
can be evaluated from equation (A.26) and (A.27) to be
1 (77;c) 27,(b)
• 9Ni
(-537) TA(c) (A.40)
where (b) and (c) are constant vectors as defined in equation (A.22). There
matrices can be taken outside the integral as well as u which is assumed
constant within the element. Equation (A.39) then becomes
a r u (..a2_5 2 + ( ay, .x . e, )d dy = U A [C] { A J. e "
OA 1 De
2 y (A.41)
where A is the area of the triangular element, CO is a matrix with the
elements given by T T
[E].= (27)2((b) (b) + (C) (C) )
b + c2
b 1 1
bib2+cic2
b1b3+C1C3 1
2 (2A) +C
2 C1 2 2 b2
2 c b2b3+C2C3
b3b1+b3C1 b3b2+C3C2 b +C
2 3 3
The second part of the integration of equation (A.30) can be
obtained by assuming an approximate distribution of the current density
within the element. The method to be considered is that in which J is
approximated by a function of the current density at the triangle vertice as
3 =
i=1
(A.44)
(A.42)
(A.43)
Fig. (A.3)
n.lrIsl I = 2U (n+r+s+2) (A.49)
286
or in the matrix form
J = (ge) fael
(A.45)
Substituting equation (A.12) and (A,44) into the part of the integration
in equation (A.30) containing the current density J gives
33 ff J A dxdy = if ( e . N! A!) dxdy (A.46a)
De -De i=1 L L .i=1 "
which can be written as the matrix form
T T ff J A dxdy = ff C{Ae} (Ne) (8e) {J} ] dxdy
(A.46b)
De De
Differentiating equation (A.46b) with respect to the element nodal poten-
tials Ae using the formula (A.38b) gives
J A dxdy = ff -e
(Re) ) dxdy De 3{9 lie} ffDe
(A.47)
The evaluation of the value of the integral of equation (A.47) is done by
dividing the triangle into three equal parts as
shown in fig. (A.3). nen the current density
is assumed constant, then 8=1. With the
assumption of a linear distribution of the
{Re} , , current d 1 .density then. 0 s = kNe). With either
of these assumptions equation (A.47), contains
elements which are a special case of the
integral
wn. w T. wk du L J
the value of which is given by 18
(A.48)
287
With a uniform current density distribution, equation (A.49) has
the form of equation CA.48) with .r and s equal to zero. Substituting for
the elements in equation (A.49) gives the value of the integral of equation
(A.47) as6
rr A jj J A dxdy = CU Ciel
.D{AT} De (A.5°)
when U is the unity matrix. If the current density is assumed linearly
distributed over the triangle elements
of equation (A.48) with s=o. Substituting
by comparision with (A.49)
A
can
CMe]
then equation
for the elements
2 1 1
1 2 1
1 1 2
be expressed
{F}
(A.47) has the form
in equation (A.47)
e Jl je (A.51) 2
-e J3
as
(A.52)
If J dxdy = -le
bae/ De 12
In general equation (A.47)
ff J A dxdy A e Dale}
where Me is square symmetric matrix.
The derivative of the element energy functional, expressed by
equation (A.30), with respect to the nodal potentials becomes from equations
(A.41) and (A.52)
9 (A). = A Fie] e {xe} _ Ae ale] {ae}
(A.53)
In equation (A.53), all the subscripts are in terms of node
identifiers, not node numbers. In order to obtain the set of equations
(A.33) the entries from each triangle expressed in equation (A,53) have
to be summated. First of all, the element node identifiers has to be re-
placed by the node numbers . The elements of thee] and CM
e] matrices
288
in equation (A.53) may also he relahled according to the numbers of the
nodes of the element considered. In the element e the node identifier
numbers y,& have corresponding node numbers (a,13) respectively. Any element
of the matrices and M labeled by y,S thus enter into the matrices formed
from setting up all the n-equations(of the n grid nodes of equation (A.33)
in the location (a,13).
A.8 THE SYSTEM MATRIX EQUATION
For each of the 2. elements, a numerical formulation can be derived
in the form of the relation (A.53). These can be assembled into a single
equation using the minimisation of equation (A.33) to obtain
ES] {A}= {I} (A.54)
Consideration of this equation and the way it is assembled (by addition of
the corresponding elements), shows that the elements of the S-matrix and I
vector are given by
and
cc , 13 =. --e A r
' a, (3 e
2 e e=1
J(3 e A M a f3 e a,13 (3 e=1 =1
(A.55a)
(A.55b)
M being given by equations (A.51) and (A.52) or the unity matrix /3.
The boundary conditions are satisfied when the following computa-
tional rules are applied:
If n is the number of a node in a Dirichlet boundary zeros are
put in the 11-th row of the S-matrices in equation (A.54) except for unity in
the diagonal position of the S-matrix, and for -.la at the I vector
Neumanes condition is implicitly satisfied in the functional formulation
of equation (A,3).
289
APPENDIX B
MATHEMATICAL METHODS FOR.CHANGING THE NONLINEAR DIFFUSION EQUATION TO THE DISCRETE. FORM
B.1 VARIATIONAL FORMULATION OF THE FINITE ELEMENT METHOD
The logical argument of the finite element method that, based on
the variational analysis, is explained in recent books of the finite ele-
ments. The way of their explanation starts by saying that the problem re-
presented by the nonlinear diffusion equation involves time as one parameter
and hence is refered to as transient or time dependent problem. If the time
term on the nonlinear diffusion equation vanishes,Poisson's equation is then
DA left. When a constant value is expressed to -5-t- the equation then represents
a steady state problem
Therefore, the partial differential equation
3 3A a 3A DA Dx u ax ay v ay -
_ o at
has a functional :4(A) in the integral form
(B.1)
j;.(A) = f2 ue 3x ay
[(2.12±)2
1-(2 )2 ] a
3t) A} dxdy (B.2)
e=1 D e •
Since o at 9A i
- used to replace -J in equation (A.30), in each
element, it follows that using the same approach of finite element method
the field has the discrete form:
Cs]. {A} --E0fa (B.3)
The formulation of the above equation has to satisfy the boundary
condition of equation (B.1). The initial conditions are satisfied when the
DA initial derivatives of the potential A, -5--t multiplied by a replaces -J.
Alternative methods are used to replace the integral form of the
functional of equation (B.2) which has the effect of reducing the argu-
ments mentioned in sec. (4.3).
290
B.2 VARIATIONAL FORMULATION OF THE DIFFUSION EQUATION USING
GURTIN T'S METHOD23
The method of deducing the variational principle for the diffusion
equation (4.2) has been proved by Gurtin in two steps. First the reduction
of the initial value problem to an equivalent boundary value problem by
replacing the diffusion equation with an equivalent integro-differential
equation which contains the initial conditions implicitly. Second, a
variational principle is derived for the reduced problem. The prime ingre-
dient in the derivation of the variational principle is the use of convo-
lutions. Following Gurtin's notation, the convolution is denoted by the
symbol *. The standardized notation of the convolution of two functions u and
U is defined by62 '63
t u * u (x,y,t) = f u(x,y,t-T) U (x,y,T) dT
It has the properties
U * u (x,y,t) = u * u(x,y,t)
(u * v) * w(x,y,t) = u * (u*w)(x,y,t) = u*u*w(x,y,t)
u * (u+w) (x,y,t) = u*u(x,y,t) + u * w (x,y,t)
u * U= 0 if and only if U E 0 or U
Moreover,.
Du Du Du DU Vu * Vu = * + *
ax ax 3y 3y
(B.4)
(B.5)
Curtin has proved that the diffusion equation (4.2) which can
be written in the alternation form
U V2 A = a at
(B.6)
has the variational functionalt (A) such. that
(A)=f iaA*A-1- U * VA * VA - 2a Ao * Al (x,y,T) dxdy (B.7)
The initial distribution of the field is also presented in equation (B.7)
by the symbol A. Thus Ao can be written mathematically as
291
A E A (x, y) 0 0
(B.8)
The method has been extended by Wilson and Nickel19 to solve the
nonlinear diffusion equation. When the relUctivityU becomes a function of
x and y, the general expression
(&) = 2 f {GA * A VA * u * VA - 2o- Ao * A} (x,y,t) dxdy (B.9) R
is the functional form of equation (B.1)
Their proof can be sununarized as follows:
Let A be the set of all functions continuous in the region R and
satisfying prescribed boundary conditions. Then by taking the variation of
equation (B.9) and applying the divergence theorem it is seen that the
incremental change of the functional 2,1 (A) vanishes over R and (o<t<00) t
at a particular. function A if and only if A is a solution of Euler's
equation
V . (u* VA) - GA GA0 =
o
(B.10)
with its natural boundary conditions.
Through the use of the identity
fu * Al (x,y,t) = f a(x,y) 9A
(x,y,T) dT
(B.11)
= a(x,y) [A(x,y,t) - Ao(x,y)]
equation (B.10) may be transformed into the transient nonlinear partial
differential equation
DA _ V . (u * VA) — 0* - 0 (B.12)
This yields equation (B.1). Hence the use of equation (B.9) is verified.
The solution is furnished when the extremum of functional (B.9) is verified
at a given time.
292
B.2.1 Gurtin's variational formulation and the finite element method
The use of the finite element method on the variational form 4
given by equation (B.9) is similar to the application of the finite element
on equation (4.13). The potential within each element can be approximated
by equation (4.14), which has the matrix form
A (x,y,t) = CNeCx,y) ) {AeW} (B.13)
where {AeW} is a column vector of the element nodal potentials
(da(x,y)) is a vector represents the'spatial approximation. It should be
noted, as mentioned in Appendix A, that it is possible to change the
assigned positions for nodes during the execution of the problem, if this
should be desired for some reason as well as allowing the physical variable
to be explicitly time-dependent62.
Differentiation of equation (B.13), with respect to the spatial
coordinates yields a column vector of potential time derivatives 'which may
be written as
{V A (x,y,t)} = cpe(x,y)] xe(01 (B.14)
e
9x)
De(x,y)] = aNe
y )
where
(B.15)
The generating functional (B. 9) can be expressed as a summation
over the elements
R (A)= y t e=1
2 fa eA e * Ae VAe * U
e * VA
e
— 2 a Aoe * Ae
}Cx,y,t) dxdy
(B.16)
where 2. is the total number of elements of the system, and Aoe is the
initial potential distribution over the element e* It is important to note
that the reluctivity of the material is different from one element to
another in the region. In the time steps the element reluctivity becomes
293
also a function of time implicitly,
The suhstitution of equations (4,-13) and CB.14) into equation
(B.16) gives
T CA) = (ACt)) * {A(0} + {A(t)} * [S]. * {A(t)}
WO) EM/1 * {A(0)} (B.17)
where,
[M1] is the conductivity matrix and its elements are given by
the equation
[11 ] = ) [me]
(B.18) e=1
wherein T _
CMe] = f x,y)) (Ne(x,y)) dxdy (B.19)
De and
the elements of the [S] -matrix are derived from the assembly
of the element matrices [Se] where
[Se] = f ue [Fe(x,y)] [Fe(x,y)] dxdy (B.20)
De The minimisation condition of sec. (A.6) when applied on equation
(B.17) gives for the first order approximation
[M1] * {A(t)} + [Si] * {A(t)} = CM1] * {A(o)}
(B.21)
then yields from equation (B.11) that equation (B.21) can be written as
[S] {A} = -[N1] t4 (B.22)
which is exactly the same as equation (4.15).
The result of this analysis shows that when finite element
analysis is applied on the variational form derived by Gurtin leads to the
same discrete equations as the case of the other methods. The mentioned
two methods used the variational calculus in different forms. However,
the discrete formulation can be obtained directly from the basic equation
when the vector analysis is used.
294
B.3 DISCRETE FORMULATION. OF THE.NONLINEAR DIFFUSION EQUATION USING THE. VECTOR ANALYSIS
The vector analysis can be used as an alternative approach to
the solution of the nonlinear diffusion equation (B.1) which can be
rewritten in the compact form
DA V . (uVA) = a (B.23)at
The problem is solved using a self-consistent approximate scheme for the
potential at the nodes of the element24
which is rather different from the
previous methods.
The method shows that when an instantaneous variation of the
potential 6 A(t) is allowed in the field at a given instant then
DA V.(UVA) 6A
t = a —
at t (B.24)
and using the identity of the vector analysis64
V.(XA) = A. VA + X V A (B.25)
Equation (B.24) can be written as
V.(6AtU VA) - UVA.V6A
t - a
DA 6A
t = o
(B.26)
Integrating equation (B.26) over the region and applying the divergence
theorem which states that
yc A .nds = f V.A dv (B.27)
Equation (B.26) becomes
f 'IV 6At.uV A
t - at S A
t) dR
R Bt
U V A t.n6 A
t d =o
• (B.28)
where n denotes the unit normal to the boundary surfaCe S on the machine
region R at a time t.
The finite element method can be used to alter'equation (B.28) to
the discrete form. The potential within each element can be approximated
as given by equation (B.13).Consequently, the instantaneous variation of
295
the potential by the value dA over the element e is
{&At} (Ne(x,y)) {0(_t)}
(B.29)
Substituting equations (B.13), (B.15) and (B.29) into equation (B.28) we
arrive after transforming the equation to the n-nodal values to
( {at} + [s] {A} ) SAt . = o
(B.30)
where the matrices [M1] and [S] are as previously defined.
Since the instantaneous variation of the potential is arbitrary,
it follows from equation (B.30) that
[S] {A} = -CM ] 1 3t
which is exactly the same as equation (4.26).
B.4 FORMULATION OF THE DISCRETE EQUATIONS USING GALERKIN'S PRINCIPLE
Galerkin's principle can be used for the formulation of the dis-
crete equations without recourse to the variational principle.
B.4.1
Galerkin's Method25
The basic idea of the method can be expounded quite briefly. Let
it be required to determine the solution of the equation
L (A) = 0 (B.32)
where L is some differential operator in two variables, which solution
satisfies homogenous boundary conditions. Assuming the approximate solution
of the problem as given by equation (B.13) over each element which has to
be chosen so that the boundary conditions are satisfied. All the functions
are assumed linearly independent over the region R and on. its boundaries L1
and L2.
.In order that A be the exact solution of the given equation, it
is necessary that LW be identically equal to zero, and this requirement,
if LEA) is considered to be continuous, is equivalent to the requirement
(B.31)
296
of the orthogonality of the expression LCA) to all function& of the system.
In order to satisfy the orthogonality.conditions the system of equations
could he obtained
L (A. (x,y)) Ni( ,y) dx1dy.=fiLqc.N,(x,y)) NiCx,y) dxdy D "
= 0 .i=1,...,n (B.33)
which serves for the determination of the parameters c..
B.4.2 Application of Galerkin's method on the nonlinear diffusion equation
Galerkin's method could be used to transform the nonlinear diffu-
sion equation to the discrete form. Let the approximating function be
assumed in the form of
A (x,y,t) = y Ni(x,y) Ai(t) 1=1
(B.34)
In the above equation Ni(x,y) are the used shape functions and are defined
piecewise, element by element. In the sumivation the appropriate function for
the particular point in space must be used65. It is a more general form of
equation (B.13). The simultaneous equations, allowing for the solution of
the n values of A.(t), are obtained typically for the point i by equating
to zero the weighted and integrated residual, resulting from substitution
of equations (B.1) and (B.34) into equation (B.33). Thus we have for the
i-th equation
f aIL u ax
v ) N.A. -(-5-2- NiN.A.] N. dxdy =: 9y
j=1 j j Dt j=1 3 3 R
(B.35)
n such equations allow the complete solution of the problem if the integral
can be. evaluated.
The first part of the integral can be evaluated by considering25
f'i OA. 4 D .0 DA .) dxdy: = f etl.t-- 4. -g6 . 1),) dxdy ) ''ax 1) - ax ay . Dy R R
,rD Cu l' ) Cu U. )1 N dxdy -I- J -Dx - Dx' 1. ay
as (B.36)
297
On the base of equation (B.36), equation.(1.35), becomes65
DN. n DN,
+ 7-4) Ai dxdy +
r ax
-R
n DN. DN. o7-- (NJ u A dxdy +
Dx oy Dy j=1 j
n DA.
fR t --j- 1 N. dxdy = 0 (B.37)
=.3 •
The value of the second integral in equation (B.37) can be
obtained by applying Green's theorem,equation (B.37) which then becomes
DN. n DN. DN. n DN.
f( D
1Vu__I al y j
) A dxdy R x ax 1,
j=1 =1
n DN. n DN. + f Ni X U Zx .y u 2, A. ds
003T Y r j=1 3 =1
n DA.
f R aN . --1 dxdy = 0 (B.38)
j=1 3 t
where P.x and Qy are the unit vectors in the x and y direction and the
integral over s is the integral over the external contour line r..
First order derivatives have to be integrated and the conti-
nuity of the functions N. has to be imposed. On the boundary points where
the value of A is prescribed by either Dirichlet's or Neumann's conditions,
the integral over the contour r is equal to zero and the boundary conditions
are justified as explained in sec.. (A.8).
The n—equations can be written down in a matrix form
r9A, ESj {A} . —[1111 {
at}
which takes the alternative form of equation (B.20) as Q DN. DN. DN. DN.
S. •,3 = f ( -4=t) dxdyax e=1 ' De
(B.39)
(B.40)
298
and M1 is given by equation (B.18)..
The summation over thek elements in the above two equations
reduces around the node to the M—surrounding elements when the first
order approximation is used. The form of equation (5.60) is seen to be
identical with,the alternative previous methods.
299
APPENDIX C
EFFECT OF THE OVERHANG LEAKAGE
The fast initial rate of decay of field current is determined
by the small leakage inductance between the field and damper windings. This
inductance must be determined accurately if the field decrement curve is
to be obtained and the contribution from the ends of a short machine (micro
machine) is vital.
Assuming that the field and the effect of the damper can be
represented in a lumped form with time constant T1
and T2, inductances L1,
L2
respectively and defining a leakage coefficient eK between these two circuits
as in ref. (53),
(C.1)
The switching operation in the field causes a current decay with two time
constants Ts
and Th where
Th
= T1 + T
2 (C.2a)
L1L2-M2
GK
L1L2
and T T
Ts
= aK T1+T2
1 2 (C.2b)
the leakage in the damper circuit being produced as a result of the slot and
overhang leakages. When the overhang leakage inductance ko is. taken into
account during the solution it can be proved that the initial time constant
has to be modified by a factor K where,
k 2 K = s o (C.3)
and ks
is the slot leakage inductance of the damper circuit.
This factor has a big effect only during the first period of
the transient and its value must become unity as the time increases. It affects
therefore the equivalent conductivities of the field circuit with a time
300
variable coefficient Kt such_ as
-t/T = 1 + (K. e (C.4)
With a reasonable. approximation, Che time constant T in the above expression
is found to be in the fort
T
T sr
(C.5)
where T is the initial time constant when the effect of the overhang. is sr
excluded.
301
Supporting Publication
(refar6nce No, 9)
1 302
11 1 CO e?
A NODAL METHOD FOR THE NUMERICAL SOLUTION OF
TRANSIENT FIELD PROBIEMS IN ELECTRICAL MACHINES
A.Y. Hannalla B.Sc. M.Sc.Fe D.C. Macdonald Ph.D.*
ABSTRACT
' The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and eleeent • methods., The finite difference method requires a'large number of nodes and is slow to converge as permeability is variable. The finite - element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient con-ditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.
I. INTRODUCTION
• The behaviour of electrical machines is controlled by the interaction of current and magnetic field in the core length. Thus a number of workers [1-41 have con-sidered a cross-section of the core allowing for vary-ing permeability and have taken end effects into account separately.
Here an attempt is made to solve the transient field problem. This involves solutions of the field at successive instants Of time and it is therefore import-ant to:
1. Define the problem with as few nodes as possible.
2. Use a method fcr the field solution which converges rapidly.
3. Make maximum use of-snarsity CS] techniques to minimize the computational time.
L. Obtain maximum accuracy for values at the nodes because these determine the currents at successive instants.
. These aims are met by formuleting the 2-dimensional problem in terms of A, magnetic vector potential, and J normal 'current density. Both are scalars; and a tri-angular grid is used similar to that of the finite element method. However, by formulating the problem at each instant in terms of Amnere'S law rather than by a variational. approach the problem of dissipation is avoided. The nodal method gives maximum accuracy at the nodes and the use of the Newton-Raphson method together with sparsity techniques gives a rapidly con-vergent (5 - 15 iterations) and swiftly calculated solutions.
II. ELECTROMAGNETIC FIELD EQUATIONS
The analysis of power frequency transient electro-magnetic fields mey be represented completely by
atV .V V A = + Gr (1) dt
Manuscript received 28th February 1975.
*The anthore eve with the Dept. of Electrical Engineer-ing, Imperial Collee, Lon-1o ::N7. Mr. Uenneile is on study leave front C■in t:Lnms CcLee.
The right-hand side of this equation may contain contributions resulting from relative motion and in the presence of closed windings, it will take a more in-volved form. Thus the current flowing will depend on the integral of dA over the coil length and on the
dt circuit impedance.
It is convenient to divide the problem into two: firstly the relationship of potential and current at each instant, and secondly the time effects. In gen-eral it is possible to regard the transient conditions as a succession of instants, at each of which the field problem may be solved. Thus it is necessary to obtain accurate solutions with minimal computing effort for
V .V VA = mj (2)
before proceeding to thelransient problem. The nodal method presented here is a solution for eqn.(2) and in later publications the extension to eqn.(1) will be . given in more detail.
Numerical methods for solving field problems give reasonably accurate results by reeking approximate representations of the field over small elements. Thus in the first-order finite element method the field is assumed uniform within a triangle, and the continuity of flux is maintained at the boundaries. The emphasis is on obtaining the correct conditions within each ele-ment as the title finite element suggests.
The finite difference method uses first order • difference approximations to relate together the values of A at the nodes of a rectangular erid. Difficulties arise, however, at irregular boundaries ana normally a very fine grid is used in their vicinity.
In the nodal method described here the emphasis is on the accuracy of values at the nodes as in the finite difference method, although the trianular erid of the finite element method is used. In transient nroblems there is continual interplay betweeen the • A-field and the j-distribution end as the current is considered to be concentrated at the nodes it is vital that the values obtained there are as accurate as possible.
III. DERIVATION OF THE NODAL EQUATIONS
The region in which thefieldiS to be found is divided into acute-angled triangles shown by continuous lines in Fig.(1) within each of which the flux density and reluctivity are considered uniform. Careful choice of triangles can ensure that a reasonable field repre-sentation is obtained, more triangles being required where field gradient is greater. The values of mag-netic potential at the nodes together with their_ co-ordinates define the magnetic field. The current density within each triangle may be considered uniform, or in eddy current problems it may be allowed to vary linearly between the nodes.
Fig. 1. The grid for the nodal 75ethod contours of integration.
(a.+b. x + c. ) - 2A i=1
lY
andwitha.,bi andc.defined as by Silvester [l],
1 = (3)
Ampere's law is then used to relate together the A values at the nodes and the current density distribu-tion. The contour of integration defines the areas over which current density is gathered to each node. It is therefore necessary and convenient to choose con-tours which:
1.'are compatible with a triangular grid,
2. are- an assembly of straight lines,
3. around adjacent nodes, touch, ie., all the region is enclosed by all the contours,
4. are chosen to give the greatest accuracy at the node they enclope.
. Collatz E63 shows that if a field is represented by a combination of linear approximations along. arbi-
- -trary lines, the point at which -the field is represent-ed most accurately is that of minimum total perpen-dicular distance from all the lines of reference (Mini-mum Euclidian distance). This point, the centre of approXimation, is thus uniquely determined. Here the. nodes are to be the centres of approximation and con-tour lines are chosen through the triangles which satisfy this condition. Thus all the four conditions above are met by the broken lines shown in Fig.1, which intersect orthogonally with the original tri-angular grid. In contrast with the finite element method this approach distributes the current density in a triangle to its nodes in a manner related to the shape of the triangle.
The importance of the contour of integration is that it determines the correct distribution of current density to the nodes within each triangle. As within each element the field is uniform and Laplace''s equation holds, the actual values of A obtained by a line integral in one element is independent of the path" and depends only on the position of the start and finish.
Thus the integral may be performed within each element in the easiest way so long as it starts and terminates at the correct points half way along the triangle sides. If (xl ,y1) , (xe,y2), (x-,ye) are the nodes, the contour around node this trifmgle starts and finishes at (x,+0.5(ze-x,),Y,4-0.5(Y1-Y4) .and (xi+0.5(xe-x1), (y„+0:5(y-Y.TO): With a linear • approximation'of A of the forth
C l x. - x 3 2
etc. and being the area of the triangle,
I = 0.5 1( x3-x2) %
ii+(y23) Aibilj () r
rt r si 3 3
j=1 1=1
i=1
Thus each node gives one equation linking together the nodal current and value of A with the values of A at adjoining nodes. The equations formed- may be summar-ised by..
(s) (A) = (I)
(5 ) The elements of A on the diagonal are
s - pp V j .(13lj + clj
.) / . (6a) j=1
' and in the pth
row of the kth column 303
Vi , V1+1 , 'plc (b b +c tb jb
j +
- 4Aj lj 3j lj ej. 4A 1 2 j+1 +1 +1
elj+1
2j+1)
(GO
. where blj and cli are the b1 and ci of the 0
th triangle.
Node k is coupieu to node p by the common side of the j and j+1 triangles around node p. There are M+1 ele-ments in the pth row or column. With careful numbering of the nodes the elements are grouped around the leading diagonal. The number of equations is minimized by solving the equations only for the nodes of unknown potential. The values .of A.= 0 at the boundary contri-bute to some of the elements in the S-matrix satisfying Dirichlet's boundary condition. Neumann's boundary condition is satisfied by the image concept of Ref.2.
The current density distribution is considered to be discontinuous, i.e., each value is associated with a given area (part of a triangle enclosed by a contour of integration). Thus the current'clensity within each triangle may be considered uniform or to vary linearly, and the .total current at a node is the sum of the 'con-tributions from the elements surrounding it.
3
2
2'
1
2 J-.) 1 1,
1 J3,-1
3 1 Ijit'94.
J3'
'1)3
J4'-,)4 7
.'.illeSb-block of the finite difference method and the contour of the nodal approximation.
IV. FINITE DIFFERENCE AND NODAL EQUATIONS
The rectangular grid of the finite difference method can be viewed as a triangular grid with selected lines removed, and these may always be replaced around any node as in Fie.2. ti the nodal method is used at node 0 in Fig.2 to form the equation corresponding to the contour shown, nodes 3, 5 and 7 are found to have no part in the S-matrix. The equation is exactly the some as would be formed by the finite difference approach, both methods using the same contour line. The equations obtained by the .two methods differ when the rcluctivity is assumed. constant only in each tri-angle rather than in each quadrant as shown in Fig.2. Equations are identical in non-permeable material.
V. um ITERATIVE PROCEDURE AND FIRST-ORDER =LET METHOD
The nodal method gives identical elements in.the S-matrix to that obtained with the finite element method end the Newton-Raphson scheme devised by Silvester et al E3] is applicable. In the finite element method contour lines are not defined, field equations being . satisfied by minimizing the sum of the rnergies stored within all the grid elements with reepect to the para-meters expeceeing the field at the grid nodes (Ritz method). The porameters are taken to be the potentials at the grid nodes. The current within the triangle is allocated to the vertices reeardlces of the trienele shape. Iaim contrerjs with the di.itribution in the nodal method, where currents are dependent on shape. Identical equations are obtained for equilateral
2.
304 IT. 71, Cf.';PY
trinncles. The finite clement method is particularly wenk in
rectansulor slots earryins uniformly distributed current, the current attributed to nodes beins depend-rut on -the trionslc:: chosen is shown in Yis.3a and b. The nodal: method distributes the currents in a much more suitable woy shown in Fis.3c. Pectansular ele-
-ments are used with odvantase in slots.
2 3
3
Fig. 3. Allocation of current to nodes around a clot carrying uniformly distributed rcr;i (a) & (b) finite elcmt-nt method (c:) nodal method.
The nodal method shown thnt it is inportant not to uue trienslcs with ansles greater than 90°, for one of the vertices of such a triangle sivos o nesa.tive con-tribution to a diunal elemont in the S-m:,tri.x. Thus it weakens the rcotrix. Also the contcur lino lies Tartly outside the triansle being cc.i:sidered or shown in Fig.4 which is clearly unsatisfactory.
,, 4 ;
Unsotisfactoy contour of.intezratipn with an obtuse-nnsled trionsle.
VI. APPLICATIOA
. . The method hen been oplAied to the solution of the field problem of a 4-pole alternator excited only with field current cLid the flux pattern is Sbown in 1; r:.5. A volution for the 2?2 roden is obtained on a CDC Coo computer ur:inc; 16.2 k word:7; (117.cludini: th.;. cCTTicl cmTiler) in en verase time per it.cs:;tiea of Us. Fourteen iterations arerequired to sive errors in -potential. 2f 0.19 x 10- '''Wb/M and errors in current of 0.15 x 10-0A. The flux density obtained in the air cap ascees well with that measured by the Gi2:1 circuit voltage of the machine. .
VII. CUIGT•USICY.:
The nodal method doscrib.-d ir n refinement of the finite element method and ovoids the distortions in current dIstrib,.;tion which e:n of},,cwioc ooec.r and ore sisnificant undr• trnnsie;It condi'ionn. ].t is also consistent with the finito dicTerence r'innonah, the only distinotion Lhe two In n'- cc nc haments over which yin iraeohstierd
Heins h..sed eel Ampene's c-in be ao doubt as to its vnlidits under 1.1.- - sient Cis:i.native con-ditions. MO format:ion O.
Fig.5. No-load -faux distribution.
an approach would oppeor to be of general value in other conditions, such no transient thermal problems when variational solutions hove been used.
It is honed tinct it will be noseible to model the behavic.nr of the solid-iron ports of machines so that transier.t behaviour cry be calculated directly without reference to s.:b-transient pt,rameters. It remains to be coon whether an accurate but simple enouc,h model can be devised so that its use will becme ccnera.
The authors sratcfmlly acknowledge the assistance of Dr. B. Adkins and C.J. Carl:enter in crony con- versation; and Er. A.B.J. Reece of G.E.C. Ltd., Stafford for supplying the machine dimensions.
REI.ITENCES
1. P. Silvester nnd Chnri, 'Finite clement solution of saturable masnotic field problems', IEEE Tri:ns. on Fewer ;Tp. and Cyst. , Vol. PAS-89, 1970, pp. 1642-51..
2. 0.1!. Anderson, 'Iterative solmtics.of finite ele-ment equations in mssnetic field problems' , IEEE Power Ens. Society, Paper C 1.4 25-7, July 1972.
3. P. Silvester, H.S. Cabayan and B.T. Esowne, 'Effi- cient techniques car finite clement analyis of • electric machines', IEEE Tra_so.. on Power App. and Syrt. Vol. PAS-92, 1973. pp. 127!c-21
4. E.F. rUCh:3 and E.A. Erdolyi,'Doterminotiou of water wheel olteraator steady state reactance', ITTEE Trans. or: Power App. end Cyr.A., Vol. FA0-:91, 197P, pp. 2510-7-
5. A. Jenninos, 'A compact sdhomc for the solution of symmetric linear simultaneous equations', Computer journal, Vol.9, 1966, pp.;':81-285.
6. T TuurAiefl-•1 analysis nod numerical. mathematids', hook, Acathuic New Yorit.1966.
3.
