Citation
An algorithm for detailed induction machine analysis without recourse to D-Q theory

Material Information

Title:
An algorithm for detailed induction machine analysis without recourse to D-Q theory
Creator:
Horak, John Joseph
Place of Publication:
Denver, Colo.
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
viii, 87 leaves : illustrations ; 29 cm

Thesis/Dissertation Information

Degree:
Master's ( Master of Science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Electrical Engineering, CU Denver
Degree Disciplines:
Electrical Engineering
Committee Chair:
Sen, Pankaj K.
Committee Members:
Roemish, William R.
Bose, Tamal

Subjects

Subjects / Keywords:
Electric machinery, Induction ( lcsh )
Electric machinery, Induction ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 86-87).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by John Joseph Horak.

Record Information

Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
34331477 ( OCLC )
ocm34331477
Classification:
LD1190.E54 1995m .H67 ( lcc )

Full Text
AN ALGORITHM FOR DETAILED
INDUCTION MACHINE ANALYSIS
WITHOUT RECOURSE TO D-Q THEORY
by
John Joseph Horak
B.S., University of Houston, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
1995


This thesis for the Master of Science
degree by
John Joseph Horak
has been approved
by
William R. Roemish
Tamal Bose
4-13 -9 5
Date


Horak, John Joseph (M.S., Electrical Engineering)
An Algorithm for Detailed Induction Machine Analysis Without
Recourse to D-Q Theory
Thesis directed by Professor Pankaj K. Sen
ABSTRACT
This thesis presents a detailed model of a squirrel cage induction machine
along with the procedures required to implement the model numerically. The
main contribution of the thesis is to set up the equations for a machine model that
fills in the gap in machine modeling between D-Q models and finite element
analysis.
The study begins with a development of how a squirrel cage induction
machine is built, stressing factors associated with stator winding design, two layer
squirrel cage rotor design, flux distribution throughout the machine, the sources
and effects of lower order harmonic flux, saturation of the machine core, and
saturation of the leakage flux paths. After machine winding and squirrel cage
design is reviewed, the equations necessary for calculating machine inductances
under the presence of saturation and harmonics are developed. Thereafter, the
differential equations that represent machine operation are provided. Finally, a
numerical methods solution technique is presented that is capable of interfacing the
machine model with various other common system components, such as line
impedances and motor capacitors. The numerical calculation procedures are
modeled after the techniques used in the software EMTP.
The model presented is not carried as far as a working program for reasons
related to the complexity and difficulty of the program required. At each step of
the solution (several thousand steps per second) it is required that the program
m


perform extensive numerical integration to find inductances, invert large matrices,
perform extensive data manipulation to set and solve the current and voltage
equations, and perform the calculations required to track flux distribution to
calculate saturation and torque. The programming effort required is left as a
future contribution to the field of motor analysis.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Sign
iv


CONTENTS
CHAPTER
1 INTRODUCTION.....................................................1
2 MODELING THE SQUIRREL CAGE INDUCTION MACHINE . 8
2.1 Introduction.....................................................8
2.2 Machine Construction.............................................9
2.2.1 Stator Construction .............................................9
2.2.2 Rotor Construction .............................................14
2.2.3 Current, Flux, and Voltage Variables .......................... 16
2.3 Flux Distribution and Machine Construction..................... 22
2.3.1 Air Gap Flux Due to Ideal Single Conductor..................... 22
2.3.2 Air Gap (Mutual) Flux Due to Stator Currents................... 28
2.3.3 Air Gap (Mutual) Flux Due to Rotor Currents.................... 31
2.3.4 Leakage Flux....................................................35
2.3.5 Saturation......................................................38
2.3.5.1 Core and Tooth Body Saturation .................................39
2.3.5.2 Leakage Flux Path Saturation....................................43
2.4 Inductance Matrix L.............................................46
2.4.1 Stator-Stator Inductances.......................................48
2.4.2 Stator-Rotor Inductances........................................49
2.4.3 Rotor-Stator Inductances .......................................51
2.4.4 Rotor to Rotor Inductances......................................52
2.5 Resistance Matrix R.............................................56
2.6 The Simplified Resistance and Inductance Matrix................ 57
2.7 Torque..........................................................60
v


3 APPLICATION OF EQUATIONS TO NUMERICAL SOLUTION 63
3.1 Introduction.................................................63
3.2 The Calculation Procedure....................................63
3.3 The Machine Inductances......................................67
3.4 Current Equations............................................67
3.4.1 Lumped R, L, and R-L Element Current Equations.............. 72
3.4.2 Lumped Capacitor Current Equations ......................... 73
3.4.3 Simple Line or Cable Current Equations -...................74
3.4.4 Induction Machine Current Equations ........................ 76
3.4.5 Current Equations for Network as a Whole.................... 77
3.5 Voltage Equations............................................78
3.6 Equations of Motion......................................... 80
3.7 Flux Wave Position and Magnitude Estimation ................ 81
3.8 Incrementing the Time Step ..................................82
4 CONCLUSIONS..................................................83
REFERENCES ..........................................................86
vi


FIGURES
Figure
2.1 Simplified Machine....................................................10
2.2 Possiblestator Winding................................................11
2.3 Flux Orientation for + Current....................................... 13
2.4 Simplified Winding Diagram of a Four Pole Machine.................... 13
2.5 Rotor Cage............................................................15
2.6 Close-Up of Rotor.....................................................15
2.7 Idealized Air Gap Flux from Simple Stator Conductor.................. 24
2.S Effect of Return Current 180 and 90 away .......................... 25
2.9 Idealized H Field for Phase A Winding of Fig. 2.2.................... 30
2.10 Concept of Current Summation for Determining Rotor Flux.............. 32
2.11 Rotor Skew ...........................................................33
2.12 Flux Dist. Seen by Stator, With and W/O Skew......................... 34
2.13 Tooth Shape With and W/O Skew ...................................... 35
2.14 Typical Leakage Inductance Paths......................................36
2.15 Areas of maximum Flux Density for 2 Pole Machine..................... 40
2.16 Flattened Flux Wave Under Saturation............................ . 41
2.17 Bu Vs Bag for Calculation of Core Flux Levels........................ 43
2.18 Tooth Tip Saturation .................................................44
2.19 Leakage Reactance Decrease During High Currents...................... 45
2.20 Rotor Voltage Drop Summation Loops and Current Nodes................. 57
3.1 R-L-C Circuit Element ................................................68
3.2 Four Node Network.....................................................79


ACKNOWLEDGEMENTS
I wish to thank my professors at the University of Colorado at Denver for
all they have taught me. I wish to especially thank the professors that I have had
in the Electrical Power field, Dr. Pankaj K. Sen and Dr. William R. Roemish.
Dr. Sen and Dr. Roemish have provided me with education that is second to none
I will benefit from and be thankful for their high quality teaching for the rest of
my working years.
Vlll


CHAPTER 1
INTRODUCTION
This study presents a relatively detailed model of an induction machine
along with the procedures required to implement the model numerically. The
study started out of a desire to develop a thorough understanding of how an
induction machine works. It grew to become an interest in determining how one
would model the induction machine or any other type of electric machine if one
wanted greater accuracy and detail than is obtainable in commonly used machine
modeling techniques, along with a desire to understand how any electrical system
is modeled numerically. The model that was desired would allow one to see
machine response via a physically accurate model of the entire machine and one
not restricted by the limitations of D-Q machine models (much less single phase
equivalent circuit models). It was desired however to do this without entering the
extreme complications of finite element analysis.
In the process of preparing this study the objective of developing a good
understanding of how an induction machine works was reached* as is hopefully
evident from the material to follow. The objective of developing a detailed
machine model as well as the numerical methods to implement the model was
obtained also, but to date the model has not be tested via a conversion to an actual
computer program. The study reviews machine design as far as necessary to
develop and present the algorithm for the detailed machine model that was
envisioned. After the equations representing machine inductances are provided, a
numerical computer algorithm for implementing the model is presented.
1


It became clear that the model developed was quite computationally
intensive and implementation in a computer program would require an extensive
programming effort. It was decided to leave verifying the accuracy the model via
implementation in an actual computer program to future work.
High detail in electric machine modeling (either synchronous, induction, or
DC type) is complicated by the fact that the machine winding to winding and even
self inductances are dependent upon rotor position. The beauty of D-Q machine
models (D-Q is the abbreviation that is used herein for the type of analysis
referred to variously as Direct-Quadrature, Parks, Reference Frame, and/or Two
Axis analysis) is that with the appropriate simplifications (which are usually quite
reasonable) the inductances appear as constants in the model. The D-Q models
are good for applications where one desires greater detail on the response of a
machine and system to which it is connected than is available in single phase
equivalent circuit analysis. As a result D-Q models have been developed and are
used widely in advanced analysis for almost any type of electric machine.
To avoid the problem of needing to provide a description of D-Q
modeling, this study is written so that it is not necessary for the reader to be
knowledgeable about D-Q models. The models are described in fairly high detail
in [1] and [2].
As useful as the D-Q machine model may be, there are limitations to the
D-Q models that may make one consider models of greater detail. The limitations
of a D-Q model includes:
Modeling machine winding imbalances and non-fundamental sinusoidal
factors is not easily done.
It is difficult to model the effects of harmonics in the line voltages,
harmonics in high frequency current transients, and harmonics in flux
2


generated by the machine windings.
Saturation modeling used in the D-Q models is not very accurate and it
is difficult to model the effects of the harmonic flux generated by
saturation.
The D-Q models are based upon a wye connected stator where the
floating neutral is effectively at ground potential. It takes special
versions of the model (e.g., [3] and [4]) to show the neutral shift of
wye connected stator machines and circulating current of delta
machines, and there are limitations to the accuracy of this special
model (it allows only for third harmonic voltage shift/circulating
current, and assumes that the third harmonic is in phase with the
fundamental).
The squirrel cage rotor cannot be directly modeled in D-Q models, so a
roughly equivalent 3 phase wound rotor must be used instead.
(Similarly, in synchronous machine D-Q analysis the damper bars are
modeled by an roughly equivalent wound coil.)
Analyzing the effects of saturation and harmonics on the induction machine
has been the object of considerable study over the years. A very small cross
section of the material that has been published is provided in the references [1]-
[13] all touch on some aspect of the analysis in a piecemeal effort. A perusal of
the material and the numerous references within these articles will show that the
analysis is piecemeal; on study touches on leakage reactance saturation; another
may study core saturation; another on some aspect of harmonics; nothing in the
list is comprehensive, not even the textbooks. Harmonics and saturation is a
complicated subject and to try to reduce it to some simple term is difficult and
inevitably introduces errors in some manner. The very existence of this situation
3


in the literature was a motivation for the model that was developed; i.e., find a
means of modeling machines in a comprehensive way.
The model herein is aimed at overcoming at least some of the D-Q model
limitations together with a providing a model that can handle machine saturation
and harmonics in a more detailed manner than D-Q analysis. This is achieved,
though, at the price of the modeling algorithm and the associated equations
becoming relatively complex. The model involves what may be called a "brute
force" tactic, using a minimum of simplifications. The model is as close as
practical to a physical model of the machine while still using an electric circuit
view of a machine (as opposed to a finite element analysis). It includes an
analysis of many of the secondary factors of machine performance that are
difficult to include in D-Q models.
It will become evident in chapters 2 and 3 that the model is relatively slow
(at least compared to a D-Q model run on the same computer) and much more
difficult to program. The relatively high amount of computations required
severely limits the application of this model.
The model developed herein is limited to the analysis of only one type of
machine, the squirrel cage induction machine. The process described for the
model could be re-written and used to model other machine types, such as the
synchronous machine or the various forms of DC machines.
A summary of the major aspects of the model that is presented follows:
It was decided that for accurate representation of the machine it is
necessary to accept the task of constantly recalculating machine
inductances as the rotor rotates. The equations needed for constant
recalculation of inductances for a very detailed physical representation
of the machine are provided.
4


To maintain the detail in the model that was desired, it was decided to
develop a fairly accurate physical representation of the machine,
including the stator windings, the rotor conductor network, the deep bar
effect, core (therefore mutual) flux saturation, and leakage flux
saturation.
A detailed physical representation of each stator phase winding is
utilized. Machine flux distribution from each phase winding is
considered separately, including a representation of stator winding flux
(i.e., space) harmonics (up to the 7th harmonic is as far as is
considered reasonable).
A detailed physical representation of the rotor is utilized. The rotor
model is not the usual "equivalent" three phase wound rotor, but an
actual model of the rotor squirrel cage network. Machine flux
distribution from each rotor conductor is analyzed including a model of
the deep bar effect. Because of the physical representation of the rotor
a more intuitive (at least as compared to D-Q analysis) modeling of the
effects of various rotor designs is possible.
A fairly detailed representation of machine non-linear permeability
(i.e., saturation) is presented. Non-linearity even at nominal operating
conditions and the localized nature of non-linearity are considered.
This includes the effects of non-linearity in:
- the leakage flux paths (i.e., at the tooth tips, where saturation is
caused by high machine currents), and
- the stator/rotor mutual flux paths (i.e., in the stator and rotor
body, where saturation is caused by high line voltages).
The model presented is capable of analyzing a wide variety of machine
5


internal and external imbalances and discontinuities, such as unbalanced
stator windings and damaged rotor bars.
A set of differential equations to represent the voltage and current
relationships in the machine is presented. The equations take the form
V = Ri + p(Li).
A numerical differential equation solution method is presented for
numerically tracking the machine response (i.e., "solving" the
equations). The calculation methods are based upon those used in the
software EMTP (Electro-magnetics Transients Program). Equations are
given for modeling lumped impedances, multi-phase power lines, and
capacitors, so that these factors which affect a machines performance
can be included in the model. A means of creating a system of
equations that represent the machine and the system to which it is
attached is presented. Given enough attention to setting up the
equations, the procedure that is presented is capable of modeling each
phase individually, modeling each stator and rotor conductor
individually, modeling line and system, impedances, line capacitors,
delta and wye voltage sources, and delta and wye machine connections;
i.e., the procedure was designed for generality.
The model herein is of fairly specialized interest so the development of a
model similar to the one herein would not be widely publicized. Therefore while
research on motor modeling did not uncover a model such as herein, one may
have been developed. Also, while the model developed is a combination of
individual facts each of which someone well versed in machine design and
numerical methods would be aware of, on its whole the model should be new
material to most people.
6


While the modeling herein is fairly detailed, it cannot be said that the
model is exhaustive and that there are not ways that the model may still be
improved. There are many simplifying assumptions used throughout the model
that affect accuracy, particularly when analyzing flux distributions. Some of the
simplifications include assuming a conductor creates a square wave flux
distribution in the air gap, approximations on flux saturation functions,
approximations for machine dimensions, ignoring higher order harmonics (the
model herein would not be suitable for analyzing voltage, current, and flux
harmonics much above the 7th), ignoring flux generated by the rotor end rings and
end components of the stator windings, ignoring the effects of slot harmonics,
ignoring the effects of hysteresis, ignoring the effects of eddy currents, and
ignoring the effects of machine heat on machine dimensions and parameters.
7


CHAPTER 2
MODELING THE SQUIRREL CAGE INDUCTION MACHINE
2.1 Introduction
Voltage and currents in an induction machine, or any electric machine,
have an interrelationship of the following form:
>/ X 0 " Y + L 'V
Vr_ 0 dt _K_
(volts)
(2.1)
where flux (X) is
L L
ss sr
L L
JS IT
Is
(webers)
(2.2)
where:
Vs = Stator voltage matrix
Vr = Rotor voltage matrix
Is = Stator current matrix
Ir = Rotor current matrix
\s = Stator flux matrix
Ar = Rotor flux matrix
Lss = flux linkages of the stator windings due to current in the
stator windings.
= flux linkages of the stator windings due to current in the
8


rotor conductors.
Ln = flux linkages of the rotor teeth and conductor bars due to
current in the stator windings.
L". = flux linkages of the rotor teeth and conductor bars due to
current in the rotor conductors.
One of the main tasks of the balance of this chapter will be to better define
and elaborate on the parameters and equations that lie underneath these
relationships and provide means of finding values for the parameters involved.
The other main tasks of this chapter are to develop methods of relating flux and
current to rotor torque, acceleration, speed, and position.
2.2 Machine Construction
In order to "flesh out" the parameters and relationships hidden beneath
(2.1) and (2.2) a short review of induction machine construction is required. The
following description tries to run a balance between providing the needed
description of construction so that currents, flux linkages, resistances, and
inductances can be derived but without providing, so much information that this
becomes a study on machine design.
2.2.1 Stator Construction
Assume a two pole, three phase machine. If the machine had only one coil
(which consists of many turns wrapped together) per phase winding it would look
somewhat like fig. 2.1. Note there is one coil fitted into each stator slot and that
there are 6 slots but only 3 coils (i.e., there are half as many coils as there are
slots). Note the coil sides are 180 apart magnetically and physically.
9


LEFT END VIEW
CROSS SECTION
AA
BB
FIGURE 2.1 -SIMPLIFIED MACHINE, SINGLE LAYER, 1 COIL PER PHASE WINDING.
In a real machine, though, there would be several coils connected in series
in each phase winding. A possible phase winding looks somewhat like fig. 2.2.
The coils of winding A, as seen from the left end of the machine, are shown.
Typically the 8 coils in the phase winding are connected in series, but a parallel or
parallel/series configuration such as 2 parallel sets of 4 coils in series, is possible.
The possible existence of parallel paths notably complicates the machine model for
computer implementation because this increases the complexity of calculating
machine flux distribution by increasing the stator current variables by the number
of parallel paths. There was no attempt herein to model parallel connected stator
coils, but the model could be modified for such a project.
The configuration shown is for a machine with a pitch factor of .99, a 60
phase "belt," and a "two layer" winding. Note that a phase winding is a series
connected group of 8 coils spread out over about 60 (referred to as a 60 phase
belt). There are some winding designs that use a 120 phase belt, but per [5] this
10


FIGURE 2.2 -POSSIBLE STATOR WINDING 8 COILS/PHASE, TWO LAYER.
60* PHASE BELT, 165* COIL SPAN, 24 SLOTS.
is uncommon. Note also that there are two coils fitted into each slot so in this
machine there are as many coils as there are stator slots. Using a two layer
winding allows coil designs where each coil actually spans a bit less than 180,
and is the normal practice in induction machine design. To avoid this becoming a
study in machine design, the explanation of why a coil pitch of less than 180 is
used will be limited to stating that it centers around reducing the space harmonics
in the stator flux field, reducing flux leakage around the ends of the coils, and
11


reducing the amount of copper needed to wind the machine. On the negative side,
if a coil does not span the full 180 that it could, less voltage is induced in it by a
rotating field than would be induced if it did span the full 180. See [5] and [6]
for a detailed discussion of why spans of less than 180 are used. The term "pitch
factor" is sometimes used in conjunction with the level to which the coils in a
machine span less than 180. Pitch factor is defined by the level of reduced
voltage induced in a phase winding by a fundamental sinusoidal flux wave. It
may be calculated from (see [5] and [6] for the derivation):
K
P
(;numeric)
(2.3)
where:
R = span of single coil; e.g., in fig 2.2 R=165.
It will be necessary to have a convention for positive rotation and positive
flux flow in the stator. Figs. 2.1 above and 2.3 below show the convention used
herein for the orientation of the a, b, and c phase flux. Positive flux was taken to
be from stator to rotor when positive current flows in the winding. The a, b, and
c phases are oriented so that if an abc sequence voltage is applied to the machine
the machine will rotate in a counterclockwise direction (which is therefore
considered the positive direction for rotation) when viewed from the "left" end as
indicated in fig. 2.1. The definition of positive flux, current, and rotation is
needed when inductances are defined.
The windings described above were for a two pole machine. A 4 or higher
pole stator would be similarly designed, but there are complications to the system.
This study will generally stay with analyzing a two pole machine in order to keep
the analysis to a minimum, but some discussion on 4 pole and higher machines is
12


90-
FIGURE 2.3 -FLUX ORIENTATION FOR + CURRENT IN PHASE WINDING AS VIEWED
FROM LEFT END PER FIGURE 2.1 AND 2.2
360* MAGNETIC
180* MECHANICAL
FIGURE 2.4 -SIMPLIFIED WINDING DIAGRAM OF A FOUR POLE MACHINE.
appropriate. In machines with more than two poles the coils associated with a
given winding take on a configuration somewhat like shown in fig. 2.4. Instead
of spanning about 180 physically as in a two pole machine, the coils span about
360/n physically, where n is the number of poles, but they still span 180
magnetically. In the figure it can be seen that two north or two south poles are
180 apart physically, but the coils sides are about 90 apart physically. The
windings associated with a given phase can be connected together to form the
13


needed number of poles by connecting them in series, parallel, or series/parallel
configurations. For instance if the machine in fig. 2.4 were real, the at and %
coils could be connected in series or parallel, depending of the appropriate voltage
rating of the windings.
Equations for flux distribution for 4 or higher pole machines are more
complicated that for a 2 pole machine since in some locations in the air gap the
stator coil produces theoretically no flux, as will be discussed in section 2.3.1.
First, the equations of air gap flux must be stated with if/then conditions in
software simulations. Second, when windings are wired in parallel for every 2
poles the number of unknown currents increases by two (i.e., for a four pole
machine where on each phase two windings are wired in parallel the possible
currents are ial, ia2, ibl, ib2, icl, and ic2, but if windings are in series ial=ia2, ibl=ib2,
and ici=ic2).
No attempt is made herein to model a 4 or higher pole machine.
However, the modeling procedure herein could be modified for such a project.
2.2.2 Rotor Construction
The "equivalent" 3 phase winding rotor model typically used in squirrel
cage induction motor models is not used in this study. A rotor model that is
physically representative of the actual machine rotor is used: i.e., the resistance of
each rotor bar, the voltage induced in the conductive path around each rotor tooth,
and the current in each rotor bar is modelled. To do this requires a large set node
voltages: one node for each end of each rotor bar. To represent the deep bar
effect each bar is divided into two halves: a outer bar (i.e., closer to rotor
surface), and a inner bar (i.e., closer to rotor center), yielding two current paths
between each end of a rotor bar. The deep bar effect is caused by flux passing
14


through the rotor bar and inducing a voltage in the bar that tends to shunts current
to the conductor surface.
The makeup of the rotor node network may be best seen by viewing figs.
FIGURE 2.5 -ROTOR CAGE, SHOWING CONDUCTORS. SPACE BETWEEN
CONDUCTORS IS ROTOR TEETH/CORE STEEL
LEFT RIGHT
FIGURE 2.6 -CLOSE-UP OF ROTOR SHOWING FLUX, CURRENTS.
15


2.5 and 2.6. Fig. 2.5 provides an overall view of the rotor, while fig. 2.6 shows
a closeup of a few rotor bars. Fig. 2.6 provides information on current, voltage,
and flux nomenclatures and conventions for positive flux and current flow (arrows
are shown in the direction of positive flow). The variables shown in the figure
are defined in section 2.2.3 below. Note in fig. 2.6 that the n* rotor tooth is
considered to be the tooth directly clockwise adjacent to the n* rotor bar when
viewed from the left end (as shown in fig. 2.1). Note also that for every physical
rotor bar there are four unknown currents that eventually need to be solved for:
l/d) ^nr
2.2.3 Current, Flux, and Voltage Variables
With the above description of the stator and rotor it is possible to provide
some details on the current, flux, and voltage variables of (2.1) and (2.2). The
fully developed equations are provided in (2.4) and (2.5) on the following pages.
Recall that the modeling herein (and therefore (2.4) and (2.5)) assumes a two pole
machine with the phase winding coils connected in series.
That there are more unknowns currents than flux equations to be solved for
in (2.5) at first appears a bit odd. But it is in (2.4), not (2.5), where there must
be an equation for every unknown current. Equation (2.5) serves to define the
voltages used in 2.4, and therefore does not violate the number of equations vs
number of unknowns rule. It should be noted that in (2.4) the lines that have "1"
four times are associated with summing currents at nodes in the rotor to zero.
These equations are needed because as the equations are set up, for each rotor bar
there are four unknown currents but only two voltage loop equations. Look ahead
to fig. 2.20 for information on rotor node and rotor current loop definitions.
It should also be pointed out that one of the current summation equations
16


associated with (2.4) must be replaced with another equation of voltage (e.g., an
equation based upon the fact that the summation of the voltage around an end ring
must equal zero) or else an erroneous / non-singular system of equations will be
set up. The can be see by realizing that if all the current summation equations in
(2.4) are added together they add to zero. By replacing one of the equations with
the voltage loop around the end ring this situation is avoided.
Some of the parameters in the list below will later be assumed to be zero.
Some of the parameters will be assumed to be effectively equivalent to one
another. Some will be determined fairly easily. However, there are several
parameters that will involve heavy amounts of computations. The determination
of these parameters will be developed in sections to follow. The reduced forms
for (2.4) and (2.5) will be provided in section 2.6.
The variables (and the associated subscripts in the definitions) in (2.4) and
(2.5) are defined below. As previously discussed the physical orientation and
numbering of the flux and conductors is defined by figs. 2.1, 2.2, 2.3, and 2.6.
Subscript Definitions:
x, y Variables that are used to represent a, b, or c where a variable
may apply to any of the three stator phases.
m Variable that may take on rotor tooth number.
n, p Variables that may take on any rotor conductor bar or end ring
section number.
i "Inner," referring to inner portion of rotor conductor.
o "Outer," referring to outer portion of rotor conductor.
Z "Left," referring to left end ring.
r "Right," referring to right end ring.
t "Tooth," referring to a rotor tooth.
17


I
d "Deep bar," referring to cross bar flux that causes the deep bar
effect.
Variable Definitions:
vx = Voltage across phase x stator winding (volts).
ix = Current in phase x stator winding (amps).
iM = Current in n* rotor bar, outer portion (amps).
iM = Current in n* rotor bar, inner portion (amps).
= Current in ntt section of the "left" rotor end ring (amps).
im = Current in n* section of the "right" rotor end ring (amps).
Rx = Resistance of phase x winding (ohms).
= Resistance of n* rotor bar, outer portion (ohms).
R^ = Resistance of n* rotor bar, inner portion (ohms).
Rrf = Resistance of n* section of the left rotor end ring (ohms).
R^ = Resistance of n* section of the right rotor end ring (ohms).
\x = flux linkages of phase x stator winding (webers).
= flux entering the m* rotor tooth from the air gap (webers).
= flux crossing the n* rotor bar (for deep bar effect) (webers).
L.t Stator flux due to Stator currents (henrvsl
Lx.x = flux linkages of winding x due to current in winding x.
Lx.y = flux linkages of winding x due to current in winding y.
L- Stator flux due to Rotor currents (henrvs')
= flux linkages of stator winding x due to current in outer
portion of n* rotor bar.
Lx.ra- = flux linkages of stator winding x due to current in inner
portion of n* rotor bar.
18



Lx.ri = flux linkages of stator winding x due to current in n* section
of the left rotor end ring.
Lx.m = flux linkages of stator winding x due to current in n* section
of the right rotor end ring.
L Rotor flux due to Stator currents fhenrvs')
Lm winding x.
= flux crossing the n* rotor bar due to current in stator
winding x.
L_ Rotor flux due to Rotor currents fhenrvs^)
Lm.no flux entering the m* rotor tooth a due to current in the outer
portion of the n* rotor bar.
flux entering the m* rotor tooth a due to current in the inner
portion of the n* rotor bar.
flux entering the m* rotor tooth due to current in the n*
section of the left rotor end ring,
flux entering the m* rotor tooth due to current in the n*
section of the right rotor end ring,
flux crossing the n* rotor bar due to current in the outer
portion of the p* rotor bar.
flux crossing the n111 rotor bar due to current in the inner
portion of the p* rotor bar.
flux crossing the n* rotor bar due to current in the p* section
of the left rotor end ring.
flux crossing the n* rotor bar due to current in the p* section
of the right rotor end ring.
mt-nl
nd-po
nd-pi
nd-pl
'nd-pr
19


to
o
V0 K 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 h
\ 0 Rb 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 h
0 0 0 0 0 0 0 0 0 0 0 0 0 0 h
0 0 0 0 K 0 Ru -K -K 0 0 0 0 0 0 0 ho
0 0 0 0 -K 0 0 0 0 0 0 0 0 0 0 hi
0 0 0 0 -1 -1 1 0 0 0 0 0 ... 0 0 -1 0 hi
0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 -1 hr
0 = 0 0 0 0 0 0 0 K 0 R21 -K ... 0 0 0 0 ho
0 0 0 0 0 0 0 0 -K ** 0 0 ... 0 0 0 0 hi
0 0 0 0 0 0 -1 0 -1 -1 1 0 ... 0 0 0 0 hi
0 0 0 0 0 0 0 -1 1 1 0 1 0 0 0 0 hr
0 0 0 0 -K 0 0 0 0 0 0 0 ... 0 ** -K L
0 0 0 0 0 0 0 0 0 0 0 0 ... -K Rni 0 0 hi
0 0 0 0 0 0 0 0 0 0 0 0 ... -1 -1 1 0 hi
0 0 0 0 0 0 0 0 0 0 0 0 ... 1 1 0 1 hr
\
K
K
A
dt
0
0
0
0
K
K
o
o
Note: The above eqns do not show that one of the current summation eqns must be replaced with an end ring voltage loop eqn.
(Eq. 2.4)
r


1 ft k6 5 'Nt 1 1 1 ft ik1 k *>** . <8 'ik . a . R k *k - .J . s 'k .J5 k ft ***

k k k
c C c ft F F F
.ft jO < 1 - ft o o <3 e o ... £ ft o o
k k k I k kl k kf k~ k2
_
s F F f F
.ft N ft 1 ft o o 5 O o a o o
k k k k kl k kf k~ k2
*5 f F 1 '5 *5 *ft *5 *ft F
.< 1 ^5 ft o a a o o a o o
k k k k k k k k~ k
g g § 1 g § § g g g
k k* i k 1 1 1 k ft k~ o k* k* a k o o
k CN cS k fS 1 * A k es k ?s k cs k s
k k0 k1. kf ft k" o k* ka o o 4 k k* o o
R R R R *? R ? a
k^ *4* ft k kf ft k~ o o k* J o - kS o o
a a ? a a s* a
^ft ft ft o o a a o o a a o o
<1 k k k k k k k~ k
<5 <5 1 <8 <8 <8 a <8
53 ft 1 L ft o o a a o a o o
k k k kl k k k k k*
k k k
* Y *Y Y
Hft a ft 1 ft o o a o a . a o o
k k k ki k k k k~ k
...
M 1) p
ft ^ft ft 1 . ft o o o o a o o
k k k 1 ki k k k k~ k~
£ 3 - U; £ u; *.3
k k6 ft k kf ft o k* ka o o a k k2 o o
ft ft
,H r*
53 ^ft 1 L o o a o o * a o o
k k k k k k k k~ k
o ft i ft ft ft ft ft ft
k' k1 k k" ft k" o o k* k^ o o a k k2 o o
*? *F ft i >ft ? ? ft ? ft
k kf ft k k" ft k" o o ks k3 o o k2 o o
ft F ft ft ft ft ft ft
1 k k'' k ft k o o k* k^ o o k2 o o 1
ii
1 ft /< 1 /< ft 0< o o a x: a /< o o a x: o o 1
21
(Eq. 2.5)


2.3 Flux Distribution and Machine Construction
Before deriving values for the parameters used in the induction matrices in
(2.2) and (2.5), it is necessary to expand upon stator and rotor construction to
determine how the flux generated by the machine currents is distributed. This will
also help in understanding the formation of the resistance matrices in (2.1) and
(2.4).
Mutual flux is defined herein as flux generated by a winding that links one
or more other windings. Leakage flux is defined herein as the flux generated by a
winding that does not reach another winding. To simplify the analysis, mutual
flux is analyzed in terms of air gap flux; it is assumed that all mutual flux crosses
the air gap, and vice versa all flux in the air gap is mutual flux. To say that all
air gap flux is mutual flux is not technically correct, as some leakage flux travels
across the air gap and back without entering a winding on the other side. The
exact path of the leakage flux is not important for the model herein because the
main effect of ignoring this leakage flux path will be to provide error in
determining the saturation levels of the machine. As modeling leakage flux as
well as the effects of saturation contains a substantial degree of estimation, the
simplification should have little effect on the model and will be considered
reasonable for the intended purpose. Therefore, air gap flux herein will be
assumed to be mutual flux.
2.3.1 Air Gap Flux Due to Ideal Single Conductor
The most basic source of air gap flux is the current in a single conductor
adjacent to the air gap. A simple plot of an ideal air gap flux distribution
resulting from current in this conductor may made if several simplifications to the
system are made:
22


Assume that the steel is infinitely permeable, so all the H field is found
across the air gap.
Assume that the conductor is infinitely thin, and that the conductor lies
directly on the surface of the steel.
Assume a flat unbroken steel surface (i.e., ignore the effects of teeth
slots, end effects, and eccentricity in the air gap).
Ignore the leakage flux, so only the air gap stator-rotor mutual flux is
shown on the flux plot.
The result of these simplifications is a square wave flux distribution in the
air gap, as shown if fig. 2.7 below. In the figure note that:
Flux magnitude in the air gap is constant except for a square wave
shaped change in direction at the conductors location and at a location
180 away mechanically from the conductor (the 180 is independent of
the number of machine poles). The flux magnitude is independent of
how close the flux is to the conductor.
Flux direction in the air gap is always perpendicular to the steel
surface. Flux does not bend in the air gap.
The total H field across the air gap induced by a conductor is equal to
1/2 the current in the conductor, irrespective of how far away from the
conductor the air gap H field is measured. The H field in the air gap is
proportionate to the current and inversely proportionate to the effective
length of the air gap, g0. The 1/2 factor arises because all flux crosses
the air gap two times.
If the current in fig. 2.7 returned in a conductor that was 180 away, the
flux distribution if fig. 2.8a results. If the current in fig. 2.7 returned in a
conductor that was 90 away, the flux distribution in fig. 2.8b results. Note the
23


9(T
27 (r
H<£>
1
1
0-----------
-1----------
2

(T 9(T 180* 270" 36(7
FIGURE 2.7 -IDEALIZED AIR GAP FLUX FROM SIMPLE STATOR
CONDUCTOR, IGNORING CURRENT RETURN PATH
areas in the air gap flux where there is no flux. When this analysis is applied to
the winding configuration of a 4 or higher pole machine, it becomes apparent that
there are some areas in the air gap where the stator winding theoretically produces
no air gap flux. This complicates programming because the analysis of air gap
flux becomes a series of if-then statements based upon determining which
windings produce flux in various portions of the air gap.
The real world is of course a far more complicated situation than shown in
24


90*
90*
0 0
0* 9ff 180* 270* 360* 0* 9(T 180 270" 360
(a) (b)
FIGURE 2.8 EFFECT OF RETURN CURRENT 180* AWAY AND 90* AWAY.
figs. 2.7 and 2.8 and this simplified analysis introduces several errors. The errors
include:
The flux level is modulated by the irregular shape of the air gap
surface. The irregularity is mainly due to conductor slots. These
modulations are usually referred to as slot harmonics. The slot
harmonics can cause notable drag on machine performance in a poorly
designed machine, resulting in the phenomena "synchronous crawling"
and "synchronous locking" (see [5] and [6] for descriptions).
The air gap surface irregularity complicates finding an effective air
gap, g0. In [5] and [6] some half theoretical/half empirical means are
provided for determining an effective air gap. They take into account
25


the effects of the irregular shape of a real world air gap using a factor
referred to as the Carter coefficient. The source of the coefficient is
conformal mapping of a simplified air gap surface, and it must be
modified by an empirical factor based upon saturation effects. The
results provide first only order accurate values for g0.
Steel is not infinitely permeable so some of the H field is developed in
the steel, especially a factor when saturation occurs. Since it is
assumed herein that all of the stator-rotor H field is developed across
the air gap, it is necessary to modify the effective air gap with
changing flux levels to represent the portion of the H field developed in
the steel. As average flux levels rise and fall the average permeability
of the machines steel changes, which changes the effective width of
the air gap. Since the flux levels around the air gap at any give point
in time vary in approximately a sinusoidal fashion, the effective air gap
will change with local flux levels, even when flux levels are very low.
When flux levels rise to cause heavy saturation the problem becomes
even more apparent.
The effect of a varying level of saturation on the air gap is
discussed in section 2.3.5 and a means of modeling the effect is
presented. As a summary to that material: To model core saturation
g0 is multiplied by a saturation function (called Sm herein) in a manner
dependent on the local air gap flux. The effect of the saturation
function is to always effectively increase the air gap to a value larger
than g0 and thereby model reduced permeability of the machine steel.
The air gap is not increased uniformly but is tied to local flux levels.
The effective air gap at the ends of the machine is difficult to quantify.
26


Flinging of the flux will tend to make the effective air gap shrink a
little at the ends of the machine. However, the effect of the fringing on
the overall effective air gap should be small.
A hot machine will have a slightly different air gap than a cold
machine. No attempt was made to determine a means of representing
the change in air gap with temperature. The appropriate coefficients of
expansion may be hard to come by. The largest effect would come
from the rotors, since they tend to run hotter than stators. A hot stator
would tend to increase the effective air gap, but a hot rotor would tend
to decrease the effective air gap.
It is not possible to have the rotor sit exactly in the middle of the stator
opening. Some errors in manufacturing and sag in the middle of the
rotor due to its bending under its own weight are bound to occur. As a
result a real world air gap is at least a small bit eccentric. Fig. 2.6 and
2.7 assume no eccentricity in the air gap. If eccentricity did exist the
flux crossover point would shift away from the point in the air gap 180
mechanically opposite the conductor and the air gap flux would not be
uniform. There is no attempt herein to model eccentricity in the air
gap. However, it could make a substantial change in machine
operating characteristics.
The square wave type sharp change in flux direction is highly idealized.
In reality it is a more sloped change around the points where flux
changes direction would occur, especially at the 180 point.
Current does not flow in an infinitely small conductor on the rotor or
stator surface. This assumption introduces error in flux distribution
and torque calculations. The nature of real world flux distribution is
27


discussed 2.7 when torque is analyzed.
2.3.2 Air Gap (Mutual) Flux Due to Stator Currents
The flux generated by winding would ideally produce only fundamental
flux waveform, but there is inevitably a harmonic content to the wave. The net
air gap H field generated by a winding is a function of what will be called herein
as the "Winding Distribution Function" and abbreviated as Wx(<£). The winding
distribution functions have the form of (units are numeric):
Wa = Nt[iVjcos(<£) + N3cos(30) + Nscos(5) + N7 cos(7<£) ] (2-6)
Wb = Ar[ATcos(0-12O) + iV3cos3(0-120) +iV5cos5(<£-120) + W7cos7 (0-120)] (2-7)
Wc = Ar[ATcos(0+12O) + W3 cos 3 (0+120) + AT5cos5(<^+120) + AT7cos7( where:
Nt = number of turns per coil times the number of coils per pole
winding, and
Nj, N3, Ns, and N7 are the winding harmonic factors that may be
determined from a Fourier analysis of the winding waveform. A
means of determining will be discussed below.
While terms above the 7th harmonic could easily be included, it does not
appear justified because they would not seem to provide real increases in
accuracy given all the other simplifications in the model (such as ignoring slot
harmonics, which can interact with stator space and line voltage harmonics to
cause substantial parasitic torques) and the errors inherent in the multiple "good
engineering estimations" needed for so many parameters herein.
The above equations indicate the same winding harmonic factors for each
28


phase. This may not be a justifiable assumption for all motors. Winding
imbalances do exist in some motors, but pinpointing the appropriate factors for
each winding would be no small task. The analysis herein is quite capable of
accepting different factors for each winding, if appropriate numbers could be
determined.
The air gap flux density that a winding will generate for a given current
level depends on the winding distribution function, saturation levels, the effective
air gap, and current in the winding. The equations that represent this are as
follows:
BJ&) =
Sn,8o
(;tesla)
(2.9)
W =
Sm8o
(tesla)
(2.10)
B&) =
p0wmc
sm8o
(tesla)
(2.11)
where:
fj,0 = the permeability of free space
Sm = a saturation function to be discussed later.
gG = the nominal air gap under unsaturated conditions.
Values for AT2.7 may be found by analyzing the flux distribution of a
winding. Assuming 1A in phase A of the machine in fig. 2.2 and using the
simplifications used to create fig. 2.7, plots of air gap H field may be made, as
shown in fig. 2.9.
29



FIGURE 2.9 -IDEALIZED H FIELD FOR PHASE A WINDING OF FIGURE 2.2
The flux shown in fig. 2.9 for was analyzed for its space harmonic factors
using a Fourier Transform software package (part of the software MathCad) to
determine the harmonic content of the waveshape and thereby determine the stator
winding harmonic factors for the winding. Through the 13th harmonic, the
factors (when using a cosine function for modeling) are:
Ni = -9.674
n3 = +2.056
n5 = -0.338
N7 = -0.138
n9 = +0.118
Nn = -0.016
n13 = +0.012
The assumed phase angle is 0 if N is positive and 180 if N is negative.
30


Note that the space harmonic factors decrease as the harmonic number increases.
Space harmonic factors above the 7th should not commonly be very large, and
determining the values above the 7th with good accuracy would require a more
rigorous winding and machine model than used to create fig 2.9. There is no
attempt herein to model the stator windings space harmonics above the 7th order.
2.3.3 Air Gap (Mutual) Flux Due to Rotor Currents
The flux generated by the current in a rotor bar may also be thought of in
terms of winding distribution factors. Current in a rotor bar also has a winding
distribution factor, but the equation is much simpler than for stator currents. One
factor that must be considered is that the flux generated by current in a rotor bar
must cross the air gap two times, which means that for 1A of rotor bar current the
H field across one air gap is about 1/2 A. Furthermore, using the assumptions of
fig. 2.7, the flux is positive for nb to 4>ab + x (mechanical degrees) and negative
for the other half of the air gap. Note fig. 2.7 was drawn for stator
currents; rotor currents would result in a similar distribution of flux. Therefore,
the rotor winding distribution factor is simply:
+1/2 for 0 = 0* to + tr)
-1/2 for = (4^+ x) to (4>A + 2x)
(numeric)
(2.12)
The net air gap B field that is induced by the cumulative effects of all the
rotor currents is a fairly simple computation when the above equation is accepted
and the simplification that current in the outer bar is as capable of producing air
gap as current in the inner bar is accepted. The result is that the H field at a
given point (0) in the air gap is a function of the sum of the currents between 0nb
and 0nb-18O (mechanical) as shown in fig. 2.10.
31


Z lr(0i>SUMMATION OF ROTOR
CURRENTS IN THIS AREA
FIGURE 2.10 CONCEPT OF CURRENT SUMMATION REQUIRED
FOR DETERMINING ROTOR INDUCED FLUX
Ignoring the effects of rotor skew, the air gap flux density due to rotor
currents is simply the H field times fi0, or:
W = l I'M) {tesla) (2.13)
where ET(<£) is determined from the conceptual picture of fig. 2.10.
It is worth noting that the 1/2 in the winding distribution function W does
not show up because the currents in EI'() will return in the other half of the
rotor, thereby doubling the H field and cancelling the 1/2 factor.
Rotor Skew Effects
If there is rotor skew there are some notable complications to the picture
given by fig. 2.10. Rotor skew is the angular twist along the rotor axis {4>s
degrees) introduced in some types of rotors (i.e., cast constructed rotors, usually
limited to smaller machines, say under 100 hp) to make the rotor teeth appear as
non-salient to the stator and reduce the amount of harmonic currents induced in
the rotor bars by stator winding space harmonics and stator slot harmonics. Rotor
skew is shown in fig. 2.11 below.
32


END RINGS
FIGURE 2.11 ROTOR SKEW
As seen by the stator, the effect of rotor skew is to make the shaip square
wave flux distribution from current in a single rotor Conductor as seen in fig. 2.7
become more sloped and to effectively reduce the flux entering the stator by the
amount indicated in fig. 2.12. In section 2.4, when the calculations for stator-
rotor inductance are presented, skew is be modeled by reducing the limits of
integration associated with stator-rotor inductance by 1/4 of the rotor skew s.
33


Bo
-Bo
Bo
-Bo
2
-x= Bo* I
WITHOUT SKEW
WITH SKEW
FIGURE 2.12 FLUX DIST. SEEN BY STATOR DUE TO 1 ROTOR CONDUCTOR,
WITH AND WITHOUT SKEW
As seen by the rotor, the effect of rotor skew is that each rotor tooth sees
the "average" air gap flux density over the angular span of the skew, as shown in
fig. 2.13 below. A detailed average would entail integrating the air gap flux over
the surface area of the rotor tooth. However, a simplified average may be used
that should provide results that are "close enough." The average flux density may
taken as simply:
BJ#) l[B( + .25s)+B(-.25s)} (tesla) (2.14)
The skew factor will be ignored herein for use in calculating saturation
levels but it will be used in section 2.4 when the calculations for stator-rotor and
rotor-stator inductances are presented.
34


V/////////////////////A -jr
WITHOUT SKEW
Bavg *(B(a)+B(b))/2
FIGURE 2.13 TOOTH SHAPE WITH AND WITHOUT SKEW
2.3.4 Leakage Flux
Stator leakage flux is flux generated by a stator winding current that links
only that stator winding and that does not enter deep enough into the rotor to
induce any voltages in the rotor bars. Rotor leakage flux is defined herein as any
flux generated by rotor currents that does not enter deep enough into the stator to
induce any voltage in the stator windings. Leakage flux paths are shown in fig.
2.14. The level of leakage flux is difficult to determine analytically. But while it
is likely that a machine designer has access to some analytical methods for
calculation leakage flux levels solely from design data, it is likely that the designer
must also use empirical methods and past experience.
Stator leakage flux parameters for the model herein may be taken directly
from the single phase equivalent circuit, and as measured values are used, the
35


FLUX CAUSES
DEEP BAR EFFECT
FIGURE 2.H TYPICAL LEAKAGE INDUCTANCE PATHS
value should be somewhat accurate. The rotor model herein, however, is very
different from the single phase equivalent circuit, which makes it difficult to use
measured rotor leakage reactance. Rotor leakage flux levels in the model herein
are best drawn from the physical parameters of the machine, as further described
in the inductance equations provided in section 2.4. The accuracy of rotor
leakage flux is suspect since physical parameters for the rotor bars is not all that
well defined as far as leakage reactance is concerned. One point that would be
good to development further for the model herein would be to define a means of
converting the measured single phase equivalent circuit rotor leakage reactance to
a parameter usable in the model herein.
Analysis of rotor leakage flux inductance is complicated by the fact that the
definition of a rotor winding is not as straightforward as in the stator. In this
study there are actually no rotor windings in the same sense as there are stator
36


windings, but instead there is a mesh of rotor conductor loops. Rotor flux as
defined herein must pass through two rotor teeth (if flux leaves a tooth it must
come back through another tooth) and therefore at least two rotor conductor loops.
Therefore, in the rotor model herein there is no such thing as leakage flux that
passes through a single rotor "winding." Rotor leakage flux is defined herein as
flux induced in the rotor by rotor bar currents and that does not enter into any
stator winding.
Whether current flows in the inner or outer portion of the rotor bar is a
function of leakage flux that passes through the rotor bar. The effect is commonly
called the deep bar effect. The deep bar effect is essentially the same phenomena
as the skin effect on conductors: changing flux levels in the conductor induces a
voltage on the bar that impedes current flow in a manner that forces the current to
rise to the top of the conductor (i.e., outer portion or portion closer to the air
gap), which in turn increases the effective resistance of the rotor and reduces the
current in the rotor.
The cause of the deep bar effect may be seen by examining the rotor
section of fig. 2.14 where flux is shown passing through the rotor bar. If the
inner and outer portion of the bar had the same current levels, the current in the
inner portion will induce more cross bar flux than current in the outer portion
because the inner portion sees a smaller net air gap than the outer portion (note
the flux path shown for the outer bar includes the tooth gap as well as the slot
gap). The net effect is flux through the bar. The flux passing through the bar
induces circulating currents in the bar and is the source of the deep bar effect.
To some extent the saturation factors Sm and S{ play a role in the level of
the deep bar effect, but the effect should usually be small (especially in
comparison to all the other sources of error in the model) and hard to accurately
37


define, so the effect of saturation will be ignored when the inductance calculations
associated with the deep bar effect are presented in section 2.4. Also, dividing a
rotor bar into 2 sections is a simplification. If the bar could be represented by 3
or more sections a more accurate representation of the deep bar effect could be
made. See [7] for a discussion.
2.3.5 Saturation
Saturation is a factor that is difficult to handle accurately because it is a
localized, non-uniform, and non-linear problem and because in any function
devised to describe the problem there are a number of parameters involved whose
values can only be estimated. To some extent it is not possible to model
saturation accurately except via finite element analysis. With the available data
the best one can really do is to "get a feel" for the effects of various levels and
forms of saturation that one expects may exist under various conditions. A saving
grace in the subject is that in most modes of operation the machine operates in a
fairly linear region of permeability and any non-linearities of the machine have
only a second order effect on machine performance. However, the intent in this
model is to create a fairly detailed representation of the induction machine, so a
method was devised to represent saturation. The method is believed flexible
enough to be modified as seen necessary for a given machine.
In the model herein the effects of saturation are divided into two
categories:
1) Saturation of the stator and rotor core and teeth bodies. This type of
saturation causes an effective increase in the air gap length and causes a
decrease in mutual inductance. It is caused by high flux levels
38


associated with high line voltages. It will be referred to herein as
mutual path saturation.
2) Saturation of the leakage flux paths, mainly the edges of the stator and
rotor teeth. This type of saturation increases the effective tooth gap (g,)
and causes a decrease in leakage reactance. It is caused by high
currents associated with high load currents or starting currents. It will
be referred to herein as leakage path saturation.
Modeling the effects of saturation has a circular/iterative nature; the flux
level is not known until the saturation level is known, but the saturation level is
not known until the flux level is known. And as mentioned above, with the
uncertainty level associated with what little data is available on the effects of
saturation, the effects of saturation is to some degree indeterminate. Therefore
there is not a good foundation for entering into an iterative procedure and trying
to be exact when the input data and saturation relationships are fairly inexact. In
the inductance calculations of section 2.4 a single iteration procedure is presented:
First determine the flux that would exist without any saturation, then use this flux
level to estimate the effects of saturation via the saturation factors Sm (for mutual
path saturation) and Sl (for leakage path saturation). An explanation of these
saturation factors will be supplied below.
2.3.5.1 Core and Tooth Body Saturation
Core and tooth body saturation is modeled by increasing the local effective
air gap gQ as the local air gap flux level rises. The function that describes the air
gap is given the name Sm.
The cause of core and tooth saturation is high line voltages, but to some
extent core and tooth permeability starts to flatten even at nominal design
39


STATOR CORE
ROTOR CORE
HIGH FLUX DENSITY PATH
STATOR TEETH
ROTOR TEETH
FIGURE 2.15 AREAS OF MAXIMUM FLUX DENSITY FOR 2 POLE
MACHINE UNDER HIGH STATOR PHASE VOLTAGE
voltages. Assume an induction motor where the air gap flux is sinusoidally
shaped and peak air gap flux is at 0 and 180. The areas of the motor that are
most likely to saturate are as shown in fig. 2.15.
Per [3] and [6], the most common location to saturate is the teeth bodies at
the peak of the air gap flux wave. The effect of saturation in this area is a
considerable flatting of the air gap flux. Fig. 2.16, derived from discussions in
[3] and [6], provides some indication of what the flattened waveform will look
like. The waveform will contain varying amounts of odd harmonics, but the third
harmonic will predominate. It is worth noting that the harmonic flux waveform
will induce harmonic currents in the rotor cage that will tend to mask the effects
of saturation as seen by the stator terminals. Irrespective of the level of
saturation, the 60 hz portion of the rotating field must remain at a level defined by
the terminal voltage (i.e., line current must rise until the 60 hz component of L
d(I)/dt is equal to the applied line voltage minus resistive losses). As saturation
rises, the non-uniform permeability, in combination with the aforementioned stator
winding space harmonic factors and any harmonic voltages present in the line
voltages, cause increasing harmonic flux levels in the machine and which in turn
40


2
~X--------------------------------------------------------------
0 50 100 150 200 250 300 350 400
*
FIGURE 2.16 FLATTENED FLUX WAVE UNDER SATURATION,
FUNDAMENTAL SHOWN FOR COMPARISON
induces increased rotor harmonic currents.
Saturation will be modeled by first determining the flux level that would
occur if there were no saturation present (BJ:
Bu = (Ba + Bb + Bc + Br) with Sm = 1 (tesla) (2.15)
Next the saturation function Sm must be determined. A simple equation
that somewhat mimics the effects of saturation is:
41


If Bu < Bo then:
Sm = .9 + .1 B
u T (numeric)
If Bu > Bo then: o B
S = .6 + .4 m u T o (inumeric)
(2.16)
where B0 is an anticipated peak flux level that would exist in the machine under
nominal operating conditions.
The final estimated effective air gap flux density under saturated conditions
is:
K = (tesld) (217)
m
Note that dividing by Sm is the same as if the air gap length had been
Sm-g0 when flux level had been originally calculated.
The equation for Sm was chosen in a fairly arbitrary fashion; other
equations could have been chosen. However, the equation is felt to represent
saturation effects on normal machines. The equation mimics the effects of the
knee of a B/H curve commonly used to represent steel saturation and the equation
is based upon the machine operating slightly saturated even in normal operating
conditions. The effects of this equation may best be seen from fig. 2.17. Note
that it will take a large value of excitation current to get a net air gap flux density
above 2 per unit.
42


2


s'
/
/
/
/
/
/
/
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Bu.
1
FIGURE 2.17 Bu vs Bag, FOR CALCULATION OF CORE
FLUX LEVELS UNDER SATURATION
2.3.5.2 Leakage Flux Path Saturation
Leakage flux path saturation is modeled by decreasing the leakage flux
level as current levels rise by the factor
The cause of leakage flux saturation may be seen in fig. 2.18. When
current in the conductors rises to high levels the leakage flux passing through the
teeth tips rises to very high levels, resulting in tooth tip saturation.
43


HIGH
CURRENT
FIGURE 2.18 TOOTH TIP SATURATION
A simple relationship that somewhat represents the effects of leakage flux
saturation is:
L; = ~ (henries) (2.18)
where:
S. = .9 + .1-L
' K
(numeric)
(2.19)
and where:
La base leakage reactance at full load conditions
44


current at full load conditions.
h =
Similar to Sm, this equation was chosen in a fairly arbitrary fashion and
other functions could be used. However, the equation above is felt to represent
what actually occurs in a machine. As shown in fig. 2.19, at 6 times normal
current the leakage flux will decrease to about 65 % of its full load value, and at
no load the leakage inductance would actually rise 10% above the full load value.
Also, the function St was chosen to be a function that increases with saturation to
be consistent with Sm, which also increases with saturation.
0123456789 10
I-
FIGURE 2.19 LEAKAGE REACTANCE DECREASE
DURING HIGH CURRENTS
45


2.4 Inductance Matrix L
Having discussed windings arrangements, the mutual flux distribution
arising from those windings, leakage reactance, and saturation effects, it is now
possible to define the mutual inductances and therefore the matrix L. This matrix
relates current in each winding to the flux that links each winding. The equation
takes the form previously given in (2.2) and (2.5).
The machine inductance matrix L is highly unsymmetric. Only the matrix
portion Lss shows any symmetry. The matrix portions Lsr and Ln are not
transpositions of one another and no values in the two are identical. The
unsymmetric nature of the matrix portion L can be seen by inspection.
As previously discussed in section 2.2.3, some of the parameters associated
with winding inductances will be assumed to be zero, others easily determined,
and some assumed equivalent to another inductance. But there are several
parameters that will involve heavy amounts of computations because of the
numerical integrations that must be done to find their values. Furthermore,
because the inductances are changing constantly, the integrations must be re-
performed at each time step of the numerical differential equation solution. This
of course is a substantial programming effort and substantially slows the modeling
down, but it allows modeling of the machine rotor and a permeability that varies
over time, space, and varying flux levels.
To some extent the relationships involved with the inductance calculations
to follow are intuitive. However, a fairly rigorous analysis of stator-stator
inductances that reduces to the equations below may be found in [1]. There is
also an analysis in [2] and [4], but they are somewhat less rigorous. As research
on machine modeling did not find an analysis that uses a rotor model quite like the
model presented herein, the stator-rotor, rotor-stator, and rotor-rotor inductances
46


are the "invention" of this study, but they are based upon the stator-stator
inductance format.
The mutual inductances relate the flux that links a winding (or a rotor
tooth) due to the flux generated by current in a conductor, ignoring leakage
reactance. The inductances involve an integration of the form:
02
Lx_y = rl\WxByd(j) (henrys) (2.20)
where:
Lx y = Flux linkages (henrys) of the x winding due to current in the
y winding.
Wx = Winding distribution function of the x winding.
By = Air gap flux density (teslas) due to current in y winding.
r = effective average radius of the air gap.
I = length of the machine air gap.
i> 2 = positive span in radians of the x winding, discussed below.
The choice of (f>1 and 2 is such that integration is over the zone of positive
flux from winding x as shown in fig. 2.3 such that integration proceeds in a
counter-clockwise direction when the motor is viewed from the left end as shown
in the fig. 2.1. Hence, for a two pole machine phase A integrations are from -x/2
to ir/2, phase B are from 5x/6 to llx/6, and phase C from x/6 to 7ir/6.
When self inductances are being examined, x=y in (2.20) and the leakage
flux effects must be considered. Leakage flux is that flux generated by a winding
that links no other winding. The equation takes the form of:
47


(2.21)
L
x-x
4>2
Lk*(rl)
WxBxd
(henrys)
where:
Lu = Leakage flux associated with x winding.
Substituting for By from (9)-(l l) with Iy = 1A and including the saturation
function yields the following:
l
1 SM
(henrys)
(2.22)
where:
k =
V-o r l
So
and for self inductances the equation becomes:
K* =
r #
L+ k[
m l
Wx()
SJM
d (henrys)
(2.23)
(2.24)
2.4.1 Stator-Stator Inductances
These inductances refer to the flux linkages that are generated in stator
windings due to currents in the stator conductors. There are 6 inductances
involved: La^, Lb.b, Lc.c, La.b, Lb^, L^, L^, Lb_c, and Lc_b. Of course La.b = Lb^
Lb.c = Lc_a, and Lb_c = Lb_c. Applying the above equations to the actual a, b, and c
stator windings for a two pole machine to determine the inductances yields the
following:
48


(henrys)
(2.25)
7x/6
k ^ ,r to wjm
%) L to
L = * + A:
v_____
7x/6
r LB uT TO .
£,. l = + £ I --------=------d (henrys)
-Vi
W t/6 ^m(^)
(2.26)
j llr/6
L = _Jf_ +k\
~ w L
TO TO
5*76 w*
TO
d (ihenrys) (2.27)
x/2
TO to
r r / f raVTV rrfcvr/ *, /y .
La-b = A,-* = j --------£-77) ^ (henrys)
-x/2 m
TO
(2.28)
7x/6
V = i- f
i( wffl
t/6 ft*
TO
d (ihenrys)
(2.29)
llx/6
L = L =
c-fl a
r W () W () n
c = f -V?-rfW (henrys) (2-30)
sL TO
2.4.2 Stator-Rotor Inductances
These inductances refer to the flux linkages that are generated in stator
windings due to currents in the rotor conductors. There are 4 inductances
involved: L^, Lx^, L^, and Lx_m.
* ^-x-no Lx-ni
These inductances refer to the flux generated in the a, b, and c windings
due to current in the n* rotor bar. Recall from section 2.3.3 that it is assumed
49


that currents in the inner and outer rotor bars have essentially the same effect on
air gap flux, at least to the level of accuracy available from the model herein.
Therefore Lx.no and may be set to the same value (L^), below.
The flux generated in the stator by currents in the rotor is determined from
an integration of the form given below. The integration is consistent with the
assumptions of positive current and flux as defined in figs. 2.3 and 2.6.
Integration proceeds in a counter-clockwise direction. To account for the effects
of skew the limits of integration are reduced by 1/4 of the rotor bar skew (<&), as
discussed in section 2.3.3. Stator and rotor winding distribution functions are as
described in section 2.3.
Wx = the stator phase a, b, or c winding distribution function.
= the n* rotor bar winding distribution function.
^ = the position of the n* rotor bar.
Subscript "nb" refers to the n* rotor bar.
If there are k stator windings (one per phase per pole pair typically, with a
minimum of 3) and n rotor bars, the integration will need to be performed kn
times at each time step of the eventual computer model.
* ^x-nb L>x-nr
These inductances refer to the flux generated in the a, b, and c stator
windings by current in the n* left and right end rotor end rings. Only air gap flux
generated by currents in the rotor bars is considered in the model; the currents in
w(4) wj#)
where:
50


the end rings are considered to generate negligible flux due to the large air gap
around the end rings and because the length of the n* section of the end rings is
small in comparison to the n111 rotor bar. Therefore these inductances may be
neglected and set to zero in the model.
2.4.3 Rotor-Stator Inductances
These inductances refer to the flux linkages that are generated in the rotor
teeth by currents in the stator windings. There are 2 inductances involved: Lm.x
and Lm.x.
The rotor winding distribution factor for the rotor takes on a different
meaning than in section 2.4.2. In this section the interest is in the net flux
generated in the rotor teeth by current in the stator windings. There is no rotor
winding per se. The inductance is simply flux induced by the stator winding
integrated over tooth area.
This refers to the flux generated by the a, b, or c stator windings that
crosses the air gap and enters the m* rotor tooth. Positive flux and current is as
previously described in figs. 2.3 and 2.6. The limits of integration are the span of
the rotor bar, and again is in a counter-clockwise direction when the machine is
viewed as in the figures. To model the effects of rotor skew a simple method of
finding average air gap flux density over the rotor bar was used, as described in
section 2.3.3.
In the equation below tooth "m" is considered the tooth that is counter-
clockwise from and directly adjacent to bar m, as seen in fig. 2.6.
d (henrys)
(2.32)
51


where
= the position of the m* rotor bar.
tw = the tooth width in radians.
If there are k stator windings (one per phase per pole pair typically, with a
minimum of 3) and n rotor bars the integration will need to be performed kn times
at each time step of the eventual computer model.
# ^nd-x
This refers to the flux generated by the a, b, or c stator windings that
crosses through the n111 rotor bar due to currents in the stator winding. It is
assumed herein, however, that only the currents in the rotor bars themselves cause
any appreciable amount of cross bar flux. Recall that cross bar flux is the source
of the deep bar effect. The inductance therefore may be neglected and set to
zero in the model.
2.4.4 Rotor to Rotor Inductances
These inductances refer to the flux linkages that are generated in the rotor
by currents in the rotor conductors. There are 8 inductances involved:
^mt-nr> ^nd-po ^nd-pi ^rd-pl> ^d Lnj-pr.
r r .
These inductances refer to rotor flux generated in the m* rotor tooth by the
current in the n* rotor bar. Each tooth sees flux entering from across the stator-
rotor air gap due to the flux generated by the current in every other rotor bar.
Each tooth also sees "leakage" flux coming across the tooth-tooth air gap from the
adjacent rotor teeth. Therefore these inductances have two parts: a part associated
with rotor-rotor mutual air gap flux, and a part that may be called leakage flux.
Recall that as set up in fig. 2.6, the n* and n* + 1 rotor teeth are on either side
52


of the n* rotor bar (e.g., the 6th and 7th rotor teeth are on either side of the 6th
rotor bar). With this in mind, when n=m or n=m+l, and involve
leakage flux from across the tooth tip air gap as well as rotor-rotor air gap flux.
For any other value of n and m, and involve only air rotor-rotor air
gap flux. It will be assumed herein that the inner and outer portions of the rotor
bar are equally effective in producing tooth tip and air gap flux. Therefore
and Lfot.nj reduce one equation for inductance when n=m, another similar equation
when n = m+1, and another equation for all other cases.
For n = m:
L = L = L .
mt-nb mi-no mt-m
pJK
S&+IJ8,
+ k

d (ihenrys) (2.33)
For n = m+1:
L = L = L
mt-nb mt-no mt-m
-k1K

- k
2 Smi4>)
dcj) (henrys) (2-34)
For any other value of n:
L = L = L = k
mt-nb mt-no mt-m

25-W
d (henrys)
(2.35)
where:
K = rotor tooth-tooth gap height.
St = rotor tooth-tooth gap width.
^mb "~ position of the m* rotor bar directly adjacent and clockwise
from the mth rotor tooth.
W* = distribution factor for the n* rotor tooth.
53


Note that the sign of the leakage term is based on assuming the positive
conventions of fig. 2.6 for positive current flow in the bar.
As there are n rotor bars and n rotor teeth, there will be n-n such
integrations at each time step. However, as each integration is a variation of 1/Sm
over a given tooth, and is always +/-.5, simplifications that reduce the
number of integrations to n can be easily seen.
These inductances refer to the flux generated in the n* rotor tooth by
current in the n* left and right end rotor end rings. Only currents in the rotor
bars are considered in the model; the currents in the end rings are considered to
generate negligible flux due to the large air gap around the end rings and because
the length of the n* section of the end rings is small in comparison to the n111 rotor
bar. These inductances therefore may be neglected and set to zero in the model.
* L'nd-po* ^nd-pi
These inductances refer to the flux that is induced across the rotor bar by
current in the rotor bars. Flux flowing in this path causes different current levels
in the top and bottom portion of a rotor bar, commonly referred to as the deep bar
effect. A discussion of how the deep bar effect would be modeled herein was
provided in section 2.3.4. The inductance will only be considered to occur when
p=m (i.e., it is assumed that cross bar flux flow is induced only by current in the
bar itself). The conventions for positive and negative flux flow are drawn from
figs. 2.3 and 2.6.
For n = p:
L, = L .
d~o nd-po
~K 1 K
Kw
(webers)
(2.36)
54


(webers)
(2.37)
^d-i ^nd-pi
pj K

For n p:
L . = L . 0 (2.38)
nd-po nd-pi v '
where:
l = rotor length.
hb = effective distance between the upper and lower rotor bars;
i.e., the effective area over which cross bar flux flows.
wb = rotor bar width; i.e. the gap over which the flux flows.
ktg = a factor to account for the effects of the tooth gap at the top
of the rotor bar, discussed below.
Note that Ld_0 is negated. This is because the flux generated by positive
current in this portion of the rotor bar causes flux to flow across the rotor bar in a
counter-clockwise direction (i.e., negative to convention for positive previously
chosen). Note that no consideration is given for the effect of saturation on these
inductances, as previously discussed. Also, the last definition above, k,g, demands
some attention. It defines the degree to which the deep bar conditions will exist.
At this time only educated engineering approximations can be made for its value,
using approximations based upon the relative size of the tooth tip air gap and the
cross bar gap.
* ^nd-pb Lnd-pr
These inductances refer to the flux generated in the n* rotor bar by current
in the p* left and right end rotor end rings. For reasons previously stated, only
currents in the rotor bars are considered in the model; the currents in the end
55


rings are considered to generate negligible flux. These inductances therefore may
be neglected and set to zero in the model.
2.5 Resistance Matrix R
The resistance matrix, which takes the form given in (2.1) and (2.4), is a
fairly involved set of equations based on Kirchoff s voltage law and current law.
The voltage and current equations need to be meshed because in the rotor there
are 1/2 as many equations for flux (and hence v because v = L(d\/dt)) as there
are rotor current variables. Assuming n teeth and n rotor bars, there are n inner
rotor bar portions, n outer rotor bar portions, n left end ring sections, and n right
end ring sections, or 4n conductors for n teeth. For n teeth there are 2n flux
equations; one for the net stator-rotor air gap flux, and another for the net cross
bar flux used to represent the deep bar effect. To uniquely solve for machine
currents the equations that represent the rotor flux and currents are supplemented
by equations that the sum the current at each rotor node to zero. The rotor loops
used in creating the voltage rotor voltage equations are shown in fig 2.20. The
nodes where current is summed to zero is also shown in the figure.
Since the rotor is assumed to be symmetrical, the resistance matrix may be
simplified a bit with the relationships:
Ro = Rno = Rmo
Ri = Rni = Rmi
Rer = Rnl = Rml ~ Rnr = Rmr
which leaves only 3 resistances in the rotor equations.
56


LEFT
RIGHT
FIGURE 2.20 ROTOR VOLTAGE DROP SUMMATION LOOPS AND CURRENT SUMMATION NODES
2.6 The Simplified Resistance and Inductance Matrix
With the inductance definitions defined in the previous sections a simplified
version of (2.4) and (2.5) is now possible, as seen on the following pages.
57


0 0 1 0 0 0 0 0 0 0 0 ... 0 0 0 0 K \
0 0 1 0 0 0 0 0 0 0 0 ... 0 0 0 0 h \
0 0 *. 1 0 0 0 0 0 0 0 0 ... 0 0 0 0 K \
0 0 0 0 1 1 0 Rer -R er -R0 0 0 0 ... 0 0 0 0 ho K
0 0 0 0 1 -R R< 0 0 0 0 0 0 ... 0 0 0 0 hi K
0 0 0 0 1 -l -1 1 0 0 0 0 0 0 0 -1 0 h, 0
0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 -1 hr 0
0 = 0 0 0 1 0 0 0 0 R o 0 -Rer ... 0 0 0 0 L _ d dt
0 0 0 0 1 0 0 0 0 -Ro R, 0 0 ... 0 0 0 0 hi K
0 0 0 0 1 0 0 -1 0 -1 -1 1 0 ... 0 0 0 0 hi 0
0 0 0 0 1 0 0 0 -1 1 1 0 1 0 0 0 0 hr 0
0 0 0 0 1 1 -*u 0 0 0 0 0 0 0 ... Ra 0 Rer -Rer L
0 0 0 0 1 0 0 0 0 0 0 0 0 ... -R0 Ri 0 0 hi K
0 0 0 0 1 0 0 0 0 0 0 0 0 ... -1 -1 1 0 hi 0
0 0 0 0 1 0 0 0 0 0 0 0 0 ... 1 1 0 1 . * . 0
Note: The above eqns do not show that one of the current summation eqns must be replaced with an end ring voltage loop eqn.
Eq. 2.39


VO
A A- K-b A-, 1 A-i6 A-16 0 0 Ai-26 A-26 0 0 L L a-nb a-nb 0 0 K
\ A-6 A-6 Lb_c 1 A-u, A-16 0 0 A-26 A-26 0 0 ^b-nb b-rtb 0 0 h
K A-, A-, A-. 1 A-i6 1 Lc-Uy 0 0 A-26 A-26 0 0 c-nb L -nb 0 0 K
A A- ^U-b Af-ff 1 1 A- Lu-ib 0 0 A/-26 Af-26 0 0 ^lt-nb A/-n6 0 0 ho
K 0 0 0 1 A- A-, 0 0 0 0 0 0 0 0 0 0 hi
0 0 0 0 1 o 0 0 0 0 0 0 0 0 0 0 0 hi
0 0 0 0 1 o 0 0 0 0 0 0 0 0 0 0 0 hr
A/ = A-fl At-b A-c 1 A-l h ^21-lb 0 0 A-26 A-26 0 0 A-6 A-i>6 0 0 A
A 0 0 0 1 o 0 0 0 A- A-,- 0 0 0 0 0 0 A
0 0 0 0 1 o 0 0 0 0 0 0 0 0 0 0 0 A
0 0 0 0 1 o 0 0 0 0 0 0 0 0 0 0 0 hr
K ^mt-a ^int-b A-c 1 Anf-16 Am-16 0 0 ^mt-Tb Am-26 0 0 L L u mt-nb ml-rib 0 0 A
K 0 0 0 1 o 0 0 0 0 0 0 0 Li-o A-,- 0 0 A
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 A
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 . A .
(Eq. 2.40)


2.7 Torque
Torque in the model is simply taken as an application of Amperes law:
force = i-B. In the case of a motor this is applied by assuming rotor current is
flowing in an infinitely thin wire on the surface of the rotor steel. For simplicity
the effects of skew on torque are neglected. The equation that represents torque
becomes:
n
Tm = Y, 0newton-meters) (2-41)
>-i
where:
rrc = effective radius of a rotor conductor network.
I = rotor bar length.
if = current in the n* rotor bar.
Bag(J = air gap flux density at i* rotor bar, at position <&., as
discussed in section 2.3.5.1
The equations of acceleration, speed and position are:
Acc = = (radians/sec2) (2.42)
dt 7+7,
m i
speed = co = j ~ dt (radians/sec) (2-43)
t
60


where:
Tt = load torque
A simple model of load torque may be:
7j = kt of (inewton-meters) (2.45)
where is approximately 2.
As previously mentioned in section 2.3.1 as a source of error in the model,
current does not flow in an infinitely thin conductor on the rotor or stator surface.
This assumption introduces error in flux distribution as well as torque calculations.
The details of the real world vs the simplified world of the model is involved
enough to deserve an a short discussion on the matter. Numerous articles and
books use the assumption that the machine currents flow on the air gap surface, so
it is likely a fairly valid simplification. However, the error in the simplification
starts to "stand out like a sore thumb" when torque calculations are performed.
Torque herein (and in many publications using the surface current concept) is
calculated by applying Amperes law, F = i-B-1, to each rotor bar. However, if
this equation was applied to the current and flux in the real rotor bar the torque
calculated would be effectively zero because the flux in a rotor bar is effectively
zero. The conductors are buried in a steel frame that shunts most of the flux
around the bar. While a great deal of flux flows around the conductors, little
flows in them. The description of the actual condition is involved enough to
prevent most authors from discussing the matter, but the matter deserves at least
some discussion for completeness.
An entry level reference on the matter of how magnetic material works is
[14]. The author of [14] provides a description of the outer shell electrons in the
outer atomic layer of magnetic material acting in cohesion to create either the
61


phenomena called "surface currents" or an external magnetic field. If the plane of
the electron spin is parallel to the steel surface it creates magnetic flux directed
out of the steel surface (i.e., the spinning electron appears as a current loop with a
magnetic field pointed out of the steel). If the plane of the electron spin is
peipendicular to the steel surface then surface currents may be created. Surface
currents are not real currents, but the effect of orbiting electrons in the very last
atomic layer of the steel surface spinning in a coordinated fashion such as to
appear as if a real current were flowing on the steel surface. Without any hard
proof to back up the statement, except basing in on the widely used simplification
of placing conductor currents effectively on the steel surface, it is assumed the
following may be taken to be true: 1) as long as the internal conductor is "close"
to the steel surface, the induced imaginary surface currents in the first layer of
atoms on the material surface will have approximately the same effect as if the
conductor were on the steel surface; and 2) torque in actuality is not a direct
interaction between the current in the rotor bars and air gap flux but more directly
it is an interaction between the air gap flux and these surface currents induced on
the outer atomic layer of the steel. As a result it is assumed herein that (2.41)
will provide a sufficiently accurate description of torque.
62


CHAPTER 3
APPLICATION OF EQUATIONS TO NUMERICAL SOLUTION
3.1 Introduction
A machine model is not too useful if the system to which the machine is
connected can not be included in the computer implementation of the model. To
determine how the equations would best be modeled in an iterative numerical type
program, the software Electoromagnetics Transients Program (EMTP) was
referenced. EMTP uses an iterative procedure that is capable of modeling the
transient characteristics of an extremely wide variety of system devices. However
machine modeling in EMTP uses D-Q analysis ([12] and [15]). The intent of the
work to follow is to show how machine modeling could be set up in a fashion
comparable with the methods used inside EMTP for modeling lumped element
devices such as short transmission lines, resistances, inductors, and capacitors.
To do this the modeling of lumped parameter devices is included in the
discussions to follow. In [15] there is a development of many of the calculation
procedures used in EMTP, and the calculations in this study used [15] as a guide.
In particular, the method of modeling lumped parameter elements and short
transmission lines/cables and the method of setting up the iterative calculations in
[15] is used below.
3.2 The Calculation Procedure
As previously discussed one of the big problems facing machine analysis
when D-Q theory is not used is that because of rotor movement inductance
63


between stator and rotor circuits are changing with time and with rotor position.
Therefore when D-Q theory is not used it is necessary to recalculate machine
inductances at each step of the solution. However, once one accepts the difficulty
of this task a great amount of detailed modeling of the machine becomes possible;
e.g., it becomes possible to create a more physically accurate model of the
machine, saturation effects may be modeled in detail, and the squirrel cage rotor
may be modeled in detail. However, as will become evident below, in acquiring
this detail in machine modeling, one faces a great amount of work required for
implementation of the machine model.
A general idea of what it takes to model the machine and the network to
which it is connected, using the model herein is as follows:
1. Assume currents, torque, and speed are known at time t-At. Assume
the flux wave position and rotor position at time t have been estimated
via extrapolation from conditions at time t-At (and possibly farther back
in time).
2. Using the above information inductances are recalculated between all
the rotor and stator circuits. This step is complicated by the need to
perform a substantial amount of numerical integration. This step is
discussed in section 3.3 below.
3. Once inductances have been calculated it is possible to set up a set of
equations (valid only for time t) for current vs voltage throughout the
system (but these equations cannot be solved yet because they require
knowledge of node voltages). This step is complicated by the need to
perform matrix inversion, with the matrix representing the system
possible being very large (a matrix well over 100x100 is likely). This
step is described in section 3.4.
64


4. Once the current equations are known it is possible to set up a set
simultaneous equations that eliminate the currents from the equations of
step 3 and hence allow one to solve for system voltages at time t. This
step is described in section 3.5 below.
5. Once voltages at time t are known it is possible to substitute the
voltages back into the current equations of step 3 and solve for network
currents at time t.
6. Once currents are known it is possible to solve for torque at time t.
This makes it possible to estimate rotor position at time t+At. This
step is described in section 3.6 below.
7. Once rotor position is known the flux wave position at the next time
interval estimated. This step is described in sections 3.7.
8. At this point the process starts over; the values at time t are all known
now, and hence they become the historical values, i.e., the system
conditions at time t-At. The next step is to find the system conditions
at the new time t. In this fashion the machine and system performance
is tracked one increment of time after another.
The procedure relies heavily on small values of At for accuracy and for
tracking currents and voltages, especially if one is expecting good resolution of
harmonic currents and voltages. If one wanted about 9-10 points in one wave of
the 7th harmonic and the fundamental was 60 hz, the At required would be about
.00025s, or about 4,000 steps per second of machine operation (i.e., 10-7-60 =
4200).
At any point where discontinuities are to be modeled the steps may be
broken and the appropriate system parameters changed accordingly.
There are sources of error in this process. The simulation process involves
65


step by step calculations using discrete values for I, V, and the differential
equations (and all the values for L, C, and R in the differential equations) to
predict values at the next discrete point in time. Most of the values, including the
differential equations, are actually changing continuously. The numerical solution
process largely assumes a linearized and predictable nature for changing values.
When a discrete time numerical process is used to track the response of a function
that is not changing per predictions a buildup of error is possible in each time
step, resulting in a solution "blowing up." The situation is sometimes referred to
as numerical instability. The calculations shown herein, as are the calculations
used by EMTP, are considered numerically stable. The solution to this error is to
reduce step size, but this will increase the number of calculations required, which
in turn introduces more of another source of error, roundoff.
Round-off error is caused by lack of precision in calculations. The large
number of calculations required for the process make it possible that the
compound effect of small errors may make the final result unacceptable. There is
not much that can be done to reduce roundoff error except to increase the number
of significant digits to which calculations are performed and results carried.
The rotor position and flux wave position estimation procedure can provide
another source of error. However, the fundamental flux wave and the rotor move
rather slow. The small At required by the model should help keep errors
introduced by this process to a minimum.
The machine and the electrical network to which it is connected will be
represented herein with a large set of node voltages, which is how EMTP models
lumped element networks. To develop the node voltages used in the equations in
this study it is first necessary to define the equations that relate current between
nodes to the node voltages.
66


3.3 The Machine Inductances
In chapter 2 the inductances that represent the machine were shown to be a
function of a large number of integrations. Given n rotor bars and a 2 pole
machine, there will be (n2)/8 + 9n + 6 integrations (from inspection of (2.40)),
assuming the simplifications of chapter 2 are used. Implementation of these
integrations may be made in a fairly organized fashion by picking a point in the
air gap and making one pass around the air gap, making the necessary summations
to the numerical integrations simultaneously as each point in the air gap is passed
by. This would require a large set of if-then type statements. However, this step
in the program will of course be computationally intensive and slow.
3.4 Current Equations
To show how the current equations are set up it will be necessary to show
how the current and voltage in a simple RLC circuit would be modeled and the
response tracked numerically in a simple lumped parameter RLC circuit. If a
device may be modeled as a series connection of lumped parameter inductances,
resistances, and capacitances, the voltage across the device and the current in the
device has the following relationship:
V = ri + p(Li) + (lip) (HQ (3 *)
where p is differentiation and 1/p is integration.
For the purpose of the analysis assume positive currents and voltages are
as shown in fig. 3.1.
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FIGURE 3.1 R-L-C CIRCUIT ELEMENT
Suppose that in a given digital simulation it is desired to find current in the
element at time t. Assume that the following are known:
Current in the element, i(t) and i(t-At)
Voltage across the entire element, v(t) and v(t-At)
Voltage across the capacitor vc(t) and v0(t-At)
R(t), L(t), C(t) and R(t-At), L(t-At), and C(t-At)
Given a small enough At, the following equation may be used to represent
the current and voltage relationships the network at t-At and t:
.5[v(r) + v(f-At)] = .5[R(t)i(t) + R(t-At)i(t-At)] +
(1/At) [L(0 i(t) L(t-At) i(t-At) ] + (3 -2)
(.25 A t)[i(t)IC(f) + i(t-At)/C(t-At)] + vc(t-At)
Before discussing the derivation of this equation, note that in the above
material there is the implication that the parameters R, L, and C may change with
time. In most applications R changes only a small amount and rather slowly, as
68


In power lines inductance may also be considered constant as long as the effects of
high frequencies can be ignored, but in application to the machine model herein
inductance is considered to change constantly. Therefore inductance will be
considered variable with time in the equations to be developed, but R and C will
be considered constants. This reduces the above equation to:
5[v(0 + v(f-A/)] = .5R.[i(t) + z(f-At)] +
(1/At)[L(f)i(f) -L(t-At) i(t-At)] + (3-3)
(.25 A t/Q[i(t) + i(t-At)] + vc(t-At)
Eqs. (3.2) and (3.3) may be derived by applying trapezoidal integration
and finite difference differentiation to (3.1). The procedure is discussed in greater
length in [15], but a quick grasp of (3.3) may be found by examining one part of
the equation at a time.
If the right hand side of (3.3) contained only the resistive element the
equation reduces to:
v(t) +v(t-Af) = Ri(t) +Ri(t-Ai) (3-4)
which is easily seen as true.
If the right hand side of (3.3) contained only the capacitive element the
equation reduces to
.5[v(0 + v(?-At)] = (.25 A t/C)[i(t) + i(t-At)\ + vc(t-At) (3-^)
As the element is considered a pure capacitance the voltage across the
capacitance, vc, is actually the voltage across the entire element. Therefore in this
case v0(t-At) = v(t-At) and the equation reduces to:
v(t) v(t-At) = (.5 At/Q [i(f) + i(t-Al)] (3'6)
69


This equation can be seen as the voltage rise across the capacitor that
results from the average current into the capacitor for the time At. This equation
may also be seen as a trapezoidal integration of the capacitive voltage equation
v = (l/p)[i/C]. The equation is reasonably correct as long as i is changing
linearly over At, which in turn implies relatively small At.
If the right hand side of (3.3) contained only the inductive element the
equation reduces to:
,5[v(r) + v(f-A/)] = (UM)[L(t)i(t) -Uf-At) i(t-At)] <3-7)
This equation can be seen as the average voltage across an inductor that
results from an a changing value of the product Li over the time At. This
equation may also be seen as finite difference differentiation of the inductive
voltage equation v = p[Li]. The equation is reasonably correct as long as v is
changing linearly over At, which in turn implies relatively small At.
Trapezoidal integration and first order finite difference differentiation
require a small At in order to obtain good numerical accuracy. Textbooks on
numerical methods typically pass over these methods rapidly, discussing them only
as stepping stones to higher order accuracy numerical methods. However, as
discussed in [15], numerical instability (i.e., the tendency of a calculation method
to "blow up" or enter into erroneous poorly damped oscillations) is a potential
problem with higher accuracy solutions and therefore are avoided even in EMTP.
Furthermore, as will be seen below, a large set of equations based upon (3.3) will
need to be solved simultaneously at each time step iteration. By keeping the order
of (3.3) low the effort of this step of the calculation will be kept to a minimum.
Eq. (3.3) is the basic equation that will be used herein to relate voltage and
currents in all network elements, including motors (eq. (3.3) is also used
70


extensively in [15]). But relating voltage and current in one element is only part
of the problem; a set of simultaneous equations using node voltage representing
the entire electrical network must be set up as well, as described next.
To set up the node voltage equations it is necessary to restate (3.3) in the
form of:
i(t) = Av(t) + History Term @.8)
where "history term" is the portion of (3.3) involving t-At, and "A" refers to an
algebraic expression of L, C, and R as appropriate. The resultant equation is:
i(t)
2m + ai+R
At 2 C
-i
v(r) + H{t-Af)
(3.9)
where H(t-At) is the History Term and is defined by:
H(t-At)
2L(t-At) At _p At 2 C i(t-At) + v(t-At) 2vc(t-At)
' 2L(t-At) ^ At At 2 C
(3.10)
This equation relates current through and voltage across a single element.
The parameter v references the voltage across the given element, but the voltage
that is eventually to be solved for is the voltage at each node in the network
relative to a common reference point, i.e., ground. To achieve this, the next step
is to apply (3.9) and (3.10) to every element in the entire network. Once this set
of equations is known the currents into each node may be summed as in the
classical node voltage approach to creating simultaneous equations. These
equations are then solved for voltage at each node.
Before proceeding one must be aware that (3.9) and (3.10) may be applied
71


to matrices as well as individual elements. The matrix approach will be needed to
represent the inductive coupling that occurs when there is inductive coupling
between the elements multi-phase system (such as power lines), and in the
induction machine representation. Showing the details of application to matrices
is rather involved, so it is being left to the reader to review [15] for more details
on the matter. However, in the sections to follow some idea of the application
should become apparent. When applied to matrices (3.9) and (3.10) take on the
form:
m =
At 2 C
I

(3.11)
where H(t-At) is the History Term and is defined by:
Hit-M) =
2L(t-At) At p
At ~2C
I(t-At) + V(t-At) -2 Ve(t-At)
2L(t-Af) A At
At 2 C
(3.12)
The next step is to show how (3.9), (3.10), (3.11), and (3.12) are applied
to the elements associated with the machine model.
3.4.1 Lumped Resistor, Inductor, or Resistor+Inductor Current Equations
It is straightforward to apply (3.9) and (3.10) to lumped parameter resistive
and inductive elements since the equations were derived with this type of element
in mind. The task is simply a matter of filling in the known values into the
equation. Series capacitance can be taken to be infinite for these elements (as
inductive elements they pass DC current indefinitely), so the value 1/C becomes
72


zero and drops out, as does ve.
(3.13)
H{t-Ai) = J_____________________
' 2L(t-Af) ^
2L(t At) + V(t-At)
At
(3.14)
At
One factor of concern in the above equations is that the resistance, and to a
smaller extent the inductance, of a conductor changes with frequency. In the
above equation this would be modeled by changing the inductance and resistance
in some manner associated with the value di/dt. The problem is not represented
in the machine model herein. The most straightforward approach is not to try to
use the model over too wide a frequency range. It is one reason one should not
try to use the model to represent harmonics much above the 7th.
3.4.2 Lumped Capacitor Current Equations
Capacitors usually have a low inductance and resistance. If the capacitor
can be viewed as an ideal device the simplified capacitor with no inductance and
resistance may be used. This means that (3.9) and (3.10) reduce to
straightforward equations involving only capacitance, which then reduce to the
following:
i(f) = fj^.1 \(t) +H(t-Af)
(3.15)
73


HQ-At) = (-1)
(3.16)
i(t-At) +
'2C
At
vQ-At)
If one wishes to model the small amount of series inductance and resistance
associated with a capacitor, one could use (3.9) and (3.10) with these parameters
included. However, this introduces the difficulty of determining vc in a separate
calculation from the nodal equations. An alternative approach is to model the
capacitor as an ideal capacitor with a separately modeled resistive/inductive
element, which creates the difficulty of additional nodes in the network.
3.4.3 Simple Transmission Line/Power Cable Current Equations
The power transmission line (i.e., the cable to the motor), for the purposes
of motor analysis, can be modeled as a simple series lumped parameter
impedance. Lumped parameter, though does not mean the same thing as in the
previous material. If there were only positive sequence currents on a transmission
line the voltage across the line may be calculated as if there were an independent
lumped impedance on each phase, but if there is any unbalance to the line voltages
or currents there will be negative and zero sequence currents as well. The
positive sequence impedance matrix must be modified to represent the effects of
having current in a phase return through the ground path instead of back through
the other 2 phases. The line impedance matrix used herein draws its elements
from the simple and classical positive and negative sequence line impedances.
The model used may be called simple due to several simplifying
assumptions: The positive and negative sequence impedance of each phase is
considered the same; the effect frequency on resistance and inductance is ignored;
phase to phase and phase to ground capacitance (and therefore charging current) is
74


ignored; and the distributed nature of the impedances will be ignored. Note this
model, therefore, would be unsuitable for high frequency transient analysis. As
the motor model is not suitable for high frequency transient analysis, it does not
seem appropriate to introduce the complexity of a complex line model herein.
Recognizing the above mentioned shortcomings of the simple positive and
negative sequence modeling of a transmission line, but considering the error as
acceptable, the impedance matrix of a transmission line (as shown in [15]) takes
on the following form:
Z,| =
z z z
s m m
Z Z Z
m s m
z_ z_ z.
(3.17)
where:
Z = (2Z +Z )/3 s \ pos zero f (3.18)
Z = (Z -Z )/3 m \ zero posJ (3.19)
If the impedance of each phase could have actually been modeled as a
lumped impedance associated with just the phase conductor, the matrix Zy would
have only diagonal elements equal to the positive sequence impedance and the off
diagonal elements would equal 0. As seen in (3.18) and (3.19) this occurs only
when Zpos = Z^, which is never the case in real world transmission lines.
It is necessary to express (3.17) in a form that can be used in (3.11) and
(3.12); i.e., the equation must be expressed in terms of separate R and L
matrices. As Zpos and Z^ is normally expressed in ohms at 60 hz, it is necessary
to divide by co to find inductance. The equation takes on the form:
75


(3.20)
R
n
x. Rm K
Rm R K
K K R
L
X xm Xm
Xm Xs xm
Xm Xm xs
(3.21)
To create the current equations that express current leaving a point on a
power line, equation (3.20) and (3.21) are substituted into matrix forms of (3.13)
and (3.14).
3.4.4 Induction Machine Current Equations
The modeling of the induction machine resistance and inductance matrices
were developed in chapter 2 and assuming these inductances have been calculated
as discussed in section 3.3, the current equations needed for digital modeling are
created by substituting the R and L machine matrices from chapter 2 into matrix
forms of (3.13) and (3.14).
While the machine equations in chapter 2 are notably different than the
lumped parameter element equations used to derive (3.9) and (3.10), it should be
noted that in (2.4) and (2.5), and (3.1), both systems of equations have the form
of v = Ai + p(Bi), and therefore the same finite difference equations may be
used to model the machine and the elements described in 3.4.1, 2, and 3 (i.e., the
differential equations for machines have the same format at the differential
equations for lines and capacitors).
As described in chapter 2 the machine inductances are changing constantly
76


and must be recalculated at each time step. Furthermore, as the machine
inductances changes, a new inversion of the impedance matrix is required at each
time step. To represent a motor requires a great many nodes: 3 for the stator as a
minimum, but the rotor requires 4n equations, where n is the number of rotor
bars. In a rotor with just 25 bars (they usually have more), the matrix forms of
(3.13) and (3.14) involves the inversion of a matrix over 100x100 in size. It can
be seen that the numerical integration and matrix inversion work is very
computationally intensive and it is obvious why a model of this detail would not
be commonly used.
3.4.5 Current Equations for Network as a Whole
Using the aforementioned equations of section 3.4 it is possible to set up
current equations for all the elements that would normally be involved in modeling
a machine and the system to which it is connected: system impedance, line
impedance, machine capacitors, and the machine itself. From these equations it
can be seen that if the voltages at two nodes are known and the nodes are
connected together by one of the devices discussed an equation is available for the
current between the nodes. Given nodes j and k and the voltages Vj and Vk, an
equation may be defined for current between the nodes of the form:
i/t) = A[Vj(t) -vt(*)] + H(t-At) (3-22)
It should be noted that the equations in section 3.4 are written in terms of
v(t)" and "i(t)" but when viewed in terms of a network of voltages and currents
v(t) becomes vk(t)-Vj(t) and i(t) becomes ijk(t).
To reduce the current equations to a more compact form it is possible to
state the current equations for the system in a matrix format:
77


(3.23)
*11 hi 1 H '4 ^12 4ik V %i Hn .
hi *22 h k = 4 4* 42* V2 + h21 H22 . H2k
Jkl *A2 *_ 4U 4*2 4tt A Hu H2k Hkk_
This matrix is more formidable than may be obvious. The elements of the
matrix involve a substantial amount of calculation. Note in particular from section
3.4.4 that a fairly large matrix is involved.
The left matrix (containing ijlc variables) above would actually be very
sparsely populated since many nodes are not connected together. In the usual case
i,* would be 0, unless there was a current injection at node kk. This would not
occur in the model herein, but it does occur in some of the modeling in EMTP.
See [15] for additional information. Also, and would be zero in most
situations.
3.5 Voltage Equations
From basic system of equation solution techniques it is known that a
system of simultaneous equations for voltage may be set up by summing the
currents at each node in a network. The currents sum to zero and as a result
current is removed as a variable in the equations. For instance assume the sample
circuit in fig. 3.2 below.
The current equations at node n must sum to zero:
L +ira A = 0
(3.24)
78


FIGURE 3.2 4 NODE NETWORK
The equations for the indicated currents are
hi = ^(v.-v,(0) + Hfi-Ai)
L = ~ v20)) + H2(t-At) (3-25)
Ini = MVn ~ V3) + H3(t~At)
Summing the 3 equations to eliminate the current variables yields:
[Ai +A2 + A3 ]vn Vi A2v2 A3v3 = Hl + H2 + H3) (3-26)
The set of simultaneous equations required to represent a machine and the
network to which it is attached takes on the form:
Matrix Derived ' Voltage Matrix Matrix Derived
From "A" Term = From "H" Term
Summations Summations
(3.27)
or
An Al2 .. .. Ai
A2i A-n .. A^
aL Ak2 - .. A&
V Sts 1
V2 II Hi
1 1 Hi
(3.28)
79


With this equation (at least after it is expanded to represent the entire
system of equations that represent the machine and the network to which it is
attached) it is now possible to solve for v(t). Once voltages in the system are
known, it is now possible to substitute back into the current equations in section
3.4 to solve for the system currents.
3.6 Equations of Motion
Having determined i(t) and B(0) in the previous equation it is now possible
to determine machine torque and acceleration at time t. With this known it is then
possible to estimate speed and finally position at time t+At.
Machine torque at time t is as defined in section 2.7:
t. ixfcm.oM e-29)
where is as defined in section 2.3.5.1 using currents calculated from the most
recent voltages, as described in section 3.5.
A simple model of load torque may be:
Tt = ktiox (3.30)
Acceleration is simply:
dus (t) Tm(t) Tft) ^ 3
dt / +J,
m 1
A simple estimation of speed at time t+At is:
a) (t+At) = a(t) + At (3.32)
80


Assuming speed over the period t to t+At is the average of co(t) and
co(t+At), position at t+At is simply:
r(t+At) = r(t) + ^L[u(f) +co(/+Ar)] (3.33)
which may be restated as:
*r(*+A0 = + (AD(0 + (3-34)
The speed and position estimation process above could possibly be
improved upon. It is based upon assuming present torque is a sufficiently accurate
representation of torque over the whole period t to t+At. Basic numerical
methods teaches that higher accuracy estimations of speed (and therefore position)
at t+At are possible if speed or torque at earlier times than t are used as well.
However, the method above is felt to be normally acceptable due to the relatively
small changes in speed that will occur within the At which would be used in the
calculations compared to the rate of change of speed of a machine. Speed changes
relatively slowly compared to the voltages and currents, so it is felt that the
accuracy to which speed is estimated is likely at least equal to the accuracy to
which current and voltages are calculated.
3.7 Flux Wave Position and Magnitude Estimation
The flux wave position and magnitude affect the saturation function. In
most conditions the saturation function has only a second order effect upon
machine performance. The fundamental period portion of the flux wave is the
predominate part of the flux wave, and this part of the wave will travel at
synchronous speed and maintain a relatively constant magnitude. Therefore it is
81


felt that a sufficiently accurate estimation of the flux wave position and magnitude
at time t+dt may be made if the flux wave shape that was calculated for the
torque calculations in section 3.6 is simply rotated as if it had moved at
synchronous speed for At time.
3.8 Incrementing the Time Step
After estimates for rotor speed and position and flux wave magnitude and
position at time t+At are made as described above, the time increment status is
moved up one notch: What was present time (t) becomes past time (t-At), and
what was future time (t+At) becomes present time. Note all conditions in past
time are known and therefore the History terms must be updated. Note also
present time conditions for current and voltage are unknown and therefore must be
solved for as described in sections 3.2.1 and 3.2.2.
The estimate for rotor speed and position and flux wave magnitude and
position at time t+At opens up the possibility of using an iterative method to solve
for speed and position: First estimate speed and position and flux wave magnitude
and position at time t+At, then increment the time status, solve for voltages and
currents, then finally determine new values of acceleration and currents. If
machine acceleration or flux wave is notably different between the two time
increments, then the position estimation may need to be updated. However, this
should be a case of "overkill" and it is not felt that the effort is warranted.
82


CHAPTER 4
CONCLUSIONS
The study presents a fairly detailed model of an induction machine that
could be used for in-depth analysis of the induction machine than in models
commonly available. The study also presents how the model would be used in
numerical computations. However, because of the very high level of
computations involved the algorithm was not converted to a working computer
program. As computer modeling sophistication is ever increasing, the model may
eventually have its place.
A summary of the major aspects of the model that is presented follows:
It was decided that for accurate representation of the machine it is
necessary to accept the task of constantly recalculating machine
inductances as the rotor rotates. The equations needed for constant
recalculation of inductances for a very detailed physical representation
of the machine are provided.
To maintain the detail in the model that was desired, it was decided to
develop a fairly accurate physical representation of the machine,
including the stator windings, the rotor conductor network, the deep bar
effect, core (therefore mutual) flux saturation, and leakage flux
saturation.
A fairly detailed physical representation of each stator phase winding is
utilized. Machine flux distribution from each phase winding is
considered separately, including a representation of stator winding flux
83


(i.e., space) harmonics (up to the 7th harmonic is as far as is
considered reasonable).
A fairly detailed physical representation of the rotor is utilized. The
rotor model is not the usual "equivalent" three phase wound rotor, but
an actual model of the rotor squirrel cage network. Machine flux
distribution from each rotor conductor is analyzed including a model of
the deep bar effect. Because of the physical representation of the rotor
a more intuitive (at least as compared to D-Q analysis) modeling of the
effects of various rotor designs is possible.
A fairly detailed representation of machine non-linear permeability
(i.e., saturation) is presented. Non-linearity even at nominal operating
conditions and the localized nature of non-linearity are considered.
This includes the effects of non-linearity in:
- the leakage flux paths (i.e., at the tooth tips, where saturation is
caused by high machine currents), and
- the stator/rotor mutual flux paths (i.e., in the stator and rotor
body, where saturation is caused by high line voltages).
The model presented is capable of analyzing a wide variety of machine
internal and external imbalances and discontinuities, such as unbalance
stator windings and damaged rotor bars.
A set of differential equations to represent the voltage and current
relationships in the machine is presented. The equations take the form
V = p(Li) + Ri.
A numerical differential equation solution method is presented for
numerically tracking the machine response (i.e., "solving" the
equations). The calculation methods are based upon those used in the
84


software EMTP (Electro-magnetics Transients Program). Equations are
given for modeling lumped impedances, multi-phase power lines, and
capacitors, so that these factors which affect a machines performance
can be included in the model. A means of creating a system of
equations that represent the machine and the system to which it is
attached is presented. Given enough attention to setting up the
equations, the procedure that is presented is capable of modeling each
phase individually, modeling each stator and rotor conductor
individually, modeling line and system impedances, line capacitors,
delta and wye voltage sources, and delta and wye machine connections;
i.e., the procedure was designed for generality.
The limitations of the model center around its complexity; the model
requires such extensive calculations that the simulation will be slow. It will be
difficult to program. The large amount of calculations involved will limit the
accuracy that can be expected just due to roundoff error. The benefits of the
model will not commonly be justified given these limitations. However, as
computer modeling sophistication is ever increasing, the model may eventually
have its place, justifying the eventual conversion of the algorithm to a working
computer simulation.
85


REFERENCES
[1] P.C. Krause, Analysis of Electric Machinery, New York: McGraw Hill,
1986.
[2] P.M. Anderson, A.A. Fouad, Power System Control and Stability, Iowa
State University Press, 1977.
[3] J.C. Moreira, "A Study of Saturation Harmonics With Application In
Induction Motor Drives," Ph.D. Dissertation, University of Wisconsin-
Madison, 1990.
[4] J.C. Moreira, T.A. Lipo, "Modeling of Saturated ac Machines Including
Air Gap Flux Harmonic Components," IEEE Trans. Industry Applications,
vol. 28, no. 2, March/April, pp. 343-349, 1992.
[5] P.L. Cochran, Polyphase Induction Motors Analysis, Design, and
Application, New York: Marcel Dekker, 1989.
[6] P.L. Alger, The Nature of Induction Machines, New York: Gordon and
Breach, 1965.
[7] S.S. Chang, J.F. Douglas, C.H. Lee, P.L. Alger, J.H. Wray, J.
Goodman, P. Jacobs, J. C. Courtin, "Symposium on Design of Double
Cage Induction Motors," AIEE Trans. Vol 72, August 1953, pp. 621-662.
[8] A.P. Russel, I.E.D. Pickup, "An Analysis of the Induction Motor," Pts 1,
2, and 3, IEE Proc, Vol 129, Pt. B., No. 5, Sept 1982, pp. 229-247.
[9] G. Angst, "Saturation Factors for Leakage Reactance of Induction Motors
with Skewed Rotor," AIEE Trans. Vol 82, October 1963, pp. 716-725.
[10] T.A. Lipo, A. Consoli, "Modeling and Simulation of Induction Motors
with Saturable Leakage Reactance," IEEE Trans, on Industry
Applications, Vol. IA-20, No. 1, Jan/Feb 1984.
[11] G.A. Covic, J.T. Boys, "Operating Restrictions for Third Harmonic
Control of Flux in Induction Machines," IEE Proceedings, Pt. B, Vol.
139, No. 6, Nov. 1992, pp. 485-497.
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