CT^
ALTERNATOR ANALYSIS AND INDUCTANCE PARAMETERS
by
JOSEPH EDWARD VANDERPOORTEN, B.S.
A THESIS
IN
ELECTRICAL EiJGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
IN
ELECTRICAL ENGINEERING
/
Approved,
^
->t-.
Chairman of the Committee
/? /
Accepted
Dean of the Gradua
December, 1976
-
T-
i^UU-ll^cH
/JO./4/
ACKNOWLEDGEMENTS
I <r. indebted to Dr. Tommy R. Burkes for his guidance in the
direction of this thesis.
I would also like to thank Dr. Stan
Liberty and Dr. Arun Walvekar for their comments and observations
on the material covered herein.
11
ABSTRACT
This thesis deals with several aspects of rotating alternators.
The subject of inductance parameter determination is discussed.
Specific emphasis is placed on methods of bounding the values of
the inductance of the damper or amortisseur winding.
In addition,
the fault analysis using the conservation of flux linkages is presented.
Numerical examples are given of fault torques and currents
for a variety of line fault situations.
Ill
TABLE OF CONTENTS
Page
li
ACKNOWLEDGMENTS
ABSTRACT
iii
LIST OF TABLES
vi
LIST OF FIGURES
vii
CHAPTER
I.
INTRODUCTION
1
II. MACHINE INDUCTANCE PARAMETERS
Inductor Interaction
4
5
Two Circuit System
5
Approximating Mutual Inductances
8
Single Phase Machine Inductance Parameters
10
Field Self Inductance
11
Phase to Field Mutual Inductance
13
Phase Self Inductance
13
Three Phase Synchronous Machine Without Damper
Windings
^^
Damper Windings
19
Damper Modeling
21
Damper Self Inductance
21
Quadrature Axis Damper to Field Mutual
23
Damper Winding Phase to Direct Mutual
23
Quadrature Axis Damper to Phase Mutual
24
Approximating Damper Mutuals
24
IV
III.
PHYSICAL REALIABILITY OF DAMPER WINDINGS
27
Lower Bound Physical Realizability of Three Phase
Machines
27
Lower Bound Physical Realizability of Damper Self
Inductances
IV.
29
Upper Bounding Damper Self Inductances
35
FAULT ANALYSIS BY CONSERVATION OF FLUX LINKAGES
39
Two Coil System
39
3 Phase Synchronous Machine Without Damper Windings . . -43
V.
3 Phase Synchronous Machine with Damper Winding
48
Non Symmetrical Short Circuits
49
Single Phase to Ground
49
Phase to Phase Fault
52
Phase to Phase to Ground Fault
54
Calculation of Fault Torques
56
CONCLUSION
67
REFERENCES
69
LIST OF TABLES
Table
^-1.
4-1.
Lower Bound Realizability Criterion for a 3 Phase Machine
With Damper Windings
28
Derivitative Values of Machine Inductance Parameters With
Kespect to Rotor Shaft Angle
58
VI
LIST OF FIGURES
Figure
Page
II-l.
Arbitrary 2 Circuit System in Space
6
II-2.
Three Coupled Inductors
9
II-3.
Single Phase Machine Model
12
II-3a. Phase A Self Inductance Variation
15
II-4.
Three Phase Machine Without Damper Windings
16
II-5.
Inverse Orientation of Flux Generated by Phase A and Phase
B Windings
18
II-5a. Phase to Phase Mutual as a Function of Rotor Shaft Angle
For a 3 Phase Alternator
20
II-6.
Three Phase Machine with Damper Winding
22
11-7.
Electrical Model of the Field to Damper to Phase
Arrangement
25
III-l.
Coupling Coefficients of Phase to Damper to Field
Arrangement
37
IV-1.
Electrical Diagram of Single Phase Machine Fault
40
IV-2.
Electrical Diagram of Three Phase Machine Fault
45
IV-lab.Three Phase Short Without Damper Windings o = 0°, 45°. . .47
IV-lc. Three Phase Short Currents for Machine Without Damper
Windings 0 = 180°
50
IV-2a. Three Phase Fault Currents for Machine With Damper
Windings 0 = 0°
50
IV-2bc.Three Phase Fault Currents for Machine With Damper
Windings 0 = 45°, 180°
IV-3.
51
Single Phase to Ground Fault Current
vn
53
IV-4-
Phase to Phase Fault Current
53
IV-5.
Phase to Phase to Ground Fault Current
55
IV-6ab.Fault Torque of Three Phase Machine Without Damper
Windings 0 = 0°, 45°
62
IV-6c. Fault Torque of 3 Phase Machine Without Dampers 0^ = 180° .63
IV-7a. Fault Torque of 3 Phase Machine with Dampers 0^ = 0°. . . .63
IV-7bc.Fault Torque of 3 Phase Machine with Dampers 0^ = 45°, 180''64
IV-8a. Single Phase to Ground Fault Torque
65
IV-8b. Phase to Phase Fault Torque
65
IV-8c. Phase to Phase to Ground Fault Torque
66
IV-9.
Tabulated Minimum and Maximum Torque for Phase A to Ground
Fault for the Full Range of Shorting Angles
66
vin
CHAPTER I
INTRODUCTION
The design and construction of synchronous rotating machines is
a well established art.
Large generation systems are built to with-
stand virtually any kind of line fault situation.
Modeling the
fault response characteristics is an important part of the overall
machine design process.
An involved part of modeling an alternator is the determination
of a set of inductance parameters to represent the electrical characteristics of the machine.
The damper winding presents specific
difficulties in the determination of its inductive parameters.
The
geometry of this "winding" does not lend itself easily to field theory
estimates of its inductance.
It can be shown by basic physical prop-
erties of coupled inductors that damper inductive parameters are restricted to a specific range of values.
This range is a function of
the other inductive parameters and the physical configuration of the
machine.
Such ranging can be useful in the determination of the in-
ductive parameters of the damper winding.
In the early 1900's a principal known as the "conservation of
flux linkages" was developed.
In short this principal states that
the magnetic flux linking any closed system having zero resistance
and no voltage sources cannot change with time.
Provided winding re-
sistances are small, this principal can be applied to calculate the
initial short circuit current response of a rotating machine.
The theorem of conservation of flux linkages is the basis of the
derivation of the well known transiant and sub transient reactance
2
parameters, x^' and x ^ " . The method described in this thesis takes
advantage of the computational capabilities of a computer.
In so
doing, some of the approximations associated with the deriviation
of transient and sub transient reactance parameters need not be made.
The results should therefore be closer estimates of the actual fault
characteristics of the machine.
An interesting question to pose is why such aspects of machine
transient analysis were not dealt with years ago.
Rotating machines
have long been in use so that the exact calculation of fault torques
and currents has been of major importance to systems design.
The
answer to this question might be summed up in what might be called
on "overkill" philosophy to machine design.
The overkill design philosophy might be stated as simply overbuilding any machine component that might be subject to stress during
a fault.
A rough modeled calculation of the maximum possible limits
of transient fault torques for machine mechanical design and fault
currents for system breaker design is sufficient to assure system
protection.
The exacting analysis of the complex field interaction
in a rotating machine is unnecessary in this case.
Obviously this
conservative approach to motor design is adequate for most alternator
applications.
Recent proposed applications of rotating machines have led to
a need for more exacting criterion in their design.
This is specific
ally important if the overall weight of the machine is a constraint.
This has opened up a need to Investigate transient models in detail.
The intent of this thesis is not to approach the overall problem
of transient modeling.
It is intended to shed some light on the
previously stated particulars so that future development of transient
modeling criterion may be more exacting.
CHAPTER II
MACHINE INDUCTANCE PARAMETERS
The inductance parameters of a rotating machine are used to
relate the electrical characteristics of the machine terminals to
2 3
the physical configuration of inductors in the machine. * In this
chapter a detailed study of these inductance parameters is presented.
The development of this material includes the basic properties of
coupled inductor systems, development of basic machine parameters
and overall final development of a machine matrix for a three phase
synchronous machine with damper windings.
The development of the inductance parameters of a machine is
relatively straight forward.
Effectively the machine windings are
modeled as simply oriented loop inductors.
The basic problem is to
model the electrical interaction as a function of the internal configuration of inductors in the machine.
In this regard a machine
can be thought of as a complex transformer which has varying inductance parameters are a function of specific machine constants and
the instantaneous rotor shaft angle.
To include saturation and resistive effects in our inductance
parameters would unduly complicate the analysis.
Past work has
shown that neglecting resistance does not significantly alter the
calculated fault current and torque response of the modeled device
4
for the first few cycles after the occurrence of the fault. Efficient
machines are typically of low winding resistance.
To neglect saturation effects leads to a linear flux current
relationship.
Straightforward methods exist for digital solution of
coupled linear equations.
It is important to note however, that
depending on the fault situation, neglecting saturation effects
may not be valid.
Inductor linearity is strongly distorted if the flux
density in the inductor achieves saturation levels.
Inductor Interaction
The electrical characteristics of inductor interaction are
well understood, provided the parameters used accurately reflect
the physical situation.
Several methods exist to determine the
exact values of inductances.
through the device terminals.
field theory techniques.
First they can be directly measured
Secondly, they can be calculated by
In order to perform such a calculation a
thorough understanding of the physical conditions in the machine
must exist.
In calculating self and mutual inductances by field theory
techniques for any but the simplest idealized coil orientations, the
exact expressions can be very involved.
For this reason inductive
parameter models for machines are derived from simple coupled inductor
configurations.
As a starting point to the development of inductive
parameters for a machine consider a simple 2 circuit system.
Two Circuit System
Consider the circuit diagram of Figure II-l. Given a certain
current I^ flowing through loop 1 a flux is generated which is a
1 /v
Figure II-l. Arbitrary 2 circuit system in space
linear function of the applied current.
^^ = L^I^
The parameter L^ is known as the self inductance of loop 1.
2-1
If this
is in some type of simple loop configuration the inductance is a linear
function of the total flux and the number of turns.
h = -17
2-2
where N^ is the loops of circuit 1
I-j is the loop 1 current
4>1 is the flux from L,
If an external flux source such as a second circuit generates an
additional set of flux lines i^2iWhich penetrates the first circuit
in the same direction as \i>-,,
*21 = "21II
2-3
the overall combined flux through the first circuit is:
1^1 = L^I^ + M^2^2
2-4
Where M-i^ is called the mutual inductance and is dependent on the
relative geometry of circuit 1 and 2 along with the permeability of
the intervening substance.
Combining flux linkage equations for both circuit 1 and circuit
2 we get
ij^l = L^ I-j + M2112
2-5
8
Approximating Mutual Inductances
At this point it is convient to deal with some of the problems
related to the determination of damper winding parameters.
Consider the coupled inductor circuit of Figure II-l redrawn in
Figure II-2 with the addition of a single turn coil Lp that occupies
an intervening space.
The total flux \\>-^2 coupling L, to L2 can be written as
*12 = V 1 2 = ^l"l2
2-6
Also the total flux !)/]„ linking L, to L Q can be written:
*1D = V l D = 'l"lD
2-7
where <^-^2 ^^ ^^^ total flux from L, that couples L2
4>iQ is the total flux from L^ that couples Lr.
^12' '^ID ^^^ ^^^ respective mutuals
I-i the current through L-,
N2, Nrj are the respective turns of L2 and L^
Because L Q occupies the space between L, and L2 the following
definition is made:
K4']2 ~ *^1D
2-8
Where K is a constant whose value is less than one and can be
approximated from the physical configuration.
Rearranging equation
2-6 the result is:
MI
*12 = l i p
2-5
Figure II-2. Three coupled inductors
10
By equation 2-8 this implies that:
^2^1
*iD = -o7
^"^^
The mutual inductance of L^ to LQ may be difficult to calculate.
Based on the accuracy of K in Eq. 2-8 a good approximation of M^^
can be made.
By eq. 2-7:
^ID-TTBy substituting this result into Eq. 2-10 one obtains:
•^iD = 17 (nn^) = -mq
^-"'^
An analogous expression can be obtained to relate Lg to L2 as
follows:
^2D
K'N^
2-13
Where K' is the analogous to the constant K in Eq. 2-8 and N-. is
the inductor turns of Li.
This technique to approximate the value of the mutual inductance
of an intervening inductor will be useful when applied to damper windings.
But first the inductive parameters of the rest of the rotating
machine will be developed.
Single Phase Machine Inductance Parameters
The interactive coil arrangement of a rotating machine is dynamic
in nature.
For this reason the inductance parameters are modeled as
functions of shaft angle.
The shaft angle is arbitrarily referred
to the phase winding of the machine.
11
Consider Figure II-3 of a single phase machine as an initial
example.
0 is defined as the instantaneous rotor angle, A is the
stator loop winding.
The same form of the flux linkage equations
in Eq. 2-5 can be applied to the machine in Figure 11-3 giving
the following result:
^A = L A ( Q ) I A ^ ^FA^^^^F
2-14
%
= f^AF^^^^A ^ Lp(0)lp
where L^^ is the stator or phase self inductance
Lp is the field or rotor self inductance
My^P is the phase to field mutual inductance
Mp^ is the field to phase mutual Inductance
In light of the simplicity of this machine model, the form of
each of these inductance parameters L A , Lp, and M«p can be easily
arrived at as functions of rotor shaft angle.
Field Self Inductance
The field self inductance is a function of the air gap variation
about the field winding.
gap is a constant.
For a cylindrical stator machine the air
For this reason the field self inductance can be
specified as a constant:
Lp(0) = Lff
For some machines the internal air gap between the field and the
stator may be a function of shaft angle due to some non cylindrical
aspect of the stator winding arrangement.
In this case the field
12
Figure II-3.
Single phase machine model
13
self inductance should reflect the added complexity.
Phase to Field Mutual Inductance
The phase to field mutual inductance is a periodic function with
respect to the rotor shaft angle.
It is a maximum at 0 = 0° when
the field winding is directly in line with the phase winding.
The
mutual inductance is zero when the field loop in perpendicular to
the phase loop at 0 = 90°. There can be no common flux linkages
between F and A in that orientation.
When the rotor is positioned
at 0 = 180°, the mutual inductance is negative the original magnitude
at 0 = 0°.
Although design criterion for machine flux linkages may vary,
a sinusoidal distribution between minimum and maximum is assumed in
the development of these inductance parameters.
For a sinusodal
flux distribution the phase to field mutual is thus of the form:
Mp- = M^p = Laf cos 0
1-15
Where Laf can be theoretically determined as the mutual inductance
when the rotor is positioned at 0 = 0°.
Phase Self Inductance
The self inductance of the phase or stator winding is a function
of the variation of the internal air gap about the rotor.
As with
any Inductor with a moving core the inductance value must always be
positive.
In the case of a single phase machine, superimposed
on
this positive level is a harmonic variation produced by the changing
air gap about the rotor.
If the rotor is cylindrical or non salient
14
pole, there is no self inductance variation.
The rotor to phase air gap is a minimum at both 0 = 0 ° and
0 = 180°. This harmonic component is double the frequency of the
shaft angle rotation. Due to the assumed sinusoidal flux distribution
about the rotor and the fact that the air gap varies at twice the frequency of the rotor the following result is obtained:
L^ = Steady level + rotor level change
= Laao + Lg2 cos(20)
2-16
where Laao is the self inductance of the phase at a shaft angle of
45° and Lg2 is the increase in phase self inductance when the shaft
angle is adjusted to 0 = 0°.
The plot of the resultant self inductance is the
periodic
function of Figure II-3a.
Having redefined the dynamic induction parameters M«p and L»
in terms of machine constants Laao, Lg2 and Laf the flux equations
can be rewritten:
ibfl = [Laao + Lg2 cos(20)]Irt + [Laf cosojlr
^
^
•"
iPp = [Laf cos0]I^ + Lff lp
2-17
A development of the parameters of multi phase machines can
now be considered in terms of those developed thus far.
3 Phase Synchronous Machine Without Damper Windings
Consider the machine diagram of Figure II-4. The phase self
inductances are identical to those of the single phase case except
for the phase angle shifts for the B and C phase windings. The
15
Figure II-3a.
Phase A self inductance variati on
16
Figure II-4.
Three phase machine without damper windings
17
resultant equations are therefore:
L^ = Laao + Lg2 cos(20)
2-18
Lg = Laao + Lg2 cos(23 + 120°)
LQ = Laao + Lg2 cos(20 - 120°)
Analogously the phase to field mutuals also incorporate the phase
shift terms for B and C windings:
^AF ^ ^FA " ^^^ ^ ° ^ ^
2-19
Mgp = Mpg = Laf cos (0 - 120°)
MQP
= Mp^ = Laf cos (0 + 120°)
As with the single phase motor example, the field winding self
Inductance is also modeled as a constant because of the cylindrical
stator configuration.
Lp = Lff = constant
The remaining phase to phase mutuals are more involved.
Consider
the relative orientation of phase A to phase B in Figure II-5.
The
current I^^ through loop A is in an inverse orientation with respect
to current Ig through loop B.
Therefore, the opposing direction of
the respective flux lines ipj^ and \p^ produces a negative mutual inductance M«g.
The action of the salient pole rotor periodically distorts the
flux lines between phase A and B.
The flux line distortion is a
function of the rotor to stator air gap geometry.
produces the phase self inductance variation.
This same air gap
Thus the magnitude
of this inductance level variation is the same for both phase self
18
Figure II-5. Inverse Orientation of flux generated by Phase A
and Phase B windings.
19
inductance and the phase to phase mutual inductance.
The graph of the mutual inductance for phase B to C is shown
as a function of shaft angle in Figure II-5a.
The phase to phase
mutuals can thus be defined in the following form:
M^g = Mg^ = -.5Lgo + Lg2 cos(20 - 120°)
2-20
'^AC " ^CA " --SLgo + Lg2 cos(20 + 120°)
Mg^ = M^g = -.5Lgo + Lg2 cos(20)
As stated previously, a variety of assumptions are made in the
decivation of these inductance parameters.
Restated these include
cylindrical stator, sinusoidal flux distribution, linear inductances,
and symmetrical rotor configuration.
The degree of accuracy and
precision of the numerical values of these inductance parameters is
dependent in part on how closely the particular machine fits those
approximations.
Damper Windings
Damper windings in synchronous machines are typically constructed
as a series of equally spaced bars placed symmetrically about the
field winding.
The bars are all shorted together at the ends of this
"squirrel cage" arrangement.
This configuration is usually symme-
trical about the rotor axis.
Unlike the other windings previously discussed, damper windings
respond significantly only under transient conditions.
They are in
fact not a winding in the usual sense at all but a metallic configuration that has important inductive properties in the fault response
20
Shaft Angle
270
36.0
-.5Lqo.
Figure II-5a. Phase to phase mutual as a function of rotor
shaft angle for three phase alternator. Mgp = -.5Lgo + Lg2 cos(20)
21
of the machine.
Direct calculation of damper winding induction parameters
by field theory techniques is a very complex problem.
This problem may
be partially avoided by relating the damper winding configuration to
the other windings in the machine.
It has been shown in Eqs. 2-12
and 2-13 that the mutual inductance of an unknown third winding can
be related to the knowns of the first two windings.
This is provided
that this third inductor occupies the region between L-j and L2.
It
can be shown that this directly applies to several damper winding
inductance parameters.
The following chapter will deal with the
problem of bracketing the values of the damper self inductances by
physical constraints.
The general model of the damper winding is now
developed.
Damper Modeling
The components of the damper inductance are modeled as two seperate inductors along the perpendicular directions which correspond
to the direct and quadrature rotor axes.
The direct axis is in line
with the rotor north and south pole and the quadrature axis leads it
by 90° as shown in Figure 11-6.
The damper is attached to the field winding in a symmetrical
structure.
The numerical form of the model of the inductance para-
meters for both field and damper are thus similar.
Damper Self Inductance
As with the field winding, the damper sees a constant phase to
damper air gap regardless of the position of the rotor.
This is due
to the fact that the stator is assumed to be cylindrical in struc-
22
Quadrature Axis
Shorted Ring for Damper Winding Bars
Direct Axis
Figure II-6.
Three phase machine with damper winding
23
ture as whown in Figure II-6.
For this reason the self inductance
of both the direct and quadrature axes of the damper winding are independent of the rotor position and designated by:
Direct Axis Damper Self Inductance = LQ
Quadrature Axis Damper Self Inductance = LQ
If the field and stator configuration are cylindrical then LQ = L Q .
In this case the air gap around the symmetrical damper winding is
cylindrical.
Quadrature Axis Damper to Field Mutual
The inductance component of the damper along the quadrature axis is
perpendicular to the field winding and will have no common flux linkages
This direct to quadrature axis mutual \nur)
is therefore by definition,
equal to zero.
Damper Winding Phase to Direct Mutual
Because the direct axis of the damper winding is in the same
orientation as the field winding its form and period is modeled
identical to that of the field winding.
Therefore the mutual induc-
tance between the direct axis damper and the phase winding is of
the analogous form:
^AKD ~ ^^^^ ^°^®
2-21a
Mg^Q = Lakd cos(0 - 120°)
M(,^Q = Lakd cos(0 + 120°)
Where Lakd is the peak magnitude of the direct damper to phase mutual
over a full shaft angle rotation.
24
Quadrature Axis Damper to Phase Mutual
The quadrature axis to phase mutual inductance models the effect
of the damper winding configuration in an orientation 90° ahead of
the direct axis.
The form of this mutual inductance parameter is
thus identical to that of the direct mutual above with the appropriate 90° phase shift.
M^^Q = Lakq cos(0 - 90°) = Lakq sine
2-21b
Mgj^g = Lakq cos(0 - 120° - 90°) = Lakq sin(0 - 120°)
M^I^Q = Lakq cos(0 + 120° - 90°) = Lakq sin(0 - 120°)
Lakq is the peak quadrature damper to phase mutual over a full shaft
angle rotation.
For a cylindrical damper rotor arrangement the peak
phase to damper mutual inductances for both quadrature and direct axes
are identical.
Therefore, it can be concluded that:
Lakd = Lakq.
Approximating Damper Winding Mutuals
The derivation previously presented on approximating mutual inductances can now be directly applied to these damper mutuals.
Con-
sider Figure II-2 redrawn in Figure II-7 where Lp is the cylindrical
field inductor, L-. is the damper winding loop inductor and L^ is
the phase inductor of the cylindrical stator.
The field to damper mutual can be approximated by Eq. 2-12 as
follows:
M
= _L^_^
•^FKD
K N^
where ^cun is the field to direct axis damper mutual
2-22
25
^
AF
Figure II-7. Electrical model of field to damper to
phase arrangement.
26
Mp^ is the field to phase A mjtual inductance
N^ is the number of phase A turns
K is the proportion field to damper coupling flux
that also couples the phase winding
This value of K was generally defined in Eq. 2-8 and is thus
defined In this field-damper-phase arrangement as follows:
*FA " "^^FD
2-23
A good approximation can be made of the 2 unknowns NQ and K
of Eq. 2-22.
The damper inductances can be thought of as an equivalent
single turn.
Therefore, the value of NQ is one.
All of the flux lines
that pass from field to phase must pass through the damper winding configuration.
Flux leakage is inherent in any coupled inductor system.
Depending on the specifics of the damper to field configuration a good
approximation may be a value of K near unity.
Similiar arguments can be made in the derivation of the damper to
phase mutual the expression is:
" A K D = " F A / ( ^ ' Np)
2-24
Where Np is the number of turns of the field winding, and K'is defined as follows:
•^^AD " *AF
^"^^
Chapter III discusses constraints that can be applied to damper
self inductance values utilizing basic properties of these coupled
inductor configurations.
CHAPTER III
PHYSICAL REALIZABILITY OF DAMPER WINDING
SELF INDUCTANCE PARAMETERS
The Inductance parameters of coupled inductor systems are by
physical law constrained to a specific range of values with respect
to one another.
If a given inductive parameter is outside of this
realizable range, the system in theory, could not be constructed.
The specific criterion that will be used as a test for the lower limit
of physical realizability is that the square of the mutual inductance
cannot exceed the product of the self inductances in a coupled inductor system.
This relationship is identical to stating that K <_ 1
where K is the coefficient of coupling.
This constraint can be
applied to any pair of windings in a machine.
Generally the application of this criterion to the case of
revolving inductors in a machine is much more complex than the
analysis of static inductor configurations.
Fortunately in the case
of damper winding inductance parameters, expressions can be obtained
which relate this constraint directly to the machine constrants.
Effectively these expressions lead to a discrete lower and upper limit
greater than zero for the damper winding self inductance values.
Lower Bound Physical Realizability of 3 Phase Machines
The model of a 3 phase machine developed in the previous chapter
included 6 self inductance parameters:
'-A' ^B' ^ C ^F' ^D* ^Q
27
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cy
s^
21
A|
Q
_l
<:
A|
cr
1
<
29
For any 2 inductor system L^ L2 the physical r e a l i z a b i l i t y criterion
''•
2
L^L2 1 M^2
3-"'
By applying Eq. 3-1 to all possible combinations of the previous
set of inductors in a 3 phase machine the 15 relations of Table 3-1
must be satisfied for the given parameter set to be physically realizable.
In general the application of Table 3-1 constraints to the full
machine can be very tedious. This is due to the fact that most relationships are functions of rotor shaft angle.
It will now be shown
that the 0 dependence can be eliminated in the case of the damper
winding parameters thus simplifying the application of Eq. 3-1.
Lower Bound Physical Realizability of Damper Self Inductances
Substituting in the 0 dependent expressions Eqs. 2-18, 2-21a,
2-21b for those Table 3-1 relationships dependent on damper self
inductances the following results are obtained.
[Laao + Lg2 cos(20)] LQ >^ [Lakd cos 0 ] ^
3-2a
[Laao + Lg2 cos(20 + 120)]Lr, 1 [Lakd cos(0 - 120°)]
^
3-2b
[Laao + Lg2 cos (20 - 120)]Ln >_ [Lakd cos(0 + 120°)]
^
3-2c
LFI-D >- ^FKD^
LQLQ >_ 0
^-^'
3-2e
For the quadrature damper self inductances the relations are:
2
[Laao + Lg2 C O S ( 2 O ) ] L Q >_ [Lakq sin o]
3-3a
[Laao + Lg2 cos(20 + 120)]Ln > [Lakq sin(0 - 120)]^
^"
3-3b
[Laao + Lg2 cos(20 - 120)]Ln > [Lakq sin(0 + 120)]^
^
3-3c
30
LpLg >_ 0
3-3d
LQLQ >_ 0
3-3e
Expressions 3-2d,e and 3-3d,e deal with machine constants involving straightforward numerical comparisonss. The relationships
of interest are therefore 3-2a,b,c and 3-3a,b,c corresponding to the
phase winding to damper mutuals.
By symmetry the configuration of each phase winding is a function
of angle only.
If a non 0 dependent expression is developed which
corresponds to Exp. 3-2a for example, it is therefore also applicable to Exp. 3-2b and 3-2c. Thus a detailed study of Exp. 3-2a for
the direct damper self inductance and Exp. 3-3a for the quadrature
damper self inductance will lead to expressions which include all
phases.
Consider first the Expression 3-2a rewritten as follows:
^A^D ^ ^AD^
which implies
M
0
I
AD ^ (Lakd cos 0)
^ .
^D - L^
Laao + Lg2 cos(20)
^^"^
The maximum possible value for the term on the right is desired
Taking the derivitative of Exp. 3-4 and setting it equal to zero
the result is:
0 =
-2Lakd^ cos 0 sin 0
Laao + Lg2 cos(20)
2 Lakd cos 0 Lg2 sin(20)
(Laao + Lg2 cos(2:))
3-5
Values of 0 which are solutions to Eq. 3-5 correspond to minima and
maxima of Eq. 3-4. Multiplying through the squared self inductance
31
term of Eq. 3-5 the result is:
2
(-2Lakd cos 0 sine) (Laao + Lg2 cos(2:))
+ 2Lakd^ Lg2 cos^ 0 sin(2e) = 0
3-6
The self impedance L^ term Laao + Lg2 cos(23) is always greater
than zero.
Therefore, the values of e which set Eq. 3-6 to zero fall
into 2 groups:
0 = 0 ° , 180°, 360°, ...n(180°) correspond to the sine terms
0 = 90°, 270°, 450°, ...n(180°) + 90° correspond to the
cosine terms
Expanding Eq. 3-6 the result is:
2
-2Laao Lakd
+ 2Lakd
2
cos0 sino - 2Lg2 Lakd
coss sin9 cos(20)
cos^ 0 sin(20)Lg2 = 0
3-7
To find which 0 values correspond to maxima of Exp. 3-4 the
2nd derivitative test can be applied.
Taking the derivitative of
the left hand side of Eq. 3-7 the result is:
-2Laao Lakd^[-sin e + cos^0]
-2Lg2 Lakd^[-sin^0 cos(20) + cos^ e cos(2e)
-2cos 0 sin 0 sin(2e)]
+2Lakd^ Lg2[-2cos 0 sin e sin(20)
2
+2cos 0 cos(20)]
3-8
Substituting in all 0 values of interest into Exp. 3-8 for one
period of shaft angle rotation 0 = 0° to 0 = 360° the results are:
at 0 = 0°:
-2Laao Lakd^ (1) -2Lg2 Lakd^ (1) + 2Lakd^ Lg2(2)
32
at 0 = 90°:
-2Laao Lakd^ (-1) -2Lg2 Lakd^ (1) + 2Lakd^ Lg2 (0)
at 0 = 180°:
-2Laao Lakd^ (1) -2Lg2 Lakd^ (1) + 2Lakd^ Lg2 (2)
0 = 270°:
-2Laao Lakd^ (-1) -2Lg2 Lakd^ (1) + 2Lakd^Lg2 (0)
So terms which correspond to 0 = 0°, 180°, 360°, ...n(180°) are
identical. They are equivalent to:
-2Lakd^ (Laao - Lg2)
3-8
Those which correspond to 0 = 90°, 270°, 450° ...n(180°) + 90° being
also equivalent are:
2Lakd^ (Laao - Lg2)
3-9
As stated in Chapter II Lg2 is always less than Laao because the phase
self inductance is always positive. Therefore, Exp. 3-9 is always
positive and Exp. 3-8 is always a negative term. This implies that
0 = 0°, 180°, 360° ...n(180°) correspond to Eq. 3-4 maxima.
A check must also be made to determine whether the maxima of
Eq. 3-4 during one shaft angle rotation correspond to absolute or
local peak values.
Substituting 0 = 0 ° into Eq. 3-4:
.
no
® " "
,
^ K D ' ( Q ° ) _ Lakd^
"-D - L^ (0°) ' Laao + Lg2
0 = 180° U•D ->
^ K D ^ (13°°)
Lakd^
L, (180°)
Laao + Lg2
33
The results are identical, therefore the ninimum realizable valje
of the direct damper self inductance is:
LQ >
Lakd^
Laao + Lg2
3-10
Similiarly it can be shown that the minimu;n realizable value
of the quadrature axis damper inductance is:
I > Lakq
^Q - Laao - Lg2
3 ,,
"^ "
A d e r i v a t i o n of Exp. 3-11 now f o l l o w s :
The physical r e a l i z a b i l i t y Exp. 3-3a i s :
>
L^LQ
M^Q2
3.33
Which implies
M
2 2
I ^ ._AQ ^ Lakq sin 0
^ -, ^
Q - L^
Uaao + Lg2 cos(20)
•^'' ^
Taking the derivitative of 3-13, setting it equal to zero, and simplifying, the result is:
2
2
Lakq Laao sin 0 cos 0 + Lakq Lg2 sin 0 cos(20)
+ 2Lakq^ Lg2 sin^ 0 cos 0 = 0
3-14
A similiar range of solutions to the LQ case is found.
0 = 0 ° , 180°, 360° ...n(180°) corresponding to sine terms
0 = 90°, 270°, 450° ...90° + n(180°) corresponding to
the cosine terms
The second derivitative of Exp. 3-14 is:
34
Lakq
Laao (cos 0 - sin^0) + Lakq^ Lg2 (cos^j cos(2:)
2
-sin 0 cos(29) -2sin0 cos9 sin(23)
+ 2 Lakq^ Lg2 (3 sin'^0 cos^O - sin^0)
3-15
Testing a full period of 0 values as in the LQ case,
at 0 = 0° Exp. 3-15 is:
Lakq^ Laao(+l) + Lakq^ Lg2(l) + 2Lakq^ Lg2(0)
at 0 = 90°:
Lakq^ Laao(-l) + Lakq^ Lg2(l) + 2Lakq^ Lg2(-1)
at 0 = 180'
Lakq
Laao(+l) + Lakq^ Lgw(l) + 2Lakq^ Lg2(0)
at 0 = 270°:
Lakq^ Laao(-l) + Lakq^ Lg2(l) + 2Lakq^ Lg2(-1)
Once again a grouping of identical solutions is obtained:
for 0 = 0°, 180°, 360°, ...n(180°) Exp. 3-15 becomes
Lakq (Laao + Lg2)
which is always positive in value.
For 0 = 90°, 270°, 450°, ...n(180°)
+ 90° Exp. 3-15 becomes
-Lakq^(Laao + Lg2)
which is always negative in value.
So this result corresponds to 0
maxima of Exp. 3-13.
Checking for relative or absolute maxima in Exp. 3-13 for 0 = 90°:
I
^
Lakq^ sin^ (90°)
D - Laao + Lg2 cos[2(90°)]
Lakq^
Laao - Lg2
35
for 0 = 270°:
I
>
Lakq^ sin(270°)
Lakq^
D - Laao + Lg2 cosL2(270°)] ' Laao - Lg2
The maxima at 0 = 90°, 270°, 450° ...90° + n(180°) are thus
identical.
Therefore, expression 3-11 is the quadrature damper
physical realizability constraint as desired.
Upper Bounding Damper Self Inductances:
It has been shown up to this point that Eq. 3-1 leads to an
easy test for lower bound of damper winding self inductances.
In-
tuitively one might expect that some type of realizable upper dound
must also exist.
Searching for such a constraint in the realtions
of Table 3-1 it is found that none of these expressions constrain
any of the self inductance parameters in the machine.
This result
is easily understood because Eq. 3-1 limits only the maximum coupling between any two inductors, not the maximum value of the self
inductance parameters involved.
In order to come up with some form of limiting criterion to
physical realizability of damper self inductances one must turn to
some consideration of the physical configuration of the winding.
Consider Eq. 3-1 rewritten in the following familiar form:
K 12
M.
'12
3-15
/Tal'-2
where K12 is the well known coefficient of coupling between inductor
L, and L2.
The coefficient of coupling, which has a value less than
or equal to one, is a function of the mutual physical configuration
36
of any two inductors L^ and L2. A theoretical maximum value of a particular unknown self inductance parameter is constrained to be the
minimum realizable value of the coefficient of coupling.
This realiz-
ability criterion is described for the general 2 inductor case as
follows:
h MAX <
M 12
Kp
2
" ~Lp
MIN
^
3-16
where all of the terms on the right are known or closely approximated.
Application of this criterion to the case of the self inductance
of the damper winding presents similiar problems to those of the
determination of the damper mutual as described in Eq. 2-22.
Consider
Figure III-l of the damper configuration, where the K terms are the
coefficients of coupling of the respective windings.
Eq. 2-23 relates
the flux from field to phase to the flux coupling field to damper.
The value of K in 2-23 is directly related to the amount of field
flux that couples the damper but fails to couple the phase winding.
The field winding is therefore more closely coupled to the damper
than to the phase winding.
The respective coupling coefficients
should therefore reflect this fact as follows:
•^FA ^ "^FD
3-17a
Corresponding to the phase A coupling coefficient the following
analogous expression can be stated:
•^FA ^ '^AD
3-17b
37
^"^
c^
6
LD 'OF
6
^
vJ
Figure III-l. Coupling coefficients of phase to damper
to field arrangement.
38
Knowing t h a t :
_ iMaf
K
AF
/TTL
A'F
M
AKD
and K
AD
3-18
/TTL
A"D
and using Exp. 3-17b the r e s u l t is
M
AF
M
AKD
3-19
' ' ^ ^
which Implies that:
M,
Lo < i-^^)'
MAF
L,
3-20
Analogously for the field to damper case:
I
< (JJ<D_ ^2 ,
-A
M
FA
3-21
Removing the 0 dependence from 3-20 and 3-21 the result is
Ln
< (^=f^)'Lff
•D
Laf
3-22a
,
,Mfkd x2 /,
^ , ^x
LQ < (-[jp-) (Laao + Lg2)
3-22b
Exp. 3-22a,b are therefore the maximum limiting criterion for
physical realizability of the damper winding inductance parameters
in an alternator.
The constraints which are placed on the damper self inductances
effectively place a lower and upper bound on their inductance value.
These constraints are completely stated in expressions 3-2d,e, 3-3d,
3-10, 3-11 and 3-22.
Thus self inductance parameters of the damper
winding can be easily tested to determine if they are physically
realizable.
CHAPTER IV
FAULT ANALYSIS BY CONSERVATION
OF FLUX LINKAGES
The study of fault analysis using the pricipal of conservation
of flux linkages is not new.
Papers dealing with some aspects of
this powerful tool date back to the early 1900s.
The statement of
this principal is straightforward and simple:
"In any closed circuit without resistance the flux linkages
must remain constant.
It doesn't matter how many secondary
circuits there are, or what the network involves, the theorem
4
is rigidly true."
In this chapter the principal of conservation of flux linkages
will be used in several ways.
First it will be applied to the general
question of short circuit analysis of rotating machines.
The short
circuit current response will then be used in the calculation of the
torque response during the initial period after the fault.
Two Coil System:
The utility of the method of conservation of flux linkages becomes apparent when the self and mutual inductances are allowed to
vary as a function of a third parameter.
The case of analyzing a two winding rotating machine is now
considered.
The electrical fault is modeled as in Figure IV-1.
problem is to calculate I«(0) and lp(0) for all 0 after t=0.
Using the flux linkage equations presented in equation 2-14:
39
The
40
(p4=o
t=0
6
\ / i Fo
Figure IV-1.
Electrical diagram of two winding machine fault
41
*A
= Lftlft + M A F I F
4-1
*F = % ' A ' Lplp
where L^ is the stator or phase self inductance
Lp is the field inductance
M^P is the stator to field mutual inductance
also it must be given that
If^ = Laao + Lg2 cos (20)
Lp = machine constant
M^P = Laf cos 0
where 0 is the designated shaft angle.
Given the initial current vector
0
I
t= 0 = 0
the initial flux vector can be calculated
i> A.
^i
Ip^ M,p(0)
Ip
Lff
0
The specific currents are now calculated as a function of shaft
angle for the above initial conditions flux vector.
The general case response equations are:
\l>f^^ = (Laao + Lg2 cos(20) ly^(0) + (Laf cos0)lp(0) 4-3a
IJ^PQ
= (Laf cos?)I^(0) + Lff lp(0)
4-3b
42
Solving for the current response lp(0) and 1.(0) the result by Eq
4-3a is:
^fi^Q - Laf cos 0 1^.(9)
Laao + Lg2 cos (23)
A^^^
4-4
Substituting this result into Eq. 4-3b
i|^« - Laf cos e lr(0)
*Fo = Laf cos 9 ( u a o > L92 cos (2o) ' ' •-" 'p^^'
Laf cos 0 rl>f^^
Laf cos^ 0 lp(0)
Uaao + Lg2 cos (2e) " Laao + Lg2 cos (20) ^ ^^^ ^F'^^
thus:
Laf cos 0 \i) Ao
^
Fo " Laao + Lg2 cos (20) " ^f'®' ^^^^
Laf^ cos^ 0 lp(0)
Laao + Lg2 cos(20) ^
4-5
So the following closed form solution for the short circuit response
is obtained:
ij^c^ Laao + ipr^ Lg2 cos(20) - ii.^ Laf cos0
'Fo
Ao
lp(0) = 'Fo
Lff Laao + Lff Lg2 cos(20) - Laf^ cos^0 lp(0)
4-7
Substituting the previous expression into Equation 4-4 a closed form
solution of the phase A current is obtained.
peated in matrix format for clarity.
The procedure is re-
Calculate the initial flux vector
from the initial current vector:
4-8
ij^ = L I
^0
""0
and thus the solution
0
0,
43
4-9
1(0) = [L(0)]"1 ^
Three Phase Synchronous Machine Without Damper Windings
The technique applied to the previous case can now by simple
extension be applied to larger systems.
to be solved is stated as follows.
The exact problem that is
Given a specific machine with an
open circuit phase windings, calculate the short circuit response for
a full shaft angle rotation if phase A,B, and C are shorted together
and the field winding is shorted to itself.
This circuit is modeled
in Figure Iv-2. The machine matrix is:
^A
"AB
"AC
"AF
"BA
4
"BC
"BF
"CA
"CB
"-C
"CF
"FA
"FB
"FC
h
L =
Where the inductances and mutuals are the following restated functions
of shaft angle.
Lp = Lff
^BA = ^ B
f^CB = ^BC
f^CA = ^AC
-.5Lgo + Lg2 cos (2e - 120°)
-.5Lgo + Lg2 cos (2o)
-.5Lgo + Lg2 cos (20 + 120°)
L^ = Laao + Lg2 cos(20)
Lg = Laao + Lg2 cos(20 - 120°)
L^ = Laao + Lg2 cos(20 +120°)
4-10
44
M^P = Mp^ = Laf cos 0
Mgp = Mpg = Laf cos(0 - 1 2 0 ^
MQP = Mp^ = Laf cos(0 + 120°)
Where Laao, Lg2, Lgo, Laf, Lff are all given machine constants
flux linkage equation for this problem is:
•
^
^B
•
M
*B
BA ""B
^c
^^AF
M
BC
M
BF
'B
=
• ' A
•"c
^CA
^CB
k
MCF
'c
*F
^FA
^FB
^FC
Lp
^F
Now consider a numerical example.
The machine constants are as follows:
Laao = 1.964 X 10"^H
Lg2 = 4.85 x 10"^H
Lgo = 1.8362 x 10"^H
Laf = 2.268 x 10"^H
Lf = 3.024 X 10"^H
For this example the field current is 200 amperes at the initial
shaft angle of zero degrees.
0
0
0
200
The current vector is as follows:
The
45
rcr
o^
>
F
-Pb0
y
o^
t=o
a^
Figure IV-2.
Electrical diagram of machine fault
'FO"(^'
46
For a shorting angle of zero degrees the initial flux vector is:
>-2 cos 0) = 4.336
i>f^^ = 200 M^p(O) = 200(2.268 x 10"^
"^Bo " 200 Mgp(O) = 200[2.268 x 10"2 cos(0-120)] = -2.268
^Co ^ ^°° ^ C F ' ° ' " 200[2.268 x 10"^ cos(0+120)] = -2.268
-2 = 6.048
I^PQ = 200 Lp = 200(3.024 x 10"^)
4-11
Thus the following matrix equation can be solved repetitively for
the current response as a function of 0.
M
AB
'^AC
M
M
M
% 4
^CA
M
^CB
M
'TA
"^FB
BC
M
M
FC
-1
AF
4-536
IA(S)
BF
-2.268
IB(0)
CF
-2.268
1^(0)
F.
6.048
lp(0)
Digital computational techniques are now used on the matrix equations
and the calculated currents are plotted over a range of shaft angles.
0^ + 0° £ 0 £ 360° + 0Q
Where 0
is the angle at which the symmetric short takes place.
the matrix listed above ij> corresponds to 0
= 0°. The resjlts are
plotted for 3 seperate initial shorting angles.
the current response for 0
For
Figure IV-la shows
= 0°, Fig. IV-lb for 0^ = 45° and Fig.
IV-lc for 0Q = 180°.
The calculated severity of the magnitude of the fault response
is similiar for all three shorting angles although specific winding
response varies as a function of shorting angle
47
LE <DEGREES>
-2000
Figure IV-la. Three phase short currents for machine without
damper windings at 0 = 0°.
2000
B
u
0.
E
in
z
cr
UJ
X
EC
-2000
•-
Figure IV-lb.
at 0Q = 45°.
Three phase short with damper windings
48
Three Phase Synchronous Machine With Damper 'bindings
Inclusion of damper windings increases the size of the machine
matrix by two.
L=
•A
^B
M BA
^KD
M
BKD "^BKQ
\ B
M^
M
k
^CF
CKD "CKQ
M,
Mr-. "F
L,
"FB "FC
M
M
FKD "\ FDKKQQ
^BKD ^CKD ^FKD ^D
^AKQ
^BKQ ^CKQ ^FKQ ^KDKQ LQ
^CA
M
FA
•B
^C
f^BC
^AF
'\F
\KD
^ K Q
'
M
Where the newly added terms are as previously stated, the following
functions of shaft angle.
'^AKD ~ ^^^^ ^ ° ^ ®
M
= Lakd cos (0 BKD
120°)
4-12
M,
= Lakd cos (0 + 120°)
CKD
^AKO ~ ^^^^
sin 0
M,
= Lakq sin (0 - 120°)
MQKQ
= Lakq sin (0 + 120°)
BKQ
The f o l l o w i n g additional machine parameter values are included for
the damper winding terms
= 1.82 X lO'^H
Lakd = 1.832 x 10"^H
LQ
Lakq = 1.773 x lO'^H
Mpi^Q = 2.268 X lO'^H
Ln = 1.861 X lO'^H
^FKQ
= \ D K Q ~- ° ^
49
Computer techniques are once again used to solve the matrix
equation for the phase currents as a function of shaft angle. The
results are plotted in Figure IV-2a,b,c.
Figure IV=2a corresponds to an initial shorting angle of 0°,
IV-2b corresponds to 0^ = 45° and IV=2c for 0 = 180°.
The results indicate that the damper windings significantly
increase the calculated fault currents.
This is in line with the
observed fault response of a machine with dampers.
Damper windings
also affect the specific shaping of the current response.
Non Symmetrical Short Circuits
The method of conservation of flux linkages can be applied to
several connection configuration of a rotating machine provided the
zero resistance assumption is maintained.
By example several possible
methods of fault analysis for non symmetrical short circuits follow.
Single Phase to Ground
Modifying the flux linkage equations to satisfy this criterion
consists of removing the row and column terms from the matrix that
corresponds to the two open circuited phases.
The two open circuited
phases cannot contribute to the maintenance of the flux vector during
fault conditions.
For example if phase A to ground fault is to be considered the
flux linkage equations are:
TEX^ST ppij VJBR^^"^
50
E <DEEREXS>
^
2000
m
i*l 0
I
t
Q:
a:
6
z
X
-2000
•
Figure IV-lc. Three phase short without damper windings
at 0Q = 180°.
HOrr
RHBUE <I>EBREXS>
H0B0
Q
3000
hi
£
2000
^
1000
u
Q:
eft-B—I—^
a
-1000
-2000
•'
-3000
Figure IV-2a. Three phase short circuit currents for
machine with damper windings at 0 = 0 ° .
51
E <DCSRCC:S>
s
a. 200B
Figure IV-2b.
Three phase short with damper windings at o = 45'
E <PEGWELfa>
X -2000
-3000 -
Figure IV-2c.
Three phase short with damper windings at o = 180°.
52
^
A
^A
^AF
4*.
'^FA ^F
YD
M
^AKD
M
FKD
^''AKQ
f-l
'TKQ
M
I
M
AKD ' FKD ^D
'V^OKQ
M
M
f*!
I
AKQ " F K Q ' \ D K Q ^Q
4*,
D
where as before IQ and IQ correspond to currents in the modeled
damper windings.
Given the machine data as used in the symmetrical
short circuit case along with the same initial conditions. Fig. IV-3
is a plot of the calculated current response for the single phase
to ground case.
Although the shaping of phase A waveform is different,
the peak magnitude of the single phase to ground current in this case
is very close to that of the symmetrical fault.
Phase to Phase Fault
Calculation of the phase to phase fault follows a similiar procedure to that of the single phase to ground case above.
The open
circuit phase cannot contribute to the maintenance of the flux vector.
As with the single phase to ground case the terms corresponding to
that winding are thus removed.
There is the additional constraint
that the shorted phase currents are identical in magnitude and opposite
in direction.
This constraint can be applied using the same flux
equation format.
The resultant flux equations for the phase A to phase B fault
are:
53
E <PESMEC5>
Figure IV-3.
Single phase to ground fault currents
RNBLC <DCBrfEC5>
^N 3 0 0 0
B
S^ 2 0 0 0
Figure IV-4.
Phase to phase fault currents
54
*A
^A
^B
*B
^^BA
4
0
=
1
M
AF
M
^AKD
^AKQ
1
M
^BF
M
'^BKD
M
0
1
1
0
0
0
*F
^FA
^FD
°
k
^FKD
*D
^AKD ^BKD
°
*Q
'^AKQ ^BKQ
0
^
^KD4
^A
^B
-
•F
'FKQ
M
KDKQ
'D
M
M
LQ
"^FKQ "^KDKQ
'Q
>
r
Where row and column 3 correspond to the I^ = lg constraint. The
term corresponding to I^^ in the current vector is ignored and should
be calculated as zero for the full shaft angle rotation.
A plot
of this current response using the above modified flux equations is
shown in Fig. IV-4.
Phase to Phase to Ground Fault
The difference between the double phase to ground case and the
phase to phase case is simply the removal of the I = I constraint
X
from the flux linkage equations.
y
Effectively this is just an extension
of the single phase to ground problem.
For the phase A to phase B to ground case the flux linkage matrix
is as follows:
^
^
B
h
^
^(
D
^A
M
"BA
^AB
I
^B
^VA ^FB
^AF
M
FB
h
M
M
"AKD AKQ
M
M
"BKD "BKQ
^FKD
^""FKQ
^AKD ^BKD ^FKD ^D
^KDKQ
M
AKQ '^BKQ ^CKQ ^KDKQ ^Q
55
Figure IV-5.
Phase to phase to ground fault current;
56
A plot of the current response for the same initial conditions
and machine parameters as previous is shown in Fig. IV-5.
These three examples illustrate some of the calculatory procedures
that may be used for the non symmetric fault conditions utilizing the
principal of conservation of flux linkages.
Calculation of Fault Torques2
Given the fault currents and inductance parameters the fualt
torque can be readily calculated.
A derivation of the fault torque
leads to the following form of the expression.
dL,
dL«
dL
Torque = 1 / 2 l^ ^
. 1/2 I2 ^
^ ... 1/2 I^ ^
dM,^
dM,,
l/2Iil2-^-l/2l2li^-.--
dM
V2I^M^-^
where L-j ... L^ represent all of the self inductance terms.
... M
.
represent all of the mutual inductance terms.
Mi2»
Applying this
general torque expression to the case of a 3(j) machine without damper
windings the result is:
9 dLft
P dLp
P dLp
Torque = 1/2 l / - ^ + 1/2 l / - ^ + 1/2 l / -.^- +
C
de
^A
d0
'
B do
dM,
9 dLp
dM-R
dM
1/2IFSI^ V B ^ ^ ^ " A'C
^
dM
I«I
AT
AF
d3 + UBIT
dM
BF
+ I.I
CT
do
^B^C
da
^\F
d3
4-14a
do
If damper windings are also to be included, the following terms must
also be added to Eq. 4-14a terms above
2 ^Ln
D
•do
dM
p dL.
KI
AKQ
. ^B^Q
I I^ 05 ^
4-14b
57
4-14a cont.
dM
dM
"^C^D
AKD
dM
+ Ir>I
B^D d
BKD
dM,
dM.
? ^ + I I ""^KD . . . "••KDKQ
d3
V ^ D ^ 0 — ^ ^D^Q — d ^
Many of these terms are zero.
The others can be easily determined
from the machine parameter expressions of Eq. 4-10, and 4-12 and
are listed in Table 4-1.
58
TABLE 4-1
dM
AKD
de "= -Lakd sin 0
^'^BKD
d0 " -Lakd sin(o-120°)
^^CKD
do ~ -Lakd sin(0 + 120°)
^^KQ _
-Lakq cos 0
do
^^BKQ
do ^ = Lakq cos (0 - 120°)
^^CKQ _
Lakq cos (0 + 120°)
do
""FKD
do
- do
''LA
do
''LB
do
do
" -2Lg2 sin (20)
do
-2Lg2 sin (20
''Lc
do
" -2Lg2 sin (2o
""BC
do
' -2Lg2 sin (2o)
''"AC
do
do
+ 120°)
-120)
= 0
59
TABLE 4-1 cont.
dM
AB
dQ- = -2Lg2 sin (20
- 120)
dM
AF
do = -Laf sin o
^•\F
~ao- = -Laf sin(e - 120)
dM
CF
de = -Laf sin(e + 120)
60
The currents calculated in Figures IV-la tnrough IV-5 previously
presented are now substituted into the above torque expression Eq.
4-14 and plotted over the same shaft angle range.
Figure 17-6 is
the plot of the calculated torque for the 3 phase machine without
dampers.
IV-6a
0Q = 0° case
IV-6b
e^ - 45° case
IV-6c
e^ = 180° case
Figure IV-7 is the plot of the calculated torque for the 3 phase
machine with dampers case (lp
- 200 Amperes)
IV-7a
0Q = 0° case
IV-7b
0Q = 45° case
IV-7C
O Q = 180° case
Figure IV-8 is the plot of fault torque for the non symmetrical
fault examples given.
IV-8a Single phase to ground
IV-8b Phase to phase
IV-8c Phase to phase to ground
Several conclusions about the calculated fault torques can be
drawn.
The torque response to a symmetric fault is independent of
the initial shorting angle.
This holds for the case of a machine with
and without damper windings.
The torque response to a non symmetrical
short however is not independent of the starting short angle.
This
61
effect is displayed graphically in Figure r.'-9 for a phase A to
ground fault.
The peak minimum and maximum torque is tabulated for
a range of initial shorting angles from 0 to 360 degrees.
(Ip^ =
200 Amperes) The maximum torque occurs when the field and phase loops
are both in line at o^ = 0°, and o
0
= 180° when the short occurs.
0
When field and phase have no common flux linkages at the time of the
short, the resultant fault torque is a minimum at o^ = 90° and 0^ =
270°.
This kind of torque response can be easily explained.
The
orientation of the field winding at the time of the short determines
the orientation of the initial flux vector.
When the initial flux
vector is in line with the shorted phase the peak value of current
induced in that phase is a maximum.
a maximum in this orientation.
Thus the fault torque will be
An analogous explanation can be
used when the initial flux vector is perpendicular to the shorted
phase winding.
In general for any shorting configuration the minimum torque
occurs at the shorting angle of minimum flux linkages, and the maximum torque occurs when the field flux linking the phase(s) are at
a maximum.
The calculation of fault currents and torques by the methods
described is a simple procedure.
In machine faults where it is
valid to consider saturation and resistive effects as being negligable
this can be a useful tool to their calculation.
62
SHHrr RNC
B000
>EGREES>
E000
X
I H000
U 2000
ZI
a
fr*
5 -H000
-E000
-B000
Figure IV-6a.
at 0Q = 0°
Fault torque 3 phase machine without dampers
SHBFT Rf
•ETSREDES >
B000
E000
X
I H000
Z
U 2000
U -2000
Z
if
fil W
Id
K
S
S
5 -H000
-5000
-BB00
Figure IV-6b. Fault torque 3 phase machine without
dampers at o^ = 45°.
63
EGRECS >
X
I H000
-8000 •'
Figure IV-6c.
at 0Q = 180°.
Fault torque 3 phase machine without dampers
a
3
-10000 •'
Figure IV-7a.
at 0« = 0°
Fault torque 3 phase machine with dampers
^
64
10000
Z
5000
K
tl N 1^ y
Figure IV-7b. Fault torque 3 phase machine with dampers
at 0^ = 45°.
0
SHPFT BNGLE .(<^EEWSr5>
I00BSB •
X
I S000
a
1
1
a
s/ H p 5 a a
-10000 -
Figure IV-7c.
at 0Q = 180°.
Fault torque 3 phase machine with dampers
65
Figure IV-8a.
iS00a
Single phase to ground fault torque
4
<I>EnREC5>
a /a a K 3 a 3
-15000 -f
Figure IV-8b.
Phase to phase fault torque
66
i 0000
DEEREErS>
•
a
Figure IV-8c.
R a
H a
Phase to phase to ground fault torque
Shaft Shorting Angle
:s'
10,01)0
en
c
ctr
O
00
o
CO
to
c
o
Q.
CO
<U
Dc;
i-10,0
o
Figure IV-9. Tabulated peak minimum and maximum torque for Phase A
to ground fault for the full range of shorting angles.
CHAPTER V
CONCLUSION
The determination of machine inductance parameters is a significant
problem to the precise calculation of fault torques and currents.
The
damper winding due to their lack of external connections and complex
configuration are a specific problem.
By making appropriate approxi-
mations with respect to the flux linking the damper winding, the damper
mutual inductances can be determined.
The values of the damper self
Inductances can be constrained to range of values greater than zero
through use of the physical realizability equations developed in
Chapter III.
Use of the principal of conversation of flux linkages and the
bounding criterion on the damper winding has useful applications
when applied to a theoretical model of a proposed machine.
The machine
windings other than those of the damper can be very closely approximated from proposed design specifications.
For this reason the bounded
values of the damper winding also bound the potential fault torque
and current response.
design.
This can be a considerable aid to machine
If the range of fault torque response is known, a closer
estimation of such physical parameters as shaft diameter, internal
bracing, etc., can be made.
Thus the size and weight of a proposed
alternator can be minimized through the more precise prediction of
the fault response by the conservation of flux linkage method.
A necessary future step in the analysis of the machine parameters
is the determination of the effect on the model of resistance and
67
68
saturation and resistance effects is commonly done to simplify the
mathematics.
If saturation is to be considered the ^ = LI relation-
ship used in this thesis must change to some form of ^i) = L(I)I.
The
added complexity this adds to the flux linkage equations would also
require a computer to solve.
If hysterisis effects are also to be
included in the saturation model some type of state equation technique
may be necessary.
REFERENCES
1.
R. E. Doherty and 0. E. Shirley, Reactance of Synchronous
Machines and its Applications, American Institute of Electrical
Engineers Transaction, 1918, Volume 37, Part 2, p. 1209.
2.
A. E. Fitzgerald and Charles Kingsley, Jr., Electrical Machinery,
New york: McGraw-Hill Book Company, Inc., 1961.
3.
Olle I. Elgerd, Electrical Energy Systems Theory: An Introduction,
New York: McGraw Hill Book Company, 1971.
4.
R. F. Franklin, Short Circuit Currents in Synchronous Machines,
American Institute of Electrical Engineers Transaction, 1925,
Volume 44, pp. 420-429.
5.
Samuel Seely, Electromechanical Energy Conversion, New York: McGrawHill Book Company, Inc. 1962.
6.
George B. Thomas Jr., Calculus and Analytic Geometry, Reading
Mass., Addison-Wesley Publishing Company, 1969, Part 1, pp. 118121.
7.
Charles A. Desoer and Ernest S. Kuh, Basic Circuit Theory, New
York: McGraw-Hill Book Company, 1969.
69
i
p