Fortescue - Method-of-Symmetrical-Co-Ordinates-Applied-to-the-Solution-of-Polyphase-Networks.pdf
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Presented
at
the
34th Annual Convention of
the American Institute
of Electrical Engineers,
Atlantic City, N. J.,
June 28, 1918.
Copyright
1918.
By A.
I.
E. E.
METHOD OF SYMMETRICAL CO-ORDINATES APPLIED
TO
THE
SOLUTION OF POLYPHASE NETWORKS
BY
C. L. FORTESCUE
ABSTRACT
OF
PAPER
In the
introduction
a
general discussion
of
unsymmetrical
systems
of co-planar
vectors
leads
to
the conclusion that they
may be represented by symmetrical systems
of the same number
of vectors, the number of
symmetrical
systems
required
to define
the
given
system being
equal
to
its degrees of freedom. A
few
trigonometrical
theorems which are to be
used
in the paper are
called
to mind. The paper is subdivided into three parts,
an
abstract
of which
follows.
It
is recommended
that only
that
part
of Part I up to formula
(33)
and the
portion dealing
with
star-delta transformations
be read before
proceeding with
Part II.
Part I
deals
with the resolution of
unsymmetrical
groups of
numbers into symmetrical groups.
These numbers may repre-
sent rotating vectors of systems
of
operators.
A
new
operator
termed
the
sequence operator is
introduced
which
simplifies
the
manipulation.
Formulas are
derived
for three-phase circuits.
Star-delta transformations
for
symmetrical co-ordinates
are
given
and expressions
for
power
deduced.
A short
discussion
of har-
monics
in
three-phase
systems is
given.
Part II deals with the practical
application
of
this method
to
symmetrical
rotating machines
operating
on
unsymmetrical
circuits. General
formulas
are
derived
and such
special
cases,
as
the
single-phase induction
motor,
synchronous
motor-genera-
tor, phase converters
of various
types,
are
discussed.
INTRODUCTION
IN
THE
latter
part of 1913 the
writer
had occasion
to
investi-
gate
mathematically
the
operation
of induction
motors
under
unbalanced conditions.
The work
was
first
carried
out, having
particularly
in mind
the determination
of the
operating
char-
acteristics
of phase converters
which
may
be
considered
as
a
particular
case of
unbalanced
motor
operation,
but the
scope
of the subject
broadened out
very
quickly and the writer under-
took this
paper
in the
belief
that the
subject
would be of interest
to
many.
The most
striking thing about the results obtained was their
symmetry; the
solution always
reduced
to the
sum of two or
more
symmetrical
solutions. The
writer
was
then led
to
in-
quire
if
there
were no
general principles by
which the
solution
of unbalanced
polyphase
systems
could be reduced
to
the solu-
1027
1028 FORTESCUE:
SYMMETRICAL CO-ORDINATES
[June28
tion of two or more balanced
cases.
The present paper is
an
endeavor
to
present
a
general
method
of
solving
polyphase
network which has peculiar advantages when applied
to
the
type
of
polyphase
networks which include
rotating
machines.
In physical
investigations
success
depends
often
on a
happy
choice
of
co-ordinates.
An
electrical network
being
a
dynamic
system
should
also
be aided by the
selection of
a
suitable
system
of
co-ordinates.
The co-ordinates of
a
system
are
quantities
which
when
given, completely define
the system. Thus
a
system
of three co-planar concurrent vectors are
defined
when
their
magnitude
and
their
angular position with respect to some fixed
direction
are
given.
Such a system may be
said
to have
six
degrees of freedom, for
each vector may vary
in magnitude and
phase
position without regard
to
the others.
If,
however,
we
impose
the
condition
that the vector
sum
of these
vectors
shall
be
zero, we
find that with
the
direction
of
one vector
given,
the
other
two
vectors
are
completely defined
when
their magni-
tude
alone is
given,
the
system
has therefore lost
two
degrees
of freedom
by
imposing
the above
condition which in dynamical
theory is
termed
a
"constraint".
If we
impose
a
further
con-
dition
that the
vectors
be
symmetrically disposed
about
their
common
origin
this
system
will
now have but two
degrees
of
freedom.
It is evident from the above definition that
a
system of
n
coplanar concurrent
vectors may
have
2 n
degrees
of
freedom
and
that
a
system
of
n
symmetrically
spaced
vectors
of
equal
mag-
nitude
has but
two degrees of freedom.
It
should be
possible
then
by
a
simple
transformation
to
define the
system of
n
arbitrary congruent
vectors by n
other
systems of concurrent
vectors
which
are
symmetrical
and have
a common
point.
The
n
symmetr
cal systems so
obtained
are
the
symmetrical
co-
ordinates
of
the given
system of vectors and
completely define
it.
This method
of
representing
polyphase
systems has
been
employed
in the
past to
a limited
extent,
but up
to
the
present
time there has been
as far as
the author is aware no
systematic
presentation of the method.
The
writer
hopes by
this
paper to
interest
others in the
application
of the
method,
which will
be
found
to
be a valuable instrument for
the
solution of
certain
classes
of
polyphase
networks.
In
dealing
with
alternating
currents in this
paper,
use is
made
of the
complex
variable which in its most
general
form
19181
FORTESCUE: SYMMETRICAL CO-ORDINA TES 1029
may be represented as a vector of variable length rotating about
a
given point
at variable angular velocity or better as the re-
sultant
of
a number of vectors each of constant
length rotating
at
different
angular velocities in the same
direction about
a
given point. This vector is represented in the text by I,
E,
etc., and the conjugate vector which rotates at the same
speed
in
the opposite direction is represented by
f,
E,
etc.
The
effec-
tive
value of the vector is represented by the
symbol
without
the
distinguishing
mark
asI,
E, etc. The
impedances
Za, Zb,
Zab,
etc.,
are
generalized expressions
for
the
self
and
mutual
impedances.
For a circuit A the self-impedance operator
will
be denoted by Zaa or Za In the case
of
two
circuits
A and
B
the
self impedance
operators
would be
Zaa Zbb
and
the
mutual
impedance operator Zab. The subletters denote the
circuits
to
which
the
operators
apply.
These
operators
are
generally
d
2
t+
2
P
(A
+
j
B)
E
PI
represents
a
vector
of length
VA2 + B2
rotating
in the
positive direction with angular
velocity
p
while
(A
-
j B)
E-i
P
is
the
conjugate
vector
rotating
at the
same
angular
velocity
in the
opposite direction. Since
E1
PI
is
equal
to
cos
pt+j
sin
Pt,
the
positively rotating
vector E
=
(A +j B)
ei
t
will be
functions of the operator,
D = dt and the
characteristics of
the
circuit;
these
characteristics
are
constants only when
there
is no physical motion. It will therefore be necessary to care-
fully
distinguish
between
Za 7a
and
Ia Za when Za has the form
of a
differential
operator. In the first case
a
differential
opera-
tion is carried
out
on the
time variable
Ta
in the
second
case
the
differential
operator
is
merely
multiplied
by
Ia.
The
most
general expression for
a
simple
harmonic
quantity
e
is
e
=
A
cos
pt
-
B sin
pt
in
exponential
form this
becomes
A
+jBEJPI+
A-j B>j
e-
A
= A cos
pt
-
B
sin
pt
+ j
(A
sin
pt
+
B
cos
pt)
or the real
part
of E which is its
projection
on a
given
axis is
equal
to e
and therefore
A
may
be taken
to
represent
e
in
phase
and
magnitude.
It should be
noted
that the
conjugate
vector
.l is equally
available,
but it is
not
so
convenient since the
1030 FORTESCUE: SYMMETRICAL
CO-ORDINA TES
[June
28
operation dt e.
Pt
gives
-
j p
e-i
Pt and the imaginary
part
of the
impedance
operator will
have
a
negative
sign.
The complex roots of unity will be
referred
to
from
time
to
time
in the paper. Thus
the complete solution of the equation
x
-1
=
0
requires
n
different
values of x, only one of
which
is
real when n
is
an odd integer. To obtain the other roots
we
have the relation
1 =
cos
2
r
r
+
j
sin 2
ir
r
-
2
-r
r
Where r is
any
integer.
We have therefore
1
.2irr
n
=
X
and by
giving
successive
integral
values
to r
from 1 to n, all
the
n
roots
of xn
-
1
=
0
are
obtained
namely,
a,
=
= cos
s+
s
2n
n
2
n
4irn
4T .
a2
=
=
CoS
+j
sin
n
6w
6
wr
a3=E = cos
+±j
sin
n
1
=
EC2J
= 1
It
will be observed that a2 a3....
a,
are respectively equal to
a,2 a,3
...
a,(.
-l)
When there is
relative
motion between the
different
parts
of a
circuit
as
for
example in rotating machinery, the mutual
inductances
enter into the equation as time variables and when
the motion is
angular
the quantities eJwt
and
e
-
jwt
will appear
in
the operators. In this case we
do
not
reject the portion of
the
operator having e-iwt as
a factor, because the equations
require that each vector shall be operated on by the
operator
as
a
whole which when it takes the form of
a
harmonic
time
function will
contain
terms with
Ejwt
and e--jw
in conjugate
relation. In some cases as a result of this, solutions will appear
with indices
of
e
which are negative time variables; in such
cases in
the final statement the vectors with negative index
should be replaced by their
conjugates which rotate in the
positive
direction.
n
-.
2r
4w 4w
1918]
FORTESCUE: SYMMETRICAL
CO-ORDINATES
1031
This paper is subdivided as
follows:
Part I.-"The Method
of
Symmetrical
Co-ordinates."
Deals
with the theory
of
the
method,
and its
application
to
simple
polyphase circuits.
Part
II.-Application
to
Symmetrical
Machines
on
Unbal-
anced Polyphase Circuits.
Takes
up
Induction
Motors,
Gener-
ator
and
Synchronous
Motor,
Phase
Balancers
and
Phase
Convertors.
Part III.
Application
to
Machines
having Unsymmetrical
Windings.
In
the
Appendix
the
mathematical representation
of field
forms
and
the derivation
of
the
constants
of different forms
of
networks is
taken
up.
The
portions
of
Part I
dealing
with
unsymmetrical
windings
are not
required
for the applications
taken
up in Part II and
may
be
deferred
to a
later
reading.
The
greater part
of
Part
I
is
taken
up
in
deriving
formulas for
special
cases
from
the
general
formulas (30)
and
(33),
and
the
reading
of
the text fol-
lowing
these equations may be
confined to
the
special
cases of
immediate interest.
I
wish
to express my
appreciation
of the valuable
help
and
suggestions
that have been
given
me in the
preparation
of
this
paper
by
Prof. Karapetoff who suggested that the
subject
be
presented
in a
mathematical
paper
and
by Dr. J.
Slepian
to
whom I am
indebted
for the
idea
of sequence operators and
by
others
who have
been interested
in
the
paper.
PART I
Method
of
Symmetrical
Generalized
Co-ordinates
RESOLUTION
OF
UNBALANCED
SYSTEMS
OF
VECTORS
AND
OPERATORS
The complex time
function
A
may be used instead of the har-
monic
time
function
e in any equation algebraic or differential
in which it appears linearly.
The reason of this is because if
any
linear
operation
is
performed
on
A
the same operation per-
formed
on
its conjugate
A
will give a result which is conjugate
to
that
obtained
from
A,
and
the sum of the two results obtained
is a solution of the same operation
performed on
A
+ X, or 2 e.
It is customary to interpret
A
and
A:
as coplanar vectors,
rotating about a common point and e as the projection
of either
vector on a given
line,
A
being a positively rotating vector
and*
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