In presenting t h i s d i s s e r t a t i o n as a p a r t i a l
fulfill-
ment of the requirements for an advanced degree from the
Georgia I n s t i t u t e of Technology, I agree t h a t the Library
of the i n s t i t u t i o n s h a l l make i t a v a i l a b l e for inspection
and c i r c u l a t i o n in accordance with i t s r e g u l a t i o n s governing
materials of t h i s type.
I agree that permission to copy from,
or to publish from, t h i s d i s s e r t a t i o n may be granted by the
professor under whose d i r e c t i o n i t was w r i t t e n , or, in his
absence, by the Dean of the Graduate Division when such
copying or p u b l i c a t i o n i s solely for scholarly purposes and
does not involve p o t e n t i a l f i n a n c i a l gain.
t h a t any copying from, or p u b l i c a t i o n of,
I t is understood
this dissertation
which involves p o t e n t i a l financial gain w i l l not be allowed
without w r i t t e n permission,
^John i/tfilliam~Eo^per_
E DETERMINATION OP THE MUTUAL SEQUENCE IMPEDANCES
OF PARALLEL TRANSMISSION LINES
7f^
A THESIS
Presented to
the Faculty of the Graduate Division
Georgia Institute of Technology
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Electrical Engineering
By
John William Hooper
June h, 1955
THE DETERMINATION OF THE MUTUAL SEQUENCE IMPEDANCES ° F
PARALLEL TRANSMISSIuN LINES
Approved ;
/
*HJL-
7/
D a t e Approved by C h a i r m a n :
J Ufl&
o^
J^SS~
ACKNOWLEDGMENT
I wish to express my appreciation to Doctor W. J.
McKune for his suggestion of the subject for this project
and for his invaluable assistance in its prosecution.
I
would also like to thank Mr. H. P. Peters and Professor
D. C. Fielder for the many helpful suggestions which they
made during the preparation of the manuscript.
Page
ACKNOWLEDGMENT
ii
ABSTRACT
vi
CHAPTER
I.
II.
III.
IV.
INTRODUCTION
METHOD OF ANALYSIS
1
,
.
3
TWO THREE-PHASE CIRCUITS, GROUND NEGLECTED .
5
TWO THREE-PHASE GROUNDED NEUTRAL CIRCUITS,
GROUND CONSIDERED
V. TWO THREE-PHASE GROUNDED NEUTRAL CIRCUITS
WITH ONE GROUND WIRE, GROUND CONSIDERED. .
VI.
VII.
TWO THREE-PHASE GROUNDED NEUTRAL CIRCUITS
WITH TWO GROUND WIRES, GROUND CONSIDERED .
CONCLUSIONS
BIBLIOGRAPHY
23
.
32
.
38
^
[+5
LIST OF TABLES
Page
F u n c t i o n a l P o s i t i v e - and N e g a t i v e S e q u e n c e M u t u a l Impedance V a l u e s F o r
A T w o - C i r c u i t P a r a l l e l - P l a n e Conf i g u r a t i o n Line
15
F u n c t i o n a l P o s i t i v e - and N e g a t i v e S e q u e n c e M u t u a l Impedance V a l u e s f o r
A Two-Circuit Triangular Configuration Line.
21
F u n c t i o n a l Zero-Sequence Mutual
Impedance V a l u e s F o r A T w o - C i r c u i t
P a r a l l e l - P l a n e C o n f i g u r a t i o n Line
28
F u n c t i o n a l Zero-Sequence Mutual
Impedance V a l u e s F o r A T w o - C i r c u i t
T r i a n g u l a r C o n f i g u r a t i o n Line
30
Figure
Page
1,
Two P a r a l l e l Three-Phase Ungrounded C i r c u i t s . .
2.
P a r a l l e l Plane C o n f i g u r a t i o n For a Two
C i r c u i t Line
13
Ohmic R e s i s t a n c e And Reactance Values As
F u n c t i o n s of Conductor Spacing R a t i o For A
P a r a l l e l - P l a n e Configuration #
17
T r i a n g u l a r C i r c u i t C o n f i g u r a t i o n For A Two
C i r c u i t Line
"
20
Ohmic R e s i s t a n c e And Reactance Values As
F u n c t i o n s of Conductor Spacing R a t i o For A
Triangular Circuit Configuration
22
Two P a r a l l e l Three-Phase Grounded N e u t r a l
Circuits
21|
Two P a r a l l e l Three-Phase Grounded N e u t r a l
C i r c u i t s with One Ground Wire
33
Two P a r a l l e l Three-Phase Grounded N e u t r a l
C i r c u i t s With Two Ground Wires
39
3«
1|.
5.
6.
7.
8.
7
ABSTRACT
The increased use during recent years of multi-circuit
transmission lines located on the same right-of-way has emphasized the need for additional information about the electrical characteristics of parallel systems.
A problem of
particular importance, because of its relation to power
system relaying, is the determination of the induced voltages in one circuit produced by currents in the parallel
circui t.
The purpose of this investigation was to derive
equations for the mutual impedances between parallel transmission circuits and thereby establish the expressions for
these induced voltages.
A detailed investigation was made of the magnetic
coupling between two parallel three-phase circuits.
Capac-
itive coupling was not considered since it is of secondary
importance.
The analysis was carried out in terms of the symmetrical components of the voltages and currents in the two
circuits, since the results are most useful when presented
in that form.
The symmetrical components of voltage induced in one
circuit were determined with positive-, negative-, and zerosequence currents flowing in the parallel circuit.
The
mutual sequence impedances between the circuits are defined
vii
as the ratios of the symmetrical components of voltage induced in one circuit to the symmetrical components of current in the other.
An application of these definitions of
the mutual sequence impedances led to the expressions for
the positive-, negative-, and zero-sequence mutual impedances between the two circuits.
The effect of the ground upon the mutual sequence
impedances was evaluated by use of Carson's Equations for
the mutual impedance between two conductors with earth return.
It was found that the mutual sequence impedances between parallel circuits could be expressed in terms of the
distances between conductors, conductor heights above ground,
earth resistivity and the ground wire dimensions.
It was also found that the mutual sequence impedances
which involved the conductor spacing could be written in a
general form which related the distances between conductors
to the ohmic impedance per unit length.
Tables and curves
for typical circuit configurations are included providing a
means for easily obtaining the mutual sequence impedances
for these configurations.
CHAPTER I
INTRODUCTION
The rapid growth of the electric power Industry during the last few years has produced many changes in utility
operating practices.
One of these changes has been the
increased use of multi-circuit transmission lines on a single
right-of-way.
New steel tower construction almost Invariably
provide facilities for two circuits.
These circuits are
installed either simultaneously or at different times, depending upon the load requirements and the economic considerations Involved.
The use of parallel transmission lines, like many
changes in operating procedure, introduces new technical
problems.
These problems are grouped into two classes:
those due to magnetic coupling between circuits and those
due to capacitlve coupling between circuits.
This paper
presents, In analytical form, the relationships between
three-phase circuits which are coupled magnetically.
Capacitlve coupling is not considered, as most present day
lines are relatively short and capacitive coupling is of
secondary importance.
The importance of these relationships Is emphasized
by a study of the past operating records of utilities with
parallel, lines.
There have been instances in which a fault
on one c i r c u i t of a d o u b l e - c i r c u i t transmission l i n e has
caused the unfaulted c i r c u i t to open. 1
This i s due to the
magnetic coupling between the l i n e s which induces unbalanced voltages.
These voltages, in turn, produce
unbalanced currents which may actuate the p r o t e c t i v e r e lays i n c o r r e c t l y ,
The s o l u t i o n of a problem of t h i s type involves the
determination of the induced voltages in one c i r c u i t produced by currents flowing in the p a r a l l e l c i r c u i t .
The
r e s u l t s are most useful i f the analysis i s made in terms
of the symmetrical components of the voltages and currents
of the system.
This r e s u l t s in expressions for the p o s i t i v e - ,
n e g a t i v e - and zero-sequence mutual impedances of the two
circui t s .
Very l i t t l e information p e r t a i n i n g to the s o l u t i o n
of t h i s problem appears in the l i t e r a t u r e . Professor Edith
p
Clarke has derived expressions for the zero-sequence mutual
impedance between two c i r c u i t s , both neglecting and considering the effect of ground.
However, she does not consider
the p o s i t i v e - and negative-sequence mutual impedances.
An
a r t i c l e by Mr. J, E. Clenr presents some experimental data
obtained from t e s t s made on an a c t u a l transmission l i n e .
Several other minor references consulted are l i s t e d in the
bibliography.
The four physical s i t u a t i o n s examined in the following
analysis a r e : (1) two three-phase ungrounded n e u t r a l c i r c u i t s
;
with ground neglected; (2) two three-phase grounded neutral
circuits with the effect of ground considered; (3) two-three
phase grounded neutral circuits with one ground wire, ground
considered; and (i|) two three-phase grounded neutral with
two ground wires, ground considered,
METHOD OF ANALYSIS
Consider
two p a r a l l e l
three-phase
l i n e s having
con-
d u c t o r s a , b , c and 1 , 2 , 3 w i t h v o l t a g e p h a s e r o t a t i o n i n
order mentioned.
the l e t t e r e d
line if
The d i f f e r e n c e
in potential
l i n e per u n i t length
terms of i t s p o s i t i v e - ,
ponents
t o ground
( o r b e t w e e n e n d s of
capacitance is negligible)
can be e x p r e s s e d
the
com-
( V a i , V a 2 and V&Q), t h e s y m m e t r i c a l components
line
(Iax>
^a2> ^ a O ^
and
m e t r i c a l c o m p o n e n t s o f c u r r e n t i n t h e numbered
11'I12'I10)
al
=
x
12zml2 + I10zml0
V
a2
X
=
I
alZll +
X
a2Z12
+
WlO
+
X
+
I
Z
a l Z 2 1 * Xa2Z22 *
+ I102m20
J
aOZ20
X
alZ01 *
I
aOZ00 *
I12222
a0 "
e
of
s
3rm"
line
as
V
V
^
of
in
n e g a t i v e - and z e r o - s e q u e n c e
c u r r e n t i n the l e t t e r e d
(I
the
I
a2Z02
+
llZmll +
{1)
llZm21 *
X
llZm01+
12 Z m02 * I 1 0 Z m 0 0
T h r e e s i m i l a r e q u a t i o n s may be w r i t t e n f o r
v o l t a g e s i n t h e numbered l i n e .
f o r one l i n e
The component
The i n t e r a c t i o n s b e t w e e n t h e l i n e s
g i v e n b y t h e t e r m s Zm]_i>
Z
ml2*
etc
«
of
"self-impedances"
( Z - ^ j Z]_2> e t c . ) h a v e - b e e n g i v e n i n g e n e r a l
by P r o f e s s o r Clarke.**
quantities,
the components
The e q u a t i o n s f o r
terms
are
these
p r e s e n t e d b e l o w , a p p l y t o t h e i n t e r a c t i o n of
the
5
numbered line on the lettered line.
The values of the mutual
sequence impedances which give the interaction of the lettered line on the numbered line can be obtained by replacing
the letter a with 1, b with 2, c with 3 and the number 1
with a, 2 with b, 3 with c.
CHAPTER
III
TWO THREE-PHASE CIRCUITS, GROUND NEGLECTED
Fig.
llel
1 r e p r e s e n t s a six conductor,
system.
only.
to r i g h t ,
The e f f e c t
Power i s assumed t o
w i t h c u r r e n t i n t h e numbered
of t h e g r o u n d i s
An a p p l i c a t i o n of K i r c h o f f ' s
lettered
para-
The two c i r c u i t s a r e r e p r e s e n t e d by t h e
t e r e d and numbered c o n d u c t o r s .
from l e f t
two-circuit,
conductors yields
Va-Val
:
the
let-
flow
circuit
neglected.
V o l t a g e Law t o
the
expressions
I ^ a ! * I2Za2 * I3Za3
(ii)
X Z
W =
hhl*
2 b2* J3Zb3
V T d = V e l * %Zo2* J3Zc3
VIh e r e
Za-j_, ^ l '
e_t;
^cl>
c.,
a r e
(5)
(6)
"t^ e c i r c u i t m u t u a l
b e t w e e n c o n d u c t o r s a and 1 , b and 1 , e t c . , when
currents are
impedances
charging
neglected.
The m u t u a l i m p e d a n c e b e t w e e n two c o n d u c t o r s m and n
w h i c h have no common p a t h i s p u r e l y r e a c t i v e and i s
the
g i v e n by
expression
Z
mn -- &mn = J 2 ' r f M I nn
(7)
The e x p r e s s i o n f o r t h e m u t u a l i n d u c t a n c e M ^ of
parallel
cylindrical
conductors i s d e r i v e d i n the
of t h e B u r e a u of S t a n d a r d s . - ^
Bulletin
The d e r i v a t i o n a s s u m e s
t h e c o n d u c t o r s a r e of n o n - m a g n e t i c m a t e r i a l ,
two
t h a t end
that
ef-
a
a1
b
b1
c
h
2
1
*
y l
>•
21
i
31
z.
3
Pig. 1
Tvio p a r a l l e l three-phase ungrounded c i r c u i t s .
fects are n e g l i g i b l e , that there i s uniform current d i s t r i bution over the conductor cross section, and that the wires
are located in a non-magnetic medium.
The mutual inductance
of two conductors of length e and spacing s m between t h e i r
centers i s given by
Mmn = 2e (in 2e. -1)
s mn
abhenries
(8)
The positive-sequence mutual impedance of the l e t t e r e d
conductors, with positive-sequence currents flowing in the
numbered conductors, i s defined as the r a t i o of the p o s i t i v e sequence voltage i n the l e t t e r e d conductors to the corresponding positive-sequence current in the numbered conductors, with
zero current in the l e t t e r e d conductors.
This r e l a t i o n s h i p i s
given as
Zmll = 3 ^ i
x
(9)
ll
where I]_-j_ i s the positive-sequence current flowing i n the numbered l i n e .
The negative- and zero-sequence mutual impedances Zm2i*
and Z Q, of the l e t t e r e d conductors are defined as the r a t i o
of the negative- and zero-sequence voltages in the l e t t e r e d
conductors to the corresponding positive-sequence current in
the numbered conductors. These impedances have the form
Zn£1 ~- IsSk
X
(10)
ll
zmoi = >21
•4.1
(">
9
If only positive-sequence currents are allowed to
flow in the numbered conductors and no current flows in the
lettered conductors equations (1+), (5) and (6) can be rewritten in the form
1
v a -v a i
(Z
11
al +
v b -v b i = I11 <Zbl
Vc-V c1x = I11
where a and a
(Z
a2
aZ a3 )
(12)
a Zb2
aZuo)
'b3
(13)
c3}
(14)
a Z
cl +
a Z
aZ
c2 *
are the symmetrical component operators.
Multiply (12) by 1, (13) by a, and (11;) by a 2 and
1
2
add, noting that V m l l = -r(Va+ aV"b + a V c ) ; the result is
3Vnii = hi
+ z
K l
b2 * Z c 3 } *
&2(Z
a2
+ Z
b
+
3
^
Zcl) * a < z a3 + zbl * zc2)]
A comparison of this with equation (9) yields
Z
mll = ¥ (Z al * z b2
Z c l ) * a ( Z a3
+
Z
+ z
c3}
Z
bl *
+ a
(Z
a2
+ z
(16)
b3
c2>]
The replacement of the mutual sequence impedances in (12)
by t h e i r values as given by equations (7) and (3), the a p p l i cation of the proper u n i t conversion f a c t o r s , and s i m p l i f i c a tion r e s u l t i n an expression for 2m^_^ involving only the
distances between conductors.
Z
-,
v3 l o g
f
mll * 0-0466 60
'a2
s
b3
s
al*
s
c l ^a3
s
b
JT
bl
b2^ Sc32
b
c2
This r e l a t i o n s h i p i s given as
S
q
b
a3
a2
S
q
3
OHMS
MILE
bl
S
c2
. .
B— * J
b3
cl
lQ
g
(17)
where the l o g a r i t h m i s to the base t e n .
Z p-| i s e v a l u a t e d i n a s i m i l a r manner to t h a t used
Ii:
i n the d e t e r m i n a t i o n of Z m l l .
(12) by 1,
i s necessary to multiply
(13 by a 2 , and (1/|) by a and add, t a k i n g note
a2Vfe + aVc).
of the f a c t t h a t V m 2 1 = j ( V t
ing e x p r e s s i o n for Z m2 i i s then o b t a i n e d
The f o l l o w from the
definition
of Z m 2 1 as g i v e n by e q u a t i o n ( 1 0 ) , and through the a p p l i c a t i o n of e q u a t i o n s
(7) and ( 8 ) .
Z m 2 1 r 0.01*66 [ f ] f e log sq a 3 s^ b 2 s^ c l
'
[60JL
a2 b l c3
s
a2
s
b l sc3 sa3 sb2
S
a l 2 Sb32 Sc22
s
cl
+
j log
(18)
OHMS
MILE
Zm0-j_ is determined by adding equations (1|), (%), and
(6) directly noting that V m0 i = i(Va + V b + V c ) and that
2
m01
=
T > Equation (19) is obtained by performing the
-11
n e c e s s a r y s u b s t i t u t i o n s for the mutual impedances as given
by e q u a t i o n s
(7) and (5) and
simplifying,
*01* 0.W[&][$ loS I* '& I?.
a2 ^b2 ^c2
s
a2 s b2 s c2 S a3 Sb3 Sc3
s
a l 2 s bl 2 s c l 2
3
loS
(19)
OHMS
MILE
The sequence mutual impedances which a r e a s s o c i a t e d
w i t h n e g a t i v e - s e q u e n c e c u r r e n t s are o b t a i n e d by a l l o w i n g
only n e g a t i v e - s e q u e n c e c u r r e n t s to flow i n the numbered l i n e s .
E q u a t i o n s (I4), (5) and (6) then become
11
Va - V a l
I
V
I12(Zbl
b "V b l
1 2 f z a l +•
r
The p o s i t i v e - ,
of
the lettered
and
(25),
conductors
a2
4. a Z b 2
i2(2cl *
negative-
a2
aZ
c2
^za3}
(20)
a Z b3 )
(21)
a
*
+
+
a
Z
and zero-sequence
a r e defined
(22)
c3}
impedances
by equations
(23), (2i|)
respectively, as
'ml2 -
V
ml2
(23)
"12
J
m22 ~
tfm22
(2^)
'12
V
J
Equations
Z
m22
anc
*
2
(20),
m02
(21)
m02
L1. «2
(25)
and ( 2 2 ) can be solved
m 0 2 ^"n a manner identical
f o r Zm\2>
to that used for
positive-sequence mutual impedances.
the
The r e s u l t s are given
as
Zml2 = 0 . ( W f ] (~x/3 log f f i J ^ L - l L j
L60 J L
a3
b2 °cl
s
al
2 s
b3
a2 °bl c3
2 s
c2
2
^ a 3 ^>b2 ^>ci
OHMS
MILE
l0
S
(26)
12
Zm22 = 0.0)466 U S
J3 loc
a3 S bl 3 c2 S a2 s b3 S cl
S
al2
S
b22
S
c32
Sa2 S b 3 S c i
^ ± 4 - j log
s
a3 Sfei S C 2
OHMS
MILE
r f 1r
sap Sh? sP2
b
Z m 0 2 = 0.0Vo6fe][3 1 o g t ,;3 °b3
2 c
'a 3 b b3 3 c3 s a2 s b2 b c2
S
al2 S b l 2 S c l 2
(27)
(28)
j l Q g
OHMS
MILE
No significance can be attached to the zero-sequence
mutual impedance between the two ungrounded circuits, since
zero-sequence currents can not flow in an ungrounded circuit.
The above equations for the mutual sequence impedances
of the two circuits provide a means of evaluating the effect
of conductor configuration upon the electrical characteristics
of parallel lines.
Pig. 2 illustrates a common conductor configuration in
which the lettered and numbered conductors lie in parallel
planes, with the center conductor of each line lying midway
between the outside conductors.
For this configuration equations (17) through (19) and
(26) through (28) may be expressed for any arrangement of circuit conductors in terms of two variables A and E, which are,
in turn, functions of the distances between conductors and
between lines.
Let X represent the distance between the two
circuits and Y the distance from the center conductor to an
outside conductor of the same circuit, as shown in Fig. 2.
13
i Ol
aQ
02
bO
c
O3
°
Fig- 2
Parallel plane configuration
for a two-circuit l i n e .
1U
With X and Y measured in the same units, there results
!
•*• T
Y 2O) VX
\/x2 *
+ ^
UY2
A = 0.0^66 log <X +
(29)
x3
x3
B
= O.Oli66 l o g ? ? * l\f
(30)
Table 1 gives the values of Zmll* Zra21 and ZmQi in
terms of A and B for each of the thirty-six possible physical permutations of circuit conductors.
The ground plane
may be considered parallel to any one of the four sides
formed by the configuration.
It may be noted that there
are only eighteen possible electrical permutations for
the conductors.
It is, therefore, possible to condense
Table 1 to half the size given; however, this would destroy
the completeness of the analysis.
Fig* 3 gives the ohmic
values of A and B as a function of the ratio of the distances
X and Y.
Calculations based on Table 1 and Fig. 3 may be extended by comparing equations (18), (17) and (19) with
equations (26), (27) and (28) respectively from which
Z
ml2 ~ " Z m21
C31
z
m22 = - z mll
(32)
Z
m02 = ~ 2 m01
^^^
where the symbol Z denotes the complex conjugate of the impedance under consideration.
^
T a b l e 1 . F u n c t i o n a l P o s i t i v e - and N e g a t i v e - S e q u e n c e M u t u a l
Impedance V a l u e s F o r a T w o - C i r c u i t P a r a l l e l - P l a n e C o n f i g uration Line.
Numbered
Conductor
Plane
Lettered
Conductor
Plane
1 2
••
3
b
a
2
1 3
0
b
&
• * •
J2A
+
JB
+•
JB
a
c
-18B
;
c
"
^gA - JA
a
b
\gA - JA
b
-.
a
b
c
a
G
I
b
a
c
:
G
a
c
a
b
GB
c
b
a
ySA - JA
a
b
G
y/3B
a
c
b
v{3A -
JA
\:
a
c
0
-+
J2A
t
c
8
4
JB
c
a
t
c
b
S
:
•••
2
c
b
: •
1 3
• -
2m2l
2ml 1
0 - J2B
l/3B
+
JB
u
+
J2A
\gA - JA
0 - J2B
-
&
- £ A
- \SB + JB
0
-
CA
•+
J2A
-V3B
+
JB
-\ffA - JA
- JA
+
JE
&
+
JB
SA
- JA
- G A - JA
0 - J2B
£A - JA
JB
0 - J2B
JB
J2A
-^B
0 - J2B
+
+
0
- JA
+
2m01
+
J5B +
JB
0 - J2B
- f l A - JA
0 + J2A
-0B
+
JB
~*T Jf
-fe-
Ji
0 +
JE
-ffi -
J|
0 + 5B
f- 4
<h - Jf
iB
2
0 * JB
-fe -$J
-
J
4
0 + JB
4
* QB
-
'!
-§B 0 +
JB
-§B-
«
0 + JB
fB -
1B
Table 1.
F u n c t i o n a l P o s i t i v e - and N e g a t i v e - S e q u e n c e M u t u a l
Impedance V a l u e s F o r a T w o - C i r c u i t P a r a l l e l - P l a n e C o n f i g u r a t i o n Line. (Continued)
Numbered
Conductor
Plane
3
1
Lettered
Conductor
Plane
a
' •
b
3
1 2
t
mll
' -
c
0B
+
c
a
0
4-
.]2B
- 10A -
JB
JA
J2A
- 0B
JB
a
- J3B + JB
0
b
G
-V3A -
\
C
- \TJB + JB
8
e
0 -
C
a
V
a
: •
•
b
-0A
-
JA
J2B
CA - JA
1
\S&
-h
J2A
4-
JB
c
b
a
s
b
c
0 -
a
c
1'
!;
a
b
0
JA
- J3A -
GB
>
2mG1
0 - j?B
JA
b
•
2m2l
J3A - JA
a
c
c
1
o
a
i
2
c
c
)
3
h
z
-
•+
4
J]
4-
J2A
-
GB
2^
0 y/3A -
JB
£B
2"
3A
0 -
e
©A -
JA
0
a
J3B + JB
a
h
b
a
-\6B
J2B
2
- iB
J
2
- iE
J
?
- j-
0 + JB
fa.,!
f -df
-fB - J§
0 + jB
J
2
GA -- JA
^ B + JB
j
-fB - j |
+ JB
0 + J2A
- i§.
0 + jE
J2B
JA
- £
c MB
0 + J2A
4-
2
2
0 + JB
2^
-§B
©B
J
J
- 0A - jA
- J3 A -
0 + J2A
2
GB
- \ 5 A - JA
+ JB
MB
GB + JB
J2B
- £B
2B "
&
^
0 +
jE
-
iB
0
1
2
Spacing r a t i o - X/Y
3
^
Fig. 3
Ohmic resistance and reactance values as
functions of conductor spacing ratio for
a parallel-plane configuration,
5
Fig, 1" illustrates a second possible conductor configuration in which the numbered and lettered conductors
are arranged in isosceles triangles.
The resultant com-
bination of the two circuits is in the form of two parallel planes.
This permits the use of the same tower
construction for the configurations of both p igs. 3 and h,
The configuration of Pig. k gives more nearly balanced selfimpedances than that of Fig. 2 because of its relative symmetry.
Equations (17) through (19) and (26) through (28) may
now be expressed in terms of two new variables E and F, which
are, in turn functions of the distances X and Y.
When the same
units are assigned to X and Y there results
E = O.Olj.66 log
x2
^.A* 2
(31;)
Y^
P = 0.01|66 log * * x
* k*
X^
Table 2 gives the values of Zm]_]_> 2m21 a n d
(35)
Z
m01
in
terms of E and P for each of the thirty-six possible physical
circuit conductor positions.
The ground plane may be con-
sidered parallel for any one of the four sides formed by the
configuration.
There are, therefore, only eighteen possible
electrical permutations.
the size shown.
Table 2 may be condensed to half
Pig. 5 gives the ohmic values of E and F as
a function of the distance ratio
X to Y.
The values of Z
^
Z 2p
and 2
np
ma
y
fee
calculated by the application of
equations (31) through (33)The conductor configurations illustrated in Figs.
2 and 1; are only two of the ones which might be used.
appear; however, to be the most practical.
They
They could be
easily mounted on towers of conventional design.
The con-
figuration of Fig. i| is inherently well balanced electrically.
It would cause some difficulty if line transposition
were attempted, but this is seldom done today on highvoltage lines.
20
i
Ob
i O
03
a O
2 O
O
Pig. k
Triangular circuit configuration
for a two-circuit l i n e .
Table 2 . F u n c t i o n a l P o s i t i v e - and Negative-Sequence Mutual
Impedance Values For a Two-Circuit Line With T r i a n g u l a r
Configuration,
Right
Conductor
Plane
Left
Conductor
Plane
1
a
2
b
3
1
a
2
c
3
1
b
c
3
1
b
2
8
3
1
c
2
s
3
1
c
2
b
3
2
a
1
b
3
C
0E
2
a
1
0
?
b
->/3F -
2
b
1
c
3
a
- I/3E
2
b
1
a
3
c
0P
2
c
]
8
3
b
0 -
J2E
2
c
1
b
3
a
( +
J2F
1
a
b
2
c
1
a
3
c
2
b
1
b
3
e
2
a
0 -
1
b
3
a
2
c
C + J2F
1
c
3
a
;:
>
1
c
3
b
3
:
•
:
0
a
•J
b
:
:
•
z
z
mll
2
m2; ]
- i/3F
-
JP
m
+
JE
/3E
+
JE
-i/3P -
3?
-*- JE
- igE
0 + J2P
< -
-02
•f
•
*
•
-
+
J2E
""^P
0
-m
+-
J2F
+
JE
$ F - JP
tt
11
't
»l
:l
M
11
n
It
n
n
u
-flp
-
JP
P
m
+
£
n
u
JE
0
+
J2F
11
n
JF
0 - J2E
i\
tT
O P - JP
»
11
•f
JE
n
tl
1/3P -
JP
r
JE
J3P - JF
v/3E
~§E7 *
0 - J2E
/3P - JP
m01
J2E
- v/3E
- \ / 3 P •f JP
- \/3P -
JP
•f
JE
•f
J2F
&
+
JE
0
-
JP
0 ... J2E
*h -Jf£
(i
"
•
tt
M
n
H
'1
ft
n
••
22
Table 2.
F u n c t i o n a l P o s i t i v e - and N e g a t i v e - S e q u e n c e M u t u a l
Impedance V a l u e s f o r a T w o - C i r c u i t L i n e With T r i a n g u l a r
Configuration. (Continued)
Right
Conductor
Plane
Left
Conductor
Plane
2
Zmll
2
m21
-/3E
m01
+ iS, -
4
OF
-
- i/3E
+
JE
i/3P ^ JP
11
+
J2P
OE + JE
it
1!
c
0 -
J2E
tt
It
2
]
- \ffF - JP
it
11
b
2
a
\Q3E +
it
2
I
1
c
c
a
2
c
1
b
(i
3
b
2
1
?: : .
3
b
2
r
1
3
c
2
a
3
c
2
2
a
2
D
1
a
1
c
2
::
b
1
G
2
a
n
3
b
]
2
3
c
1
a
3
c
1
3
a
3
3
3
• '
b
••T:
'
•
JP
-f
-i/3F 0 -
JE
JP
J2E
n
JE
0 + J2P
M
-
J2E
0
* •
J2F
G
+
J2F
'•: -
J2E
it
tt
$&
+
JE
tt
Tl
c
- SP
-
JE
-/3E
•;
M
1
b
- QE
+
it
II
b
1
a
\5F -
,
b
1
c
0 + J2P
0 -
a
3
C
1
b
0 -
2
].-
3
C
1
a
2
b
3
a
1
c
2
c
3
a
1
b
2
c
3
"!:.
1
a
•
,
•
-
+
JE
/3P -
JP
+
JE
JE
->/3P -
JP
JP
^E
JE
ft
tt
J2E
tl
ft
0 + J2P
n
H
tt
tl
II
tt
n
tl
n
It
J2E
4-
GF -
JF
I/3E
+
JE
/JF
-
JF
G
-
JF
^E
+
JE
->/3E + JE
-G"F
m.
JF
-\/5B 4 JE
23
O
U
d>
ft
o
Fig. 5
Ohmic resistance and reactance values
as functions of conductor spacing r a t i o
for a triangular circuit configuration,
CHAPTER IV
TWO THREE-PHASE GROUPED NEUTRAL
CIRCUITS, GROUND CONSIDERED
Two t h r e e - p h a s e
are i l l u s t r a t e d
lines
only.
t h r e e w i r e c i r c u i t s w i t h ground
i n Fig. 6.
C u r r e n t s flow I n t h e numbered
Power flow i s from l e f t
The e f f e c t
return
to r i g h t .
o f t h e g r o u n d upon t h e m u t u a l
sequence
6,7
impedances i s e v a l u a t e d by use of Carson's E q u a t i o n s
,
w h i c h g i v e t h e m u t u a l i m p e d a n c e b e t w e e n c o n d u c t o r s m and n
w i t h ground r e t u r n a s
Z
= 0.00159 f - 1.63xlO"f'
mn-g
hm
t
h n
fir*
vp
(36)
\-> i
4.j2rrfxlO"3[o.7Ull(log §1 - lo ? |/f) * 2.1j.7l5 +
+0.00026 ^
rnm-g
\
h n
$
OHMS
MILE
= 0 . 0 0 1 5 9 f - 1 . 6 3 x l 0 " 6 hm f\f| 4- J 2n-fxl0~ 3 ( 3 7 )
[0.7l|ll(log€
- l o g \ F ) + 3.h3kkk
• 0.00026
hm
$]
OHMS
MILE
where p i s the r e s i s t i v i t y in ohm meters,
f is the frequency in cycles per second,
hm & hn a re the heights of conductors m and n in feet,
Sis the conductor spacing in feet.
This relationship includes the f i r s t
son's functions P and Q.
use of only the f i r s t
two terms of Car-
Wagner and Evans° point out that the
term in the functions yields negligible
a
al
I
b
,
•
^b1
,
1
C
,
-iX
3
1
|CX
_.
•
ll
2
= _ _ _
l*
a I ill 11 n i /1 in mi 11 ii 111 ininn
Pig. 6
Tvio p a r a l l e l three-phase grounded
neutral c i r c u i t s .
e r r o r a t frequencies of s i x t y cycles or l e s s .
S u b s t i t u t i o n of equation (36 J i n t o the general expressions for the mutual sequence impedances shows t h a t the
terms involving p o s i t i v e - and negative-sequence currents
and voltages ( i . e . Z
, Z „, Z
, Z ^)
are exactly the
same as for the ungrounded l i n e s (equations (17) through
(18) and (26) through (27).
The values of the terms i n -
volving zero-sequence voltages produced by p o s i t i v e - and
negative-sequence currents are a l t e r e d by the effect of
ground and a r e , r e s p e c t i v e l y ,
W g = W 1 0 _ 6 ) < l o ) ( ^ 8 - 9 " i 48.*)
(38)
2
(hu •* a ti£ + a h o )
Z
m02-g = W
(fa
1 0
"6'
C
ro>(j|)(U8.9 - ^ W>98)
(39)
+ a 2 h 2 + ah )
where Z __ and Z _ a r e t h e v a l u e s f o r
mOl
m02
t h e u n gQ r o u n d e d
lines
as given by equations (19) and (28).
The values of p o s i t i v e - , negative- and zero-sequence
voltages produced i n the l e t t e r e d l i n e by zero-sequence curr e n t in the numbered l i n e may be found in general terms by
replacing 1-^ I,, and I- in equations ([;), (5), and (6) by I
and then proceeding in a manner analogous to t h a t used for
the other sequence impedances.
Replacement of the mutual
impedances by t h e i r values as given in equation (36) and
applying the definitions
v
mlO
^mlO-g = j
7
Z
_ Vm20
m20-g - —
0
7
- VmOO
^mOO-g
f—
the following e x p r e s s i o n s a r e obtained
Zml0
-_ 0.01+66 ( f ) { £ l o g ^ b l S b 2 S b 3 ^ . l
60
b
o lbc2 bc3
Sbl S b2 Sb^ 3 e l S c 2 3 c ? H i o . 6 ) ( f
s
al2
s
a2 2
s
) (
^
) ( kQ_9
Q g
(4Q)
_
a32
j 1+8.98} (ha + ahb + a 2 h c )
W g
= °- 0 ^ 6 6
(
^
) (
^ l 0 S S ^ S ^ 3^3
Sbl S b 2 S b ? S e l S c 2 3 C ?
S
a l 2 Sbl2 Sol2
-
6
f
(|
So V?
+
J
l o
S
(
^>
_
j 1+8.98 ) ( h a * a 2 h f e +. a h 0 )
Equations
2
( 3 6 ) a n d ( 3 7 ) may b e r e w r i t t e n i n t h e form
ml0-g = W
-i.
O 1,_
1 0
6
"
O
p
_i_ Q '—Vl
) <£><#><
60
^ - 9" J ^.98)
(1+2)
^ . 9 - J 1.-8.98)
(1+3)
i
(h B + ah>. + a ^ h „ )
Z
m20-g = " W
1
0
( h a + a 2 h b *• a h c )
H£><l/?H
•57
27
where Z -,>> i s a function of the q u a n t i t i e s B and F given by
P i g s . 3 and St r e s p e c t i v e l y .
The functional form of Z m l 0
i s tabulated i n Tables 3 and L\ for the conductor arrangements of F i g s . 2 and l\_.
The r e l a t i o n s h i p for the r a t i o of zero-sequence
voltage in one l i n e to zero-sequence current in the other
9
line i s given by Clarke and i s repeated below for completeness .
Z
m00-g = 3 2 ^ T . g
(kk)
is evaluated from equation with term ^ ( n m * ^n^
1
replaced by 7-(ha -<• h^ +. h c + h-j_ + h2 + ho) and S^n by the
where 2-y__
geometric mean distance between conductors (S a -y S a 2 Sao S^i
S
b2 S b3 8 cl S c2 S c 3 } '
T a b l e 3 . F u n c t i o n a l Z e r o - S e q u e n c e M u t u a l Impedance V a l u e s
For a Two-Circuit P a r a l l e l - P l a n e C o n f i g u r a t i o n Line,
Numbered
Conductor
Plane
Lettered
Conductor
Plane
1 2
a
b
c
fB
a
c
b
fa
b
a
c
b
c
a
3
„
*mlO
2
1 3
fa
S
2B
-
32 s
0
4-
jB
-
j B
^
4B
c
b
a
fB
a
b
c
fB
a
c
b
2e
b
a
c
B
4-
JB
b
c
a
fB
-
•
c a b
2
-
0 +- jl
c a b
1 3
4B
?
0 + jB
c
b
a
-
5B
-
a
b
c
-
GB
- j^B
a
c
b
fe
2
jiB
jiB
J
2
b
a
c
0 + jB
b
c
a
JB - j ^ B
c a b
c
b
0 +• jB
a
-§B 2
J
4B
2
Table 3.
F u n c t i o n a l Zero-Sequence Mutual Impedance Values
For a T w o - C i r c u i t P a r a l l e l - P l a n e C o n f i g u r a t i o n L i n e . (Continued )
Numbered
Conductor
Plane
3
1
Lettered
Conductor
Plane
a
b
c
f 8 " j^
a
c
b
§B - j^B
b
a
c
0 + jB
b
c
a
fB - ]lB
c
3
1 2
a
b
0 4- j B
c
b
a
a
b
c
a
c
b
b
a
c
b
c
a
c a b
3
2
1
'mlO
^P
SR
fB - | B
0
&
+
JB
-
'a 8
ilp,
0 + JB
c
b
a
fB - jlB
a
b
c
fB
9
a
c
b
f3 . jl B
b
a
c
0 + JB
b
c
a
c
c
a
b
b
- j^B
fB" J
5
0 +- jB
a
B -jfe
Table i|. F u n c t i o n a l Zero-Sequence Mutual Impedance Values
For a Two-Circuit T r i a n g u l a r C o n f i g u a r t i o n L i n e .
Right
Conductor
Plane
Left
Conductor
Plane
1
a
2
b
3
c
0
1
a
;-:
c
3
b
0 + jP
1
b
2
c
3
a
- f P - j^p
1
b
2
a
3
e
1
c
2
a
3
t
1
c
2
b
3
-jr-jjF
§p - J|P
fp - J|P
2
a
1
fc 3
2
a
1
c
2
b
i
2
b
2
•
J
mlO
*•
jP
c
0 4 jP
3
;
0 + jP
c
3
S
- fp - j^p
]
8
3
•'
- Ip - j§p
c
1
8
3
2
c
1
b
1
a
3
b
2
1
a
3
c
2
1
b
3
0
1
b
3
a
2
c
1
c
3
a
2
L
- fp - jlF
f p - j^p
b
2
a
fp - j§p
1
c
2
:
fp - j^p
<lp- jip
"-
0 + jP
0 + jP
a
- f7 - jlp
31
Table [>. F u n c t i o n a l Zero-Sequence Mutual Impedance Values
For a Two-Circuit T r i a n g u l a r C o n f i g u r a t i o n L i n e . (Continued)
Right
Conductor
Plane
Left
Conductor
Piano
a
1
b
a
1
c
3 b
:i
c
b
l
;
•
3
3
:i.
•
•
•
c
0 <• jP
b
0 4- jF
2
V
- ^F
2
G
- Qp _ t i p
-
j^F
i
a
2
b
§F - j | F
§F
- a^p
3
c
i
b
':
3.
3
a
2
b
1
c
0 + jF
3 a
2
c
i
i
0 + jF
3
b
z
c
i
3
b
:
1".
1
c
:
c
:
:
a
1
3
c
2
b
2
a
3
2
a
?.
•
:
!
- §P > j|P
-
| F
~ j|P
b
^
- jlF
]
a
fF " j|F
b
1
3
0 * JP
3
c
1
b
0 + jF
b
3
c
I
a
- §F - J^F
2
b
3
-":
1
c
-
2
c
3
a
i
b
fF - jlF
2
c
;.
b
1
8
fF- j ^
| P
~ j|F
CHAPTER V
TWO THREE-PHASE GROUNDED NEUTRAL CIRCUITS
WITH ONE GROUND WIRE, GROUND CONSIDERED
Fig. 7 i l l u s t r a t e s
two t h r e e - p h a s e
ground w i r e and ground r e t u r n .
circuit
only.
c i r c u i t s w i t h one
C u r r e n t s flow i n t h e numbered
Power flow i s from l e f t
to r i g h t .
The e v a l u a t i o n of t h e m u t u a l s e q u e n c e i m p e d a n c e s
g r o u n d w i r e s i s b a s e d on t h e a s s u m p t i o n t h a t t h e ground
with
wires
and t h e e a r t h a r e c o n n e c t e d i n p a r a l l e l a t e a c h e n d of t h e
length of line
considered.
s o l i d l y grounded a t each
The g r o u n d w i r e s a r e u s u a l l y
tower.
An a p p l i c a t i o n o f K i r c h o f f ' s
formed b y t h e l e t t e r e d
V o l t a g e Law t o t h e l o o p s
c o n d u c t o r s and t h e e a r t h and t h e g r o u n d
w i r e and t h e e a r t h , w i t h c u r r e n t s
t h e numbered l i n e s , r e s u l t s
1^, I2
an
^ I 3 flowing i n
i n the expressions
v a -v a i = i l Z a l . g • i 2 z a2 _ g , i 3 z a3 _ g v b -v b i = i 1 z bl _ g
v
+
c-vel = Ilzcl-g +
0
= I l z wl-g
+
i2zb2.g
V m
.
g
(U5)
i 3 z b3 _ g - iwzbw_g
(kb)
X Z
2 o2-g * I 3 Z c3-g " I « z cw-g
U7)
x
2zw2-g + ^ w e - g " Vww-g
(US3
+
Equation (2+8) may be solved for the current Iw in the
ground wire.
The r e s u l t is given as
J
w
= I l Z w l - R + I2Z W 2_ R •*• l3Z W 3_p.
ww-g
( 1^9)
33
a
"~~ia^
-•I
71- 1
12
1
£--
% l2 h-1*
-1 HI i in in 111 mi in /11111 in
Fig. 7
Two p a r a l l e l three-phase grounded n e u t r a l
c i r c u i t s with one ground w i r e .
2
The s e q u e n c e m u t u a l i m p e d a n c e s Zmii-w
Z
m01-w
a r e
found
from e q u a t i o n s
only positive-sequence
(1$)
m21~w
through
(l\J)
and
when
components of c u r r e n t a r e a l l o w e d
t o flow i n t h e numbered l i n e s b y a method s i m i l a r
u s e d f o r t h e s y s t e m w i t h no g r o u n d w i r e s .
to
that
These a r e
Z
mll-w = Zmll "
<50)
P
P
fcaw-e: +aZbw-g *a Zcw-g ) (Zwl-g + a
3zww-g
z
w2- g + aZw3-g)
Z
ra21-V = Z m21 "
2
2
(2
a Z
aZ
)(2
fa Z
wl-g* w2-.g-^ w3-g aw-gbw-g^ aZ cw-g>
32
J
ww-g
Z
r Z
m01-w
m01 "
2
(Z
2
Z
)(Z
a 2
aw-g + bw-g* cw-g
wl-g+ w2-g+ a2 w3-g }
(51)
(
^2}
_
-^ ww-g
z
The s e q u e n c e m u t u a l i m p e d a n c e s Zm]_2-w>
Z ra o2-w
w
^th only negative-sequence
m22-w
and
components o f c u r r e n t
i
••.'
t h e numbered l i n e a r e found to be
z
ml2-w = zml2 ~
(53)
^zwl-g4azw2-g^zw3-g)
J
Z
(Z
aw-g +a 2 b w - g * a
Z
cw-g>
ww-g
m22-w = Z m?2 "
<^>
P
a2z
( Z w l - g * a Z w 2 - g +* Zw
w3-g^(Zaw-g * a
37
ww-g
P
Z
bw-g+aZcw-g)
Z
m02--w =
Z
m02 "
^
*
C
^6)
(
^7)
J
( z w l - g * aZw2-g * a \ zw3-g) ^zaw-g ^Zbw-g *Zcw-g)
ri
32
ww-g
The s e q u e n c e m u t u a l i m p e d a n c e s
z
mlo-w
an(
^
Z
m20-w
given in equations
(56) and
(57) a r e e v a l u a t e d b y a l l o w i n g
only zero-sequence
currents
t o flow i n t h e numbered
Z
mlO-w -
^wl-^
Z
Z
lines.
mlG "
w2-g + Z v Q - g X z a w - g + a Z b w - g + a
2
Z
cw-g)
3Z
J
2
m20-w -
Z
ww-g
m20 "
2
(Z n
+Z 0
* 2 _ )(Z
+ a 2.
+aZ
)
w l - g * w2-g + w3-g
aw-g +
bw-g *
cw-g y
3zww-g
The r e l a t i o n s h i p
tage i n the l e t t e r e d
numbered l i n e
f o r the r a t i o of z e r o - s e q u e n c e
line to zero-sequence
i s g i v e n by C l a r k e
Z
mOO-w -
Z
current in
volthe
and i s r e p e a t e d h e r e
as
(
m00 "
'58)
V z wl-g * z w 2 - g + z w 3 - g ' l z a w - g + z b w - g + z c w - g )
J
ww-g
The n u m e r i c a l d e t e r m i n a t i o n o f t h e q u a n t i t i e s
tions
(50) t h r o u g h
(^$)
can be f a c i l i t a t e d
of
by t h e u s e of
equathe
relationship
Z
aW-g
+
aZ
bw-g
*• * \ v -
B
= " (U8.9X10-6)
(59)
l ^ y ) ( j £ ) ( h a + ah b
S
y-j 0.279J-! [ l o g
a 2 h c ) + ( 0 . 2 1 | 2 ) ( ^ ) ( l o g Jew) *•
bw
+
bw SCVJ
b
+
0.00013 \ j |
aw
rt
ha
+
ah b + a h c )]
The e q u a l i t y of e q u a t i o n (59) h o l d s i f the p o s i t i o n
of the o p e r a t o r "a" i s i n t e r c h a n g e d w i t h the p o s i t i o n of the
o p e r a t o r "a " throughout
and
the
.^UUUIJ e
c 4q uuaa tu ii ou n
i i (59)
\J1I
aiiu
ULizj udiiosot aa uni cj oe os SC
and S b w a r e
u
interchanged.
The e v a l u a t i o n of the terms of the form -rfZaw-aZ
bw-s
2
cw-g)
ma
y
be
simplified! by the use of the d e f i n i -
tion
z
aw-g + Z bw-g + Z cw-g
The average impedance 2>^_
=
32
(60)
OT-g
i s e v a l u a t e d by use of e q u a t i o n
(36) w i t h ^ ( h m + h n ) r e p l a c e d by ^ ( h a + h b + h c + 3h w ) and
the d i s t a n c e S rnn by the geometric mean d i s t a n c e
(Saw Sbw
Sew * •
I t i s noted from e q u a t i o n s
(50) and ( £ l ) and (S3)
and
(5U) t h a t the ground wire has no e f f e c t upon the mutual sequence
impedances Z m l l - W , Zml2-w>
2
m2l-w
and
Z
m22-w
i f
the
ground wire
i s symmetrical t o e i t h e r the numbered or the l e t t e r e d
circuit.
S i m i l a r l y , from e q u a t i o n s (52) and (55)* i t i s seen t h a t the
e f f e c t upon Z m oi-g and Zmo2-n- i s z e r o i f the ground wire i s
symmetrical to the numbered c i r c u i t .
effect
In a l i k e manner the
of the ground wire upon Z m io-w and Zm2o-w i s zero when
37
the ground wire is symmetrical to the lettered circuit and
is small in any case.
TWO THREE-PHASE GROUNDED NEUTRAL CIRCUITS
WITH TWO GROUND WIRES, GROUND CONSIDERED
The method o f e v a l u a t i o n o f t h e m u t u a l s e q u e n c e i m p e d a n c e s of a s y s t e m made up of two t h r e e - p h a s e
two ground w i r e s ,
circuits
such as i l l u s t r a t e d i n F i g . 8, i s
t o t h a t u s e d f o r a s y s t e m w i t h one g r o u n d w i r e .
i n the ground w i r e s , w i t h c u r r e n t s
t h e numbered l i n e ,
around
Ix,
1^
a
The s o l u t i o n s
1^ =
z
for
the ground wire c u r r e n t s
^ d 1^ f l o w i n g
I
=
1^ and I
v v - g ( I a 2 w l . g * I2zw2-g * I3zw3"g^
lZvl-g *
J
2Zv2-g +
I
z
:
m01-wv ^i-^h o n l y p o s i t i v e - s e q u e n c e
=
z
m l l "3H^ Z aw-g
M
(20
* Z^
•jg* a v - g * b v - g
+
+
a2Z
ground.
are
wv-g
!61)
z
m
ii-wv»
"
ww-,g
t,2\
\u /
Z
Z
m21-wv
components of
+
aZ
)
cv-g'
bw-g
+
a
Z
and
current
a r e a l l o w e d t o flow i n t h e numbered l i n e a r e found
mll-wv
voltages
3Zv3-p:)
The s e q u e n c e m u t u a l i m p e d a n c e s
Z
in
wv-g vw-g
vw-g(^wl-g + ^ V - ^ + ^3^3-^)
v
Z
Z
Z
Z
wv-g vw-g
v v - g ww-g
I 2
T Z
( I
Z
1 V 1 - R * 2 v2-g »
3 v3-Rj
z
currents
a r e e v a l u a t e d b y t h e summation of
ww-g v v - g ~
(I
similar
The
t h e l o o p s formed b y t h e g r o u n d w i r e s and t h e
and
cw-g) "
t o be
(6^)
39
Ta 1
I
Jo*
_a
2l
-13
ix
i2
I3-IV-IW
nut/n/Trrmn/n7tminuil
Fig. 8
Two p a r a l l e l three-phase grounded n e u t r a l
c i r c u i t s with two ground w i r e s .
~
1
2
Z
m21-wv =
C
m21 " 3 H ( Z a w - g
Tiff
2
a
Z
bw-g
+
+
Z
a2
cw-g)_
O
^H^av-g +
Z
4
=
m01-wv
2
M (7
"^H
a
2
+
bv-g
aZ
cv-g)
- G
m01 " J H ( Z a w - g 7
a v - g *•
cw-g)
^ )
-
(66)
a 2 Z v 2 - g + aZv3-g) -
(67)
7
+
bv-g
Z
bw-g
)
cv-gJ
where
0 = (Zvv.g)(Zwl_g
+
(Z
a
2
(Zww.gJfZvi.g
+
M
wv-g)(Zvl-g +
=
(Zvw-g>(Zwl-g +
11
" (^-g^vv-g
a2z
a2Zw2_g
w2-g -
"
aZ
V 2 - g •*
Z
W v-g
Z
aZ
azw3.g)
+
v3-g)
«3-g'
(68)
vw-g>
If only negative-sequence
c u r r e n t s flow
i n t h e numb-
ered l i n e s ,
t h e s e q u e n c e m u t u a l i m p e d a n c e s Z m ]_2- WVJ
and 2 m Q2-wv
of
equations
(69) t h r o u g h
z
ml2~wv = 2 m l 2 " l3H
H(Zaw-g +
iL(Z
. aZ,
. a2Z
)
+
+
3H a v - g
bv-g
cv-g'
7
m22-wv = z m22 - ^ ( z a w - g
(Z
Z
av-g +
m02-wv
=
3^Zav-g
Z
+
a
Z
b v - g +*
aZ
2
bv-g *
+ a
(71) a r e d e t e r m i n e d
bw-g
Z
m22-wv
+
a
2
as
cw-g)~
bw-g * a Z e w - g ) ' "
(70)
cv-g>
m02 " " J E ^ a w - g
Z
a2
z
+
Z
bw-g *
2
cw~g^ "
t?1 >
cv-g^
where
P
-
Z
v v - g ( Z w l - g H- a Z w 2 . g , a 2 Z w 3 . g ) - Z^_g
(zvl-g *
Q
z
aZ
v2-g
+
ww-g<zvl-g +
a
Z
aZ
v2-g +
(72)
v3-g>
\Z . aZ 0
4. a Z _ )
w l - g «w2-g -1"
w3-g
a
2
v3-g)
"
Z
vw-g
(73)
IPThe s e q u e n c e m u t u a l i m p e d a n c e s i n v o l v i n g
sequences as given i n e q u a t i o n s
ilk)
through
mined by a l l o w i n g o n l y z e r o - s e q u e n c e
t o flow i n t h e numbered
Z
mlC-wv
=Z
Z
"3H< av-g
aZ
(76) a r e
components o f
deter-
current
line.
R
mlO-g ~ " J S ^ a w - g
+
zero-
+
bv-g
a£
r
2
+a Z
bw-g
cw-g) "
+
aZ
C?b)
Z
~ cv-g)
Z
m20-wv z Z m20-g~ - ^ f ^ a w - g *
T
2
z
+ a z
*3H^ av-g
bv-g + aZcv-g)
a
Z
bw-g *
2
m00-wv = z mOO-g~ T g ( 2 a w - g * 2 b w - g
T
*3lT a v - g + z b v - g + z c v - g '
+
Z
aZ
(75)
cw-g>~
cw-g) "
(76)
where
R = 2 v v „g(Z w ]__g + 2 w 2 - g •+
( Z _
vl-g
T
=
Z
+j . Z
n
v2-g
+
Z
w3-g) "
z
wv-g
(77)
+j » Z _
v3-g'
ww-g^zvl-g *
(Zwl-g
2
w2-g *
Z
Z
v2-g
4
Z
v3-g) "
Z
vw-g
(78)
w3-g^
The expression for Zmoo-wv tabulated in equation (76) is
11
given by Clarke ' and is repeated here for completeness.
Lawrence and Povejsil 12 suggest the desirability of
obtaining the relationships given in equations (63) through
(65), (69) through (71) and (7k)
through (76); however, they
conclude that the solution of the problem would be very complicated.
It will be noted from the preceding work that while
the results are cumbersome, the method of attack is straight-
forward and is, in essence, merely an extension of the
techniques used on the more simple cases.
The problem in-
volving n ground wires could be solved in a manner similar
to that outlined above.
CHAPTER VII
CONCLUSIONS
The mutual sequence impedances of parallel transmission lines can be expressed in terms of the distances
between conductors and the conductor heights above ground.
The terms in the expressions for the mutual sequence impedances which involve the conductor spacing can be written
in a general form which relates the distances between the
conductors to the ohmic impedance per unit length.
These
general variables are plotted in Figs, 3 and $ for various
ratios of conductor spacing and for line configurations as
in Figs, 2 and br.
The ground has little effect upon the positive- and
negative-sequence voltages induced in the lettered circuit
by positive- and negative-sequence current In the numbered
circuit.
It does, however, effect the zero-sequence
voltages
induced by positive- and negative-sequence currents In the
numbered line and also all components of voltage Induced by
zero-sequence currents In the numbered line.
The effect of
the ground upon 2 , , 2 0 , 2 .
and Z^can be reduced
mol-g* mo^-g' mlo-g
m^o-g
to zero by equalizing the heights of the circuit conductors
in which voltages are Induced.
The equalization of heights
will not eliminate the zero-sequence voltages In one circuit
caused by zero-sequence currents In the parallel circuit,
since this expression involves the average value of the mut-
ual sequence impedance of two lines with a common earth return.
The effect of ground wires upon the mutual sequence
impedances can be made zero by arranging the ground wires
symmetrically with respect to either the numbered or the
lettered circuit, except for the impedances involving either
zero-sequence voltages or currents in the two circuits.
The
corrections in the latter case can be eliminated in the instances involving positive-sequence currents or voltages by
making the ground wires symmetrical to the numbered circuit.
The effect of ground can be eliminated in the case of zerosequence currents in the numbered line by making the ground
wires symmetrical to the lettered circuit.
B I B L I O G R A P H Y
Barnes, H. C. and A. J. McConnel, "Some Utility Ground
Problems,*' American Institute of Electrical Engineers
Transactions, No. 55-59, (To Be Published).
Clarke, Edith, Circuit Analysis of A. C. Power Systems,
New York: John Wiley and Sons, Inc., 1950, Vol. I
p. kOk*
Clem, J. E # , 'Reactance of Transmission Lines With
Ground Return," American Institute of Electrical
Engineers Transactions,
Vol. 5cTJ September, 1931,
pp. 901-91O1.
Clarke, op. cit., pp. 3&3~k33
Bulletin of the Bureau of Standards, Vol. l\, No. 2,
pp. 302-306.
Clarke, op. cit., pp. 372-3^2
Carson, John R,, "Wave P r o p o g a t i o n i n Overhead Wires
With Ground R e t u r n , " B e l l System T e c h n i c a l J o u r n a l ,
Vol. 5, 1926, pp. 539^5FIi~
Wagner, C. P. and R. D. Evans, Symmetrical Components,
New York: McGraw-Hill Book Company, I n c . , 1933, p.HDpl.
C l a r k e , op. c i t . , p . IjOlj
loc.
cit.
loc,
cit.
Lawrence, R. P . and D. J. P o v e j s i l , " D e t e r m i n a t i o n of
I n d u c t i v e and C a p a c i t i v e Unbalance for Untransposed
T r a n s m i s s i o n L i n e s " American I n s t i t u t e of E l e c t r i c a l
E n g i n e e r s T r a n s a c t i o n s , Vol. 7 1 , P a r t I I I , pp. 5I47-55I4..