IMPEDANCE CALCULATION OF CABLES USING
SUBDIVISIONS OF THE CABLE CONDUCTORS
by
Kodzo Obed A b l e d u
B.Sc.(Hons.), U n i v e r s i t y o f S c i e n c e and Technology, Kumasi, 1976
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF APPLIED SCIENCE
in
THE FACULTY OF GRADUATE STUDIES
(Department o f E l e c t r i c a l E n g i n e e r i n g )
We a c c e p t t h i s t h e s i s as conforming
to t h e r e q u i r e d
standard
THE UNIVERSITY OF BRITISH COLUMBIA
September, 1979
(c) Kodzo Obed A b l e d u , 1979
In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r
an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t
the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y .
I f u r t h e r agree that permission f o r extensive copying o f t h i s t h e s i s
f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r
by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n
o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my
written permission.
Department n f
£ U F C T K ( ^ * U -
The U n i v e r s i t y o f B r i t i s h
2075 Wesbrook P l a c e
Vancouver, Canada
V6T 1W5
Date
S€PTe/Wfe<S«
6 K G t r ^ R i H q
Columbia
^ ,
ABSTRACT
The impedances o f c a b l e s =are some o f t h e parameters needed
f o r v a r i o u s s t u d i e s i n c a b l e systems.
I n t h i s work, the impedances o f c a b l e s a r e c a l c u l a t e d u s i n g
the s u b d i v i s i o n s o f t h e c o n d u c t o r s ( i n c l u d i n g ground) i n t h e system.
Use i s a l s o made o f a n a l y t i c a l l y d e r i v e d ground r e t u r n formulae t o
speed up t h e c a l c u l a t i o n s . The impedances o f most l i n e a r m a t e r i a l s
are c a l c u l a t e d w i t h a good degree o f a c c u r a c y b u t m a t e r i a l s w i t h h i g h l y
nonlinear p r o p e r t i e s , l i k e s t e e l pipes, give large d e v i a t i o n s i n the
r e s u l t s when they a r e r e p r e s e n t e d
by t h e l i n e a r model used.
The method i s used t o study a t e s t case o f i n d u c e d sheath
c u r r e n t s i n bonded sheaths and i t g i v e s v e r y good r e s u l t s when compared
w i t h t h e measured v a l u e s .
i i i
TABLE
OF
CONTENTS
ABSTRACT
TABLE
OF
i
CONTENTS
i
LIST
OF
TABLES
LIST
OF
ILLUSTRATIONS
i
i
i
• V
v i
ACKNOWLEDGEMENTS
v i i i
LIST
i
1.
OF
SYMBOLS
INTRODUCTION
1.1
x
1
A B r i e f Review
Impedances
of Methods
for the Calculation of
Cable
1
o
2.
3.
1.2
Skin
1.3
A Brief
1.4
Scope
THEORY
and P r o x i m i t y
Effects
Explanation
of
of
t h e Method
of
Subconductors
4
the Thesis
AND T H E
5
F O R M A T I O N AND S O L U T I O N
2.1
Subdivisions
of
2.2
Assumptions
2.3
Loop
2.4
Formation
of
Impedance
2.5
Bundling
of
the Subconductors
2.6
Reduction
of
the Large
2.7
The C h o i c e
2.8
Including
OF E Q U A T I O N S
6
the Conductors
^
^
Impedances
of
Subconductors
^
Matrix
i n t h e Impedance
Impedance
and C o n s t r a i n t
the Constraint
*
'
'
on the Current
Path
15
i n the Matrix
Solution-
,
3.1
Return
i n Neutral
Conductors
3.2
Return
i n Ground
Only
3.3
Return
i n Ground
and N e u t r a l
3.4
Use of A n a l y t i c a l Equations
"
Matrix
on t h e R e t u r n
RETURN PATH IMPEDANCE
Matrix
13
20
Only
20
20
Conductors
f o r Ground
22
Return
Impedance
•
•
. 2 2
iv
3.5
3.6
3.7
3.8
4.
25
R e p r e s e n t i n g the Ground as One U n d i v i d e d Conductor Model I I I
26
Mutual Impedance between a Subconductor and Ground w i t h Common
Return i n Another Subconductor
28
Comparison o f Model I I I w i t h the T r a n s i e n t Network A n a l y z e r
Circuit
RESULTS
4.1
5.
Model U s i n g Ground Return Formulae D i r e c t l y w i t h the
Subconductors - Model I I
31
33
Comparison of the Method o f Subconductors w i t h Standard
Methods
33
4.2
Comparison o f Ground Return Formulae
37
4.3
Comparison o f R e s u l t s from the D i f f e r e n t
4.4
Reproduction of Test Results
49
4.5
P i p e Type Cables
54
CONCLUSIONS
LIST OF REFERENCES
Models
. . . . . .
47
63
64
V
LIST
OF
TABLES
TABLE
4.1
Variation
of
4.2
Variation
Number o f
Proximity
o f t h e I m p e d a n c e o f t h e C i r c u i t o f F i g . 4.3
S u b d i v i s i o n s , Showing the I n c l u s i o n of Both
Effects i n the Calculations
4.3
Self
Impedance
Impedance
of
with
Ground
Ground
and Other
Formulae
4.4
Mutual
Impedance
Between
4.5
Comparison
4.6
Induced
4.7
Impedance
of Various
Currents
of
Pipe
Return
Path
of
Subdivisions
34
with the
S k i n and
37
as C a l c u l a t e d Using
Two U n d e r g r o u n d
Subdivided
Conductors
^
Models
i n Bonded
Type
t h e Number
0
Sheaths
Cables
~,->
for Various
Degrees
of Magnetic
Saturation
4.8
4.9
-
~
J
Zero Sequence Impedance Measurements
i n a Pipe with Pipe Return
Impedances
Concentric
-
on Three
Cables
of Cables i n Magnetic Pipes Represented
Pipes of D i f f e r e n t P e r m e a b i l i t i e s
Enclosed
J
o
a s Two
^
vi
LIST OF ILLUSTRATIONS
FIGURE
1.1
Current d i s t r i b u t i o n i n s o l i d round conductors due to skin
and proximity effects
3
Current d i s t r i b u t i o n i n the subdivided conductors of the
model
5
2.1
Subdivision of the main conductors
6
2.2
C i r c u i t of two subconductors with common return
7
2.3
Geometry of Subconductors
8
2.4
I l l u s t r a t i o n of the reduction process.
2.5
A two-w;ir'e r e t u r n ' c i r c u i t
2.6
A two-wire c i r c u i t
conductor
1.2
£, k, q.
15
:
with, common r e t u r n
1'6
in a third _
16
3.1
Subdivision of ground into layers of subconductors
21
3.2
Model with, only subconductors and ground return
3.3.
Model with ground represented as only one conductor
27
3.4
A c i r c u i t of two conductors with common ground return
29
3.5
A c i r c u i t of one conductor and the ground with common return
.26
in another conductor
29
4.1
A return c i r c u i t of two conductors far apart
4.2
Variation of impedance with the number of subconductors
35
4.3
A return c i r c u i t of two conductors very close together
35
4.4
V a r i a t i o n of the impedance of a buried conductor with
depth of b u r i a l
Cross sections of buried conductors for ground return impedance
calculations •
4.5
- . . . 33
3'8
40
4.6
Comparison of calculated s e l f impedances of a ground return loop • 43
4.7
Comparison of calculated mutual impedances between two buried
conductors
46
4.8
E l e c t r i c a l layout of the induced sheath current test
50
4.9
C i r c u i t diagram of the induced sheath current test
51-
vii
4.10
V a r i a t i o n o f magnetic p e r m e a b i l i t y o f s t e e l p i p e w i t h c u r r e n t
i n the p i p e
57
4.11
Shape o f m a g n e t i z i n g curve d u r i n g one c y c l e
57
4.12
L i n e a r i z e d m a g n e t i z i n g curves
62
ACKNOWLEDGEMENTS
I would l i k e t o express my thanks t o my s u p e r v i s o r ,
Dr. H.W.
Dommel, f o r h i s h e l p
suggestions
throughout t h i s work and f o r the t i m e l y
and c o r r e c t i o n s he made.
A l s o , I w i s h t o convey my g r a t i t u d e
to Mr. Gary Armanini of B r i t i s h Columbia Hydro and Power A u t h o r i t y , f o r
making h i s r e p o r t and t e s t r e s u l t s a v a i l a b l e f o r use i n t h i s work.
I am a l s o very
g r a t e f u l to the Government of the R e p u b l i c o f
Ghana f o r f i n a n c i n g my e d u c a t i o n
For reading
a t the U n i v e r s i t y o f B r i t i s h
through and c o r r e c t i n g the s c r i p t s , I would
to thank Ms. M a r i l y n Hankey o f the F a c u l t y o f Commerce.
b e a u t i f u l l y done by Mrs.
Engineering;
I
Columbia.
Shih-Ying
do a p p r e c i a t e
like
The t y p i n g i s
Hoy o f the Department o f E l e c t r i c a l
i t v e r y much.
ix
L i s t o f Symbols
B
flux density
D„
£q
D
P
d i s t a n c e between conductors £ and q
e
=2.71828
p i p e diameter
f
frequency
g
s u b s c r i p t denoting
GMR
geometric mean r a d i u s
h
depth o f b u r i a l o f conductor
i, I
I
p
current
pipe current
j
complex o p e r a t o r
i,j,k,£,n
/-I
subscripts
km
kilometres
m
= /(jyu/p)
m
metres
M
=
ground
inductance
q
s u b s c r i p t denoting
r,R
resistance, radius
v,V
voltage
X
reactance
Z
impedance
return
path
K ,
Bessel
log
Common l o g a r i t h m
£n
Natural logarithm
Hz
hertz
U
a b s o l u t e p e r m e a b i l i t y o f f r e e space = 4irx 10 ^ H/m
Q
functions
(base 10)
(base e)
X
y
y
tj)
=viQy
r
,
relative
permeability
permeability
flux
¥
flux
linkage
ir
=3.1415926...
y
=0.5772157...
ft
ohm
(JJ
=2lTf
=
Eulers
Constant
1
Chapter
1.1
A Brief
Review
For
input
propagation
lines
of Methods
the analysis
parameters
studies
which
of
D.M.
a r e found
calculations
Schelkunoff
f o r the Calculation
transmission line
the lines.
conductors
reasonably
cable
Simmons
systems
[1]
h a s done
systems.
the basic
surge
effects
lines,
pipes
between
and
data.
analysed
by many
are often
analysis
of
used
For single-cored
a comprehensive
one o f
studies,
i n the publication
and which
Impedances
induction
as other
been
Cable
Fault
impedance
have
resulted
for distribution
(such
accurate
i n many h a n d b o o k s
[2]
of
of
systems,
and the c a l c u l a t i o n of mutual
a l l require
work
INTRODUCTION
i s the impedance
Underground
The
of
and p a r a l l e l adjacent
fences)
1
authors.
standard
i n
charts
impedance
(coaxial)
and h i s r e s u l t s
cables,
are widely
used.
Carson
derived
turn.
the
equations
Smith
used
effects.
and
of
many
Another
[3],
of
the formulae
used
[ 2 2 ] , who h a v e
has been
of
specific
shape,
which
approach
used
and others
used
effects,
i n this
also
ground
r e -
have
calculated
namely:
ground)
systems.
calculations,
the impedances
accounts
have
i n distribution
i n impedance
(including
automatically
i s also
[4,5]
[9]
cables with
used by C o m e l l i n i ,
calculated
a l l the conductors
and Wilcox
underground
cables
f o r two i m p o r t a n t
approach
Wedepohl
et a l .
concentric neutral
dividing
This
of
Lewis,
by
above.
[19],
f o r the impedance
to correct
Talukdar
Pollaczek
and Barger
impedances
In
are
[10],
into
skin
and
et a l .
of
factors
proximity
[7]
and
Lucas
transmission
lines
smaller
conductors
f o r t h e two e f f e c t s
thesis.
Cables
with
mentioned
sector
2
shaped
conductors
uniform
or
properties
conductors
across
the
of
any
cross
irregular
section
cross
s e c t i o n or
of
non-
can e a s i l y
be h a n d l e d
with
this
method.
Skin
1.2
and
Proximity
The
r e s i s t a n c e of
determined
because
the
from
direct
wire.
In
distribution
by
the
the
of
"skin
case of
of
over
the
in
the
direct
wire
current,
cross
the
of
to
distributed
alternating
phenomenon,
of
in
neighbouring
these
bution
in
effect,
other
the
is
and
across
there
s e c t i o n of
conductor.
a
This
to
or
the
flow
conductor.
circular).
either
opposite
The
conductors
symmetrical
is
on
current
the
the
type
of
cross
exists
a
is
easily
material
section
of
nonuniform
conductor
which
phenomenon
is
is
caused
called
in
the
causes
This
around
In
conductors
a
sides
a distortion
axis
two-wire
of
the
of
arises
c l o s e by.
distortion,
the
effect",
line,
Changing
in
the
unlike
symmetry
for
conductors
due
of
due
the
instance,
which
face
the
currents
current
that
to
to
d i s t r i skin
conductor
of
Figure
s k i n and
1.1.
proximity
effects
in
round
(if
more
current
each
other
sides.
phenomena
illustrated
called "proximity
current-carrying
first
not
conductor
tends
are
uniformly
current
presence
on
transmission line
effect".
Another
the
is
current
variation
a
p h y s i c a l dimensions
current
the
Effects
conductors
3
^>
Current
Distributions
( a ) Skin Effect
Figure
(b) P r o x i m i t y E f f e c t
Current
1.1
distribution
proximity
The
conductors
amount
the
of
uneven
causes
direct
self
internal
linkage
more
current,
in
distribution
power
thus
much w i t h
both
-
of
extent
impedance
the
while
pronounced
for
current
the
inductance
conductor
with
current
additional
higher
The
very
solid
round
conductors
due
to
skin
and
loss
across
above
i n c r e a s i n g the
the
that
cross
s e c t i o n of
produced
effective
by
a.c.
an
the
equivalent
resistance
of
conductor.
The
the
in
effects
to
inner
the
part
which
of
towards
of
the
the
conductor
conductor.
This
surface
reduces
decreases
the
conductor.
depends
size
density
the
above
o n how
the
proximity
effects
pronounced
conductor
alter
they
and w i t h
effect
depends
closer spacings
between
mainly
the
are.
the
on
values
Skin
effect
frequency
the
conductors).
of
-
geometry
i t
the
varies
increases
(being
Bessel
to
alternating
functions
current
widely
found
drical
conductors
effect
is
more
derived
[17]
in
in
most
hand
close
in
the
at
higher
1.3
thesis
seeks
of
the
is
to
Figure
correct
makes
cables
both
of
Method
the
Enrico
work
take
of
into
both
any
of
into
the
of
these
main
the
above
Dividing
the
conductors
1.2.
the
cables
and
cylin-
hand,
are
in
proximity
factor
tables
customarily
size
proximity
and
are
effects
frequency
surge
et
a l .
account
and
used
usually
important
especially
studies.
[7]
it
is
shown
simultaneously
This
of
is
Subconductors
is
done
by
subconductors,
The
and
work
in
that
by
by
the
main
finding
bundling
described
i t
calculating
dividing
c y l i n d r i c a l shape,
conductors.
on
approximate
power
of
method
other
[18]
resistance
This
correcting
large
switching
subconductors
impedances
are
in
effect.
at
Comellini
effects
this
and
transmission line.
smaller
impedance
based
of
the
and
skin
even
the
formulae
for
Explanation
and m u t u a l
give
to
for
impedance
self
complicated
coaxial
On
Charts
needed
to
as
[1,9].
analyze.
increases
analytically.
e x p e c i a l l y when
impedances,
frequencies
calculate
skin effect
underground
c a l c u l a t i o n of
conductors
to
This
to
analyzed
otherwise
together.
possible
the
difficult
general,
A Brief
to
being
calculations
In
is
due
used
literature,
are
from
In
laid
the
are
in
the
them
to
this
reference.
current
into
parallel cylindrical
distributions
shown
in
subconductors
Figure
1.1
by
those
Conductor s
Approximate Current
Distributions
(b) Proximity Effect
(a) Skin Effect
Figure
1.2
Obviously
w i l l
depend
on
Current d i s t r i b u t i o n
of the model
the
accuracy
the
degree
This
Thesis
of
to
be
in
subdivided
expected
discretization,
from
such
and hence
conductors
a
on
representation
the
number
of
subconductors.
1.4
Scope
of
The
developed
in
the
porated
theory
using
model.
into
computing
calculating
fictitious
•Analytically
the
model
to
'return
derived
reduce
the
the
impedances
path'
ground
number
which
return
of
from
allows
subconductors
more
formulae
are
subconductors,
type
layers
permeability
of
cables
pipe
depending
on
are
modelled
material,
the
degree
with
of
by
treating
each
layer
saturation.
the
steel
having
a
is
flexibility
then
incor-
storage
time.
Pipe
concentric
a
for
pipe
as
different
and
6
Chapter
T H E O R Y AND T H E
2.1
Subdivisions
In
a main
and
the armour.
the
described below,
Each main
subconductors
values
at higher
before
these
makes
2.1
present,
tried
show
suggests
can be used w i t h
Subdivision
the cable
a number
considered
conductor
of
parallel
a cylindrical
formulae
by Lucas
simple.
and Talukdar
a large
that
i s
the neutral
into
inductance
c a l c u l a t e d by them
shapes
Figure
have been
of
The c h o i c e o f
the derived
which
core
i s divided
(Figure=2.1).
frequencies,
other
each
and i f
conductor
f o r the subconductors
resistance values
EQUATIONS
Conductors
as i s the sheath,
the subconductors
shapes
F O R M A T I O N A N D S O L U T I O N OF
the
the model
conductor,
cylindrical
for
of
2
deviation
further
Other
[22] b u t
from
rese.arch
shape
measured
i s
needed
confidence.
of main
conductors
Assumptions
2.2
It
i)
ii)
Each
i s assumed
subconductor
The magneitc
that:
i s uniform
permeability
of
and homogeneous
throughout
a subconductor
i s constant
i t s
length;
throughout
7
the
whole
that
i i i )
There
iv)
2.3
A l l
Loop
To
the
two
path,
or
a
of
in
is
of
uniform
current
are
derive
loop
the
formed
by
current,
but
may
be
different
from
subconductor;
of
Figure
alternating
other
Impedances
fictitious
lation
any
of
subconductors
loops
q,
cycle
distribution
in
each
subconductor;
and
parallel.
Subconductors
impedances
any
2.2.
two
The
conductor
the
subconductors,
subconductors,
return
chosen
of
path
for
the
% and k,
can e i t h e r
voltage
be
first
with
one
a
consider
common
return
of
the
subconductors
measurements
and
the
calcu-
inductances.
Figure
2.2
Loops
formed
common
Writing
the
loop
by
two
subconductors
with
a
return
equations
for
subconductor
Z
gives:
(2.1)
where
= voltage
R^
i
drop
per
unit
length
of
=
r e s i s t a n c e per
unit
length
of
=
resistance per
unit
length
of
,
i ,
=
currents
in
subconductor
subconductor
return
subconductors
path
£ and k
respectively
AC
JO
M
= mutual
inductance
between
loops
formed
by
subconductors
J6K.
£ and
N
current
phasor
= number
For
ac
in
(2.1)
value
jcol.
i
k
with
of
return
steady-state
are
Figure
2.3
path.
To
subconductor
common
subconductors
Figure
the
a
£ and
shows
(N=2,
by
the
Geometry
the
derive
returning
in
for
conditions,
replaced
2.3:
return
the
in
cross
of
£,
the
phasor
k
in
Figure
2.2)
instantaneous
values
subconductors
s e c t i o n of
inductance
q.
q
two
formulae,
V
and
voltage
I
and
v
and
by
the
subconductors
and
(£,k,q).
such
consider
current
I
in
The
flux
B
Total
flux
per
density
radius
length
*
linkage
in
the
elemental
of
loop
k^
due
to
q(equiv)
of
loop
k^
due
to
calculation
(=r^e
The
two
yldr
_ _yl_
2irr
2TT
I
in
i i s :
(2.4)
2
return
r
current
I
in
q
is:
D.
(2.5)
q(equiv)-
q( Q ly)
the
u
equivalent
^ r q ^ ^
,
fluxes
are
the
radius
of
subconductor
geometric
mean
additive;
hence
ZTT
r
q
for
inductance
radius)
the
total
flux
linkage
&q
*£k -
The
mutual
t
o
}
inductance
M£k
where
2V
£k
yr^=relative
<5r
* Kk
dr
kq
r
current
r=D
£k
e
thickness
(2.3)
yi
linkage
of
6r
2irr
r=D
cylinder
Sr
yi
where
£ is:
(2.2)
2Trr
r = r
outside
.2irr
6<j> = B
Flux
r
yi
=
unit
Flux
B at
£k
(M£^)
y
between
£n
2T:
permeability
Iq
loops
kg
of .the
q
exp(-y
/4)
rq
& and k
is
+ .IS.
return
path.
i s :
(2.6)
'
therefore
(2.7)
To
current
derive
the self
linkage
of
£q
r=r
linkage
loop
t h e same
,
pi
.
r£
= ~
££
2
and p
(2.8)
I
returning
i n q i s :
£q
pi „
(2.9)
q(equiv)
k
t
linkage i s :
£q
7 ^
£
£n
IT./
,
r£ .
exp( - r - )
are the relative
rq
i n £ i s :
£(equiv)
q(equiv)
flux
I
13_
£n
2TT
£^ d u e t o c u r r e n t
D
u
ul
• £ — d r = ^— £n
2irr
2TT
the total
*
where
£ we c o n s i d e r
£^ due t o c u r r e n t
£(equiv)
£q
r=r
loop
uldr
2irr
of
D
Hence
of loop
path.
Flux
Flux
inductance
^
exp (
)
(2.10)
ZTT
permeabilities
of
subconductors
£ and
respectively
The
self
inductance
(
M
^£) of
D
M
2.4:
=
££ h i
Formation
of
Writing
the
subconductors
n
=
27
£n
£
Impedance
the loop
gives
£q
D
loop
£q
£ i s
y
r£
,
y
rq_
(2.11)
q
Matrix
equations,
using
a s e t of l i n e a r
(2.1),
equations
(2.7)
and (2.11),
for a
11
V
J
11
Z
In
llln
llll
Z
lnll
Z
lnln
lnkl
llkm
11
Inkm
In
Z
^lkl*"
Z
(2.12a)
V.
kl
J
Z
km
i.e.
[V] = [ Z
where
b ± g
"klkm
klll
kmkm
Z
kmll
kl
' • • kmln
km
(2.12b)
] [I]
th
V.. refers to the voltage on the i
subconductor of the
Ji
.th
.
, „
2
main conductor.
I.,
1
1
i s the current i n the i * " * subconductor of the j*"* main
conductor.
Z..
jimn
i s the mutual impedance between the loops formed by
1
1
1
1
1
the i * " * and n*"* subconductors of the j " * and m"^ main
conductors respectively.
i
The p a r t i t i o n i n g of the matrix [ Z ^ g ]
n
(2.12a) groups the equa-
tions of the subconductors within each main conductor together.
The resistances and inductances i n the impedance matrix [ Z ^ g ]
are
constant, but since the current d i v i s i o n among subconductors changes
with frequency, skin and proximity effects are accounted f o r .
Normally the large impedance matrix i s not of d i r e c t i n t e r e s t .
Instead, the matrix giving voltages on the main conductors i n terms of the
12
currents
the
i n these
large
impedance
equivalent
of
(2.13)
tribution
which,
to
of
below.
equations
the impedances
to
of
can be obtained
Mathematically,
'(2.12a)
Practically,
i n a l l subconductors
voltages
i t
with
the
this
(2.12a),
a current
satisfies
i s
conditions
i s equivalent
achieve
from
to
redis-
distribution
the
voltage
on the subconductors
forming
any main
conductor
are
hence:
= V.0
J2
the current
subconductors
I.
J
=
= V.
= V.
jn
j
i n any main
into
which!.it
= I... + I
+
J l
J2
Bundling
currents
in
] by reduction.
shown
by
which
(2.13).
V.,
J l
2•5
i s needed,
the algebraic
currents
The
Also,
solving
and (2.14)
of
conductors
matrix
when m u l t i p l i e d
condition
equal;
main
of
(2.13)
conductor
i s divided.
. . .
the currents
i n the
Thus:
+ I.
(2.14)
jn
the Subconductors
i n the Impedance
Expressing
the voltages
they
i s accomplished by
carry
i s t h e sum o f
on the main
Matrix
conductors
the use of
i n terms
equations
of
(2.13)
the
and
(2.14)
(2.12a).
Consider
subconductors
(a)
the f i r s t
as shown
Subtracting
subsequent
leaves
(-
i n
main
conductor;
assume
i t
i s subdivided
into
n
(2.12a).
the f i r s t
equations
the left-hand
illustrated i n
equation
( i . e . row-1 i n 2.12a)
( i . e . row-2
side
(2.15)).
of
to
row-n)
the other
of
that
equations
from
main
equal
the
conductor
to
zero
13
(b)
By w r i t i n g
conductor
to
first
I
instead
( i . e . row-1) ,
equation
Corresponding
1^^
an e r r o r
has been
errors
i n the f i r s t
made
of
since
are introduced
into
a r e removed
by
whole
matrix
[Z,
the subsequent
from
subtracting
of
^12
+
the
+
^ilYlJ'H
I-^I^l
errors
. ]
bigJ
equation
adding
These
main
a l l the other
the f i r s t
(n-1)
^j_n'
+
equations.
column
columns
^im^ln^
+
•••
•••
+
first
of
of
the
that
conductor.
These
(c)
of
two s t e p s
The same
are i l l u s t r a t e d i n equation
steps,
conductors.
(a)
These
: i n g the voltages
and ( b ) ,
give
(2.16).
are c a r r i e d out on the other
a set of
linear
on the conductors
equations
i n terms
of
main
(2.15)
express-
the t o t a l
currents
til
in
"
these
l '
v
conductors
Z
0
l l l l
?
and i n t h e 2nd t o n-
1112-
?
S.211
0
?
\
Z
due to
; - '
; ;
l l k l
Z
•
•
C
llkm
V
hi
•
lnln<
(2.15)
>
k l l l
z
k l k l
?
klkm
\
*
0
(2.15)
|
•
0
The
lllnj
subconductors.
kmll '
symbol
" 5"
f
denotes
the operations
C
kmqn
= Z
kmqn
(a)
—Z
(for
the elements
and ( b ) ,
klqn
— Z
m,n^l)
^kmkl '
• *°kmkm
which have been
and the general
kmql
changed
i n
term i s :
(2.16)
14
2.6
Reduction
The
exchanging
equations
in
of
the Large
equations
(2.15)
the positions
of
(2.15)
Impedance
of
Matrix
are rearranged
rows
and columns
corresponding
f o r the reduction
i n such
to the main
a way t h a t
conductors
come
the
by
"bundled"
f i r s t ,
as
shown
(2.17)
1
V,
k
0
J
l l l l
J
l l k l
J
k l l l
J
klkm
=
12
0
or
process
(2.17)
km
i n abbreviated
form,
V
A
B
0
C
D
(2.18)
From
Hence
(2.18)
,-1,
V=(A-BD
C)I
the desired
[Z„]
impedance
=
Reference
(2.17).
Using
(2.19a)
[A -
[8]
Gaussian
last
row and going
been
reduced
BD-1C]
[Zc]
i s
(2.19b)
provides
a more
elimination
up u n t i l
to zero,
matrix
efficient
on t h e m a t r i x
the submatrix
achieves
the
[B],
reduction.
way o f
finding
(2.17),
as shown
[Zc]
starting
i n (2.18),
from
from
the
has
just
15
The
stored
in
d e s i r e d impedance m a t r i x
[A*]
in
[Zc]'corresponds
to
the
submatrix
(2.20).
1
V,
k .
(2.20)
0
12
_ 0
An
i l l u s t r a t i o n of
Figure
2.7
The
^
2.4
Choice
Illustration
and
Obviously,
2.2
of
w i l l
the
path
influence
return
s h o u l d be
Figure
2.5.
this
path
is
zero.
final reduction
of
the
Constraint
the
geometry
the
values
removed
To
the
on
and
obtained
by
I
reduction
the
l o c a t i o n of
for
requiring
illustrate
Return
this,
the
that
km
stage
J
is
in
Figure
process
Path
the
return
inductances.
the
consider
shown
current
the
path
The
in
influence
through
circuit
Figure
shown
this
in
2.4.
Figure 2.5
Writing
V
l
=
( R
t w o - w/i r e r e t u r n ,
A
the loop
1
equation
for F i g . 2.5 gives:
+
h
+
c i r c u i t
( R
2
^X22-X12>
+
2
h
21
<' >
s i n c e 1^=1
V
Introducing
For
l
"
( R
1
+
R
2
+
a fictitious
equivalence
Figure
J
( X
11
return
+
X
22 ~
path
o f t h e two c i r c u i t s
2.6
A, • t w o - w . i r e .
in
a
t h i r d
2 X
12
gives
2
h
} )
a configuration
of
Figure.2.6.
we r e q u i r e :
c i r c u i t
with,
conductor
22
<' >
common
r e t u r n
17
The loop e q u a t i o n s
V
a
= (%
o f Figure,2.6 a r e :
+ J ^ - X ^ ) ) ^
+ <R
+ i(X
+ (R
+
q
j (
X
q q
-X
l q
))I
+ (X
q
1 2
-X
2 q
) I
f e
>
V, = ( R
2
2 2
-X
2 q
))I
b
q
+ i(X
-X
q q
2 q
))I
q
+ (X
1 2
-X
l q
)
I
(2.24)
a
Imposing the c o n s t r a i n t t h a t the c u r r e n t i n the r e t u r n path i s
zero means t h a t
I
I
q
= - I
a
From e q u a t i o n s
V
= 0 = I
I
+
(2.25)
b
(2.26)
b
(2.24) and (2.25)
- V
a
a
b
=
[R
+ j ( X
x
( X
1
X
- 12- 2q
Using
1
- X
) ]
l
q
) - ( X
1
2
- X
l
q
) ] I
a
-
[R +
2
j(X
2 2
-X
2 q
)
(2 27)
h
-
(2.26) i n (2.27) g i v e s
V
a
_
V
b
=
[ R
1
+
R
2
+
( X
J H
+ X
22"
2 X
12
) ]
X
(
a
2
#
2
8
)
which i s i d e n t i c a l to (2.22) d e r i v e d u s i n g F i g u r e 2.5.
Therefore,
of any convenient
i t i s t h e o r e t i c a l l y p o s s i b l e t o choose a r e t u r n path
shape and l o c a t i o n f o r the i n d u c t a n c e
c a l c u l a t i o n s as ...
long as a zero c u r r e n t c o n s t r a i n t i s imposed on such a path.
considerations discussed
should
Nevertheless,
i n s e c t i o n 3.6 would r e q u i r e t h a t the r e t u r n path
be c y l i n d r i c a l i n shape, have a s m a l l r a d i u s , and be p l a c e d a t a
s m a l l d i s t a n c e below the e a r t h s u r f a c e n o t f a r from the c a b l e s and other
conductors.
18
2.8
I n c l u d i n g the C o n s t r a i n t on the C u r r e n t i n the M a t r i x S o l u t i o n
Equation
(2.20) g i v e s the v o l t a g e s on the main conductors
w i t h r e s p e c t to the r e t u r n path) i n terms of the c u r r e n t s i n these
(measured
conductors.
In p r a c t i c e , however, v o l t a g e s are measured w i t h r e s p e c t to the l o c a l ground
(or n e u t r a l conductor
or s h e a t h ) .
I f the c o n s t r a i n t on the c u r r e n t i s
i n t r o d u c e d , t h i s changes (2.20) i n t o the form:
J
ll
J
lk
J
k-l,l
J
k-l,k
(2.29)
V.
k-1
V.
J
k
L_-
•••
k l
J
k-1
-I
kk
k-1-
k
Since
£
I
o
1=1
and
I - - l
=
( 2 . 3 0 )
0
iL
r
l
-
2
...-I ^
t
This gives:
z
i r
z
Z
i k
lk-1
Z
lk
( 2 . 3 1 )
V.
k-1
Z
k-l,l
Z
kl
Z
Z
kk
k-l,k
Z
k-l,k-l
Z
Z
k,k-l" kk
k-l,k
k-1
Z
I f conductor k r e p r e s e n t s the l o c a l ground (or n e u t r a l conductor
or
the sheath) w i t h r e s p e c t to which a l l v o l t a g e s are measured, then sub-
tracting
the e q u a t i o n f o r V
from the other equations accomplishes
rC
and g i v e s :
this
19
V
1
-V
r-
k
*
lk-1
l l
Z
(2.32)
V
where
Z
-V
k
J
. . = Z. . + Z, , IJ
kk
neutral
Z
matrix
conductor
or
kl
J
k - l , k - l
k-1
2Z.,
ik
13
The
(or
k-1
is
the
impedance m a t r i x
sheath).
which
implies a
local
ground
20
Chapter
In
practice,
transmission
RETURN PATH
there
are three
return
i n neutral
(ii)
return
i n ground
only;
(iii)
return
i n ground
and n e u t r a l
Return
i n Neutral
Each
conductor,
sheath,
(2.7)
subconductors.
process,
i f
voltages
from
provided
with
Model
1:
results
phase
there
i n any
to
and ground
wires
only);
Only
conductor
a r e used
to
or sheaths
obtain
i s zero
pipes
conductors.
and i s d i v i d e d
to neutral
i n Ground
of
to
i s represented
as i n S e c t i o n
form
2.1.
the impedance
can be " e l i m i n a t e d "
the impedance
the phase
matrix
on i t ,
or
In
that
i t
The
i f
formulae
the
reduction
relates
any
so
separate
of
i n the
fact,
process,
as a
matrix
which
currents.
i n the reduction
voltage
the
other
desired,
i s connected
i n parallel
Only,
i s considered
subconductors
i n any lower
choice
decreases
were
path
conductor.
subconductors
This
for the return
(including
or neutral
and (2.11)
The ground
layers
ground
pipe
so d e s i r e d ,
another
layer.
Conductors
The n e u t r a l s
that
Return
cases
or
can also be eliminated
3.2
into
conductors
as a r e the c o r e s ,
equations
conductor
IMPEDANCE
system:
(i)
3.1
of
3
appears
as shown
layer
by using
i n Figure
are chosen
reasonable
a s o n e moves
obtained
as a separate
farther
a depth
3.1.
from
equal
to
and i s
subdivided
The d i a m e t e r s
to be twice
because
away
conductor
that
the current
the
density
the cables.
3300/,rp/f
of
of
previous
i n the
Reasonable
metres.
the
(a)
( b )
Figure
3.1
Subdivisions
of
ground
into
layers
of
subconductors
22
p = ground
f
=
frequency
The
reduction
the
3.3
of
ftm,
If
is
in
a system
in
the
of
subconductors,
section
reduced
arrangements
Ground
the
in
is
eliminated
leaving
matrix.
3* 1 ( a )
the
The
and
(b)
in
ground
the
return
difference
is
between
discussed
in
The
as
is
through
a set
ground
is
of
one
k
both
such
Analytical
To
represent
the
number
subconductors.
conductors
impedance
must
be
directly
Equations
were
industry.
derived
These
over
flat
When
these
to
earth
same
the
of
Equations
considered,
to
reduce
for
by
J.R.
which
is
i t
the
ground
equations
values
for
and
retained
return
is
are
i t
or
systems
better
amount
of
to
circuits
of
where
many
cables
or
the
ground
return
and
overhead
used
assume
that
the
conductors
are
of
ground
to
return
in
the
located
cables,
impedances
time.
transmission
a n d h a s an u n i f o r m
underground
divided
computing
are widely
applied
desired.
be
and
are
the
must
[11]
extent
in
it
Carson
in
as
sub-
Impedance
calculate
storage
the
into
eliminated
eliminated
adequately,
In
conductors,
subdivided
is
Ground Return
return
infinite
equations
true
ground
can be
and n e u t r a l
which
conductor
of
a large
neutrals
ground
conductors
3.4
Use
The
conductors
process.
mations
thus
Figures
reduction
lines
2.8,
impedance
and N e u t r a l
return
modelled
conductors .
into
and
Hz.
shown
in
in
4.2.
Return
system
the
in
as
as
included
results
section
ground,
process
implicitly
resistivity
can be
..
power
in
air
resistivity.
useful
approxi-
obtained.
23
With
in
overhead
the
ground
equal
cables,
to
the
a
earth's
1.0
of
and
Z
paper
is
that
S
which
l i e below
when
these
equations
l i e
above
the
now
the
are
ground
ground
applied
surface
at
are
to
used
under-
heights
burial.
[10],
variation
surface
m)
images
later
the
conductors
However,
these
depth
underground,
about
image
calculations.
In
the
lines,
of
Carson
ground
relatively
the
=
(1+C)
=
the
ground
showed
return
small
return
for
that
for
conductors
impedance
with
the
depths
usual
impedance
(Zg)
buried
distance
of
can be
below
burial
calculated
Z°
g
(3.1)
o
where
Z^
ground
extend
= a
Reference
[10]
gives
C
2K0
and
reference
[12]
where
K5,
m&
2rrr
that
is
in
if
the
earth
a l l directions
circular
factor
conductor
which
symmetry
accounts
located near
were
around
modified
r
=
the
exists,
for
ground
the
fact
that
surface
(jm)
Z
2
log(l/m)
for
small
m
(3.2)
as:
Ko(mr)
KL(mr)
,„
m = /
are
to
as:
gives
° _
g
so
correction
the
impedance
indefinitely
conductor
C
return
(3.4)
Bessel
internal
conductor
functions
radius
of
the
insulation)
earth
(i.e.
outer
radius
(i.e.
of
the
as:
24
p = ground
to =
2iTf,
resistivity
f=frequency
y = magnetic permeability of ground
Carson's formula for overhead conductors cannot be used for c a l culating the s e l f impedance of underground
conductors.
Equations (3.1) i s
used for this purpose, but the mutual impedances are calculated using the
overhead formula - which i s known to give good approximations for buried
conductors at power frequencies [ 2 0 ] .
Equations for c a l c u l a t i n g the s e l f and mutual impedances of underground conductors have also been derived by F. Pollaczek involving i n f i n i t e
series [19].
of underground
Closed-form approximations to the s e l f and mutual impedances
conductors v a l i d for a wide range of values of the parameters
involved have been derived by Wedepohl and Wilcox [9].
These equations (3.5),
given below, are accurate up to frequencies of approximately 160 KHz for
separations of approximately 1.0m between the conductors, and to approximately
1.7 MHz
i f the separation i s only 30 cm.
Thus
very accurate approximations
can be obtained for most p r a c t i c a l cases of cables l a i d i n the same trench
to quite high frequencies. These equations are:
Zs = ^
{ -An
M
(YmD
Z
i k = *f£
i-
£ n
+ T2 " T3 n^} fl M
}
+
J
,
" f m£ }
(3.5a)
ft/m
(3.5b)
where Z , Z-y^ are s e l f and mutual impedances of ground return path respectively,
s
(ft/m)
Y = Eulers constant = 0.5772157
h = depth of b u r i a l of conductor
(metres)
I = sum of depths of b u r i a l of conductors i and k (metres)
25
r
= outer
D
M
IK.
radius
of
conductor
= d i s t a n c e between
(metres)
conductors
i
and k
(metres)
m = /jtju/p
p = earth
Equations
|mD_^J
(3.5)
resistivity
are valid
< 0.25 for mutual
For
the range
i n fim
f o r the range
|mr|
ik
impedance
and
impedance.
|mD., | > 0 . 2 5 r e f e r e n c e
[9]
IK.
-£/(a2+m2)
J
< 0.25 for self
suggests
/(a2+m2)
-V
+
2TT
|a|+/(a +m )
2
the
integration:
-£/(a2-rm2)
-e
exp(jax)dx
2/(a +m )
2
2
2
(3.6)
where
x = horizontal
modulus
V=
of
3.5
Model
Model
II:
formulae
loop
A very
treats
through
formulae
be
Using
used
to
may b e
used.
Return
i
each
Formulae
subconductor
(Figure
case.
If
of
Directly
the depths
i
and k
of
burial
and uses
and mutual
the results
which
with
the
Subconductors
the a n a l y t i c a l ground
as an i n s u l a t e d
3.2)
and ( 3 . 8 ) below,
conductors
and k.
simple model which uses
calculate the self
(3.7)
the difference
conductors
Ground
the ground
i n this
equations
of
d i s t a n c e between
conductor
the available
impedances.
a r e needed
a r e found
with
the
ground
Equations
f o r power
return
return
return
( 3 . 5 ) may
frequency
i n many h a n d b o o k s
only,
[24,
25],
Figure
3.2
The
z
impedances
i i
=
Z..
l
where
Z^
GMR^
R^
D k
R
i
+
Z^
+
J ( ° -
j(0.1736
1
7
3
of
6
l o
are
the
self
circuit
s £MR7
+
+
ground
return
in Figure
°-
4892
3.2
are:
fi/km
)
0.4892)
and m u t u a l
=
r e s i s t a n c e of
=
d i s t a n c e between
view
the
of
between
ground
mean
radius
3
7
<-)
ft/km
the
fact
be
conductors
series
are
of
One
that
i
(3.8)
the
i
and
it
(=0.0592
i
k
and
ft/km)
(m)
(m)
Conductor
equations
at
high
approximations
used,
respectively,
(ft/km)
Undivided
inaccurate
if
path
conductor
conductor
as
impedances
return
conductor
Ground
c a l c u l a t i o n s may
infinite
and
°ik
geometric
In
the
log
=
III
separations
subconductors
60 Hz
r e s i s t a n c e of
Model
if
g
at
=
Representing
obtain
R
only
g
3.6
impedance
+
= R
k
and
Rg
Model with
would
be
used
for
ground
frequencies
are
used,
or
advantageous
and
return
for
costly
if
most
wide
to
of
27
the
elements
of
the matrix
[Z,
.
1 of
equation
(2.12a)
(2.11).
This
could be
calculated
bxg
with
of
the simpler
a fictitious
lated.
return
The ground
subdivided
the
equations
i n this
subconductors
With
ground
is
path
then
case)
this
conductor
2 above
ground,
that
effect
but
caution,
unless
more
The
ground
return
matrix
i n
to which
i t
are ignored.
main
region,
frequencies
c a n be shown
than
eddy
must
of
only
1
In
KHz
this
skin
model
be used
conductor
between
which would
above
effects
this
introduction
are
calcu-
( n o t i.
the ground
and
below.
ground
reference
for
circulate in
and returns
[20]
i t
the case of
approach
the results
that
current
advantage
formulae
1
the
the inductances
impedances
currents
up t o
involves
as one a d d i t i o n a l
conductor
frequency
at higher
pronounced
into
eddy
i s negligible
the lower
answers,
respect
considered
approach,
flows
In
with
a r e c a l c u l a t e d as shown
current
line.
and
and the mutual
i f
this
(2.7)
must
gives
a 500 kV
very
be i n t e r p r e t e d
i n the
ground.
t h e more
overhead
with
is
3.3
Model with
ground
i n one row and one column
represented
as only
one
some
much
complicated
of
(2.12a).
Figure
shown
accurate
i n the conductors
that
through
has been
effect
i s
the
conductor
the
28
In
the
addition,
fictitious return
there
path.
can be
nearly
halved
values
of
parameter
the
by
~ V
1
to
be
(3.5),
in
delays
for
the
the
use
of
circuit
l l
Z
1N
IN
Z
NN
the
of
choice of
used
equation
equations
J
distances
|mD^|
also
"
the
this
frequencies.
loop
to
locating
for
the
as
centrally
and
Writing
freedom
The
approximations,
much h i g h e r
is
path.
equation
This
thereby
more
of
(3.5)
reduces
the
g i v i n g more
accurate
complicated formula
Figure
Z
in
location
3.3
(3.6)
gives:
l g
=
v„
N
V
in
which
ground
ig
with
equation
the
Z.
g _
refers
to
The
the
mutual
in
Equation
next
z
gl
common r e t u r n
(3.5).
ground.
J
q.
(3.5)
Z
gN
z
M
Ng
gg
il
!_ g.
impedance between
This
is
s e c t i o n 3.7
cannot
valid
shows
be
only
how
Z
(3.9)
N
subconductor
i
and
calculated directly
when
is
the
common r e t u r n
derived
using
by
is
equations
(3.5).
3.7
The M u t u a l Impedance B e t w e e n a S u b c o n d u c t o r
Return i n Another Subconductor
Consider
The
loop
impedances
Figure
may
J
(3.4)
i n which
be w r i t t e n
Hg
the
and
Ground w i t h
common r e t u r n
is
the
Common
ground.
as:
J
12g
(3.10)
Z
L
J
91
2 1
s
22g
29
t
(ground
return)
*1
v»
Figure
A l l
Carson's
or
the
consider
return
is
Two c o n d u c t o r s
return
impedance
Wedelpohl'.s
Now
common
3.4
terms
in
(3.10)
¥
if
Q
common
ground
can be
c a l c u l a t e d by
Figure
(3.5)
using
equations.
a similar
conductor
circuit
in
i n which
the
2.
»
:
> Vn
with
(ground)
(reiurn )
Figure
3.5
Circuit
with
of
common
one
conductor
return
in
a
and
the
second
ground
conductor
30
The
loop
equations
may b e w r i t t e n
-112
J
as:
lg2
(3.11)
J
The
third
subscripts
The
term
and
(3.5) are equivalent
gg2.
i n equations
(3.10)
and (3.11)
Z.
=Z . , „ i s t h e o n e o f i n t e r e s t
lg2
gl2
and
Va
= \
Vb
= -V
I
-
2
~ V
The c i r c u i t s
return.
of Figures
(3.4)
(3.12)
2
(3.13)
+
g
Substituting
(3.14)
IX)
these
•v - v"
x
t h e common
i f
2
- ( I
here.
denote
2
into
Z
l l g
equation
Z
12g
(3.10)
gives
12g
Z
Z
I,
1
22g
(3.15)
-
"
V
1
V
2
-
-Z
-
V
2
Z
llg
Z
22g
+ Z
-Z
12g
22g~
2 Z
12g
Z
22g
-I
22g
Z
g
-I,
1
12g
(3.16)
From
(3.12)
identical,
and ( 3 . 1 3 ) ,
12g
i s evident
J
that
22g
The
lg2
mutual
Z
22g
Z
equations
(3.11)
and (3.16)
are
(3.17)
12g
impedances
be c a l c u l a t e d u s i n g
(3.17),
(Z^g)
(note
required
that
Z.
i g
common
g
hence:
Z
fore
i t
Z
return).
i n equation
(3.9) can there-
=Z.
q i s the f i c t i t i o u s
i
where
g
q
31
Thus
(3.9)
one
results
column.
ground
i n the use of
If
return
II
since
of
the matrix
3.8
using equation
series
impedance must
Comparison of
It
the method
would
III
should be noted
that
(TNA)
[26],
where
decoupled
from
therefore
e l i m i n a t e s the need
the phases.
On a t h r e e
A V
A V
=
Z
ca
z
3
A V
Z
ab
Z
bb
Z
cb
ab
the procedure
b
AV
c
aa
=
only
one row and
forms
be f a s t e r
phase
Network
for
of
than
every
the
model
element
then
Analyzer
i n Model
III
of
the ground
i n c l u d e d as an e x t r a
the ground
return
Circuits
i s related
transmission lines
the impedance
Z
Z
Transient
return
i s
conductor
i n every
cc
impedance i s ;
(3.19)
)
as:
, -z
Z
ab
Z
bb
Z
cb
m
-Z
ca
mutual
ca
m
ba
(3.18)
the average
m
-Z
m
ac
-Z
be
m
-Z
m
-Z
Z
m
m
m
m
m
m
-Z
-Z
Z
cc
I
+1,
a
and
line.
ac
be
to
on the
z
be
-z
a
w i l l
the Transient
transposed,
can be w r i t t e n
~AV
integral)
to be evaluated
= f (z , + z, + z
m
(3.18)
i s
III
for
equation
line,
ba
the line
formulae
of
case,
to model
aa
c
that
i s
Z
a
b
It
phase
AV
Scjuation
model
used i n representing three
Network Analyser
Assuming
with
the matrix
infinite
have
i n the l a t t e r
Model
return
(or
be used,
the i n f i n i t e series
. ]
big
i n forming
the ground
the i n f i n i t e
[Z,
(3.17)
+i
b
c
32
where I
a
+ 1^ + I
t h i s case.
c
= I g i s the c u r r e n t i n the e x t r a conductor,
ground i n
The ground r e t u r n formula w i t h i t s pronounced frequency
i s then o n l y used
for
i n the l a s t
column.
dependence
A l l o t h e r elements Z ^ - Z
a
Z , -Z a r e c a l c u l a t e d w i t h ground i g n o r e d . Furthermore,
ab
m
.
0
.
0
m
,
i f the l i n e i s
t r a n s p o s e d , the d i a g o n a l elements Z ^ - Z ^ , e t c . , become e q u a l to the p o s i t i v e
sequence impedance, and a l l o f f - d i a g o n a l elements Z
-Z
, e t c . , become z e r o .
'33
Chapter
Comparison
4.1
This
account
of
two
The
by
of
section
subdividing
conductors
d.c.
the
Method
shows
the
placed
r e s i s t a n c e of
RESULTS
4
of
how
Subdivisions
s k i n and
conductors.
two
each
metres
apart,
is
Standard
proximity
The
conductor
with
effects
impedance
as
shown
of
in
are
taken
return
Figure
and
0.0417 fi/km
a
Methods
the
into
circuit
is
4.1,
calculated.
frequency
is
60 Hz.
2000.0mm
Figure
The
4.1
large
A
return
separation
effect
negligible.
The
for
using Bessel
functions.
used
The
This
by
for
taken
subdivisions
value
of
of
Z=
0.0887
as
the
the
Figure
number
between
increase
in
The
the
two
two
conductors
conductors
r e s i s t a n c e due
UBC/BPA l i n e
to
for
apart
makes
proximity
skin effect
constants
is
program
corrected
[16]
is
this.
corrected
is
c i r c u i t of
the
+
impedance
j
exact
impedances
4.2
shows
subdivisions.
It
0.7901
fi/km
reference
shown
the
is
i s :
in
value.
Table
By
4.1
using
are
impedance
variations
seen
the
that
exact
various
numbers
of
obtained.
as
a
function
reference
of
values
the
are
34
Table
4.1
Variation
No.
may
keep
Errors
1
0.0833
0.7913
0.0377
6.1%
0.2%
7
0.0855
0.8016
0.0480
3.6%
1.4%
19
0.0878
0.7944
0.0408
1.0%
0.5%
37
0.0883
0.7923
0.0387
0.5%
0.3%
61
0.0885
0.7914
0.0378
0.2%
0.2%
Reference
0.0887
0.7901
0.0365
0.0%
0.0%
as
t h e number
of
subdivisions
i f
an e r r o r
i n storage
subdivisions
The
close
divisions
in
Subdivisions
n/km
be a p p r o p r i a t e
of
of
Subdivisions
subdivisions
compared
t h e Number
internal
ft/km
of
brought
with
X
X
t h e number
savings
use
Impedance
R
ft/km
approached
to
of
of
two c o n d u c t o r s
together,
i s increased.
However,
i t
as low as p o s s i b l e .
Nineteen
subdivisions
one p e r c e n t
time
2 of
forming
as shown
result
i s
tolerable.
from
the return
i n Figure
o n t h e two c o n d u c t o r s .
calculations
published
Chapter
and computing
X
keeping
i s
best
Substantial
t h e number
of
down.
are used
with
of
R
charts
done
using
and t a b l e s
reference
[17].
to
4.3.
The
standard
correct
circuit
of
Various
impedances
methods
Figure.4.1
numbers
of
sub-
calculated
which
for proximity
involve
effect
are
as
are
the
shown
35
0-032 -
NUMBER
Broken
Figure
4.2
OF
lines
Variation
SUBCONDUCTORS
are the reference
of
impedance
with
values.
•'
t h e number
of
subconductors
i
l<
27.02 mm
Figure
4.3
A return
According
circuit
above
circuit
to reference
of
[17],
two c o n d u c t o r s
very
the a . c . resistance
close
of
together
the
return
i s
r
= R'
x
~y
(A.D
36
where
R'=a.c.
resistance
R"/R'
A
and
Tables
ratios
the
reference
calculated
circuit
in
j
the
0.1340
pares
i t
ft/km.
In
for
This
using
effect
complicated
formulae
of
referred
to
division
among
many
the
is
and
case
in
most
parallel
in
the
is
seem to
the
conductors
conductors
is
correction
as
the
from
methods
two
cable
same
or
systems
in
adjacent
useful,
and
Charts
r e s i s t a n c e and
The
skin effect
inductance
impedance
only,
of
i s :
the
in
three
Table
of
Z=0.1048
4.2
which
only
need
two
the
or
from
+
com-
division
where
many
method
reasonably
is
accurate
of
otherwise
the
as
the
the
not
known
and
or
charts
current
a priori.
cables
of
use
hand,
spacing)
known
corrections
conductors
subdivisions,
be
the
other
three
delta
not
this
by
derived
On
or
current
calculations,
conductors
tables"
using
ducts),
a value
in
impedance
(flat
cities
gives
value
above.
to
gives
subdivisions.
"factor
Also,
and
for
of
or
arrangement
involved
From the
respectively.
reference
limited
be.
effect
factors
as mentioned
not
ratio.
inductance.
various
for
only.
ft/km
p a r a l l e l conductors
are
very
0.95
corrected
correcting
[18],
the
proximity
0.1410
made
conductor
above
j
used
are
subdivisions
common, f o r m s
when
resistance
for
and
conventional
charts"
of
1.18
obtained
"estimating
method
+
skin effect
effect
the
proximity
those
proximity
4.3,
= 0.0887
above
with
be
for
holds
[17],
to
Figure
Z
Applying
= proximity
similar equation
of
are
corrected
Where
(as
pipes
subdividing
results.
is
run
the
37
Table
4.2
Variation
with
of
of
Both
Skin
X
0.0966
0.1474
0.0429
7.8%
10.0%
19
0.1010
0.1394
0.0349
3.6%
4.0%
37
0.1022
0.1370
0.0325
2.5%
2.2%
61
0.1026
0.1361
0.0316
2.1%
1.6%
0.0887
0.1410
0.0365
15.4%
5.2%
0.1048
0.1340
0.0295
0.0%
0.0%
of
return
[19],
these
Ground
effect
have
been
Wedepohl
to
as g i v e n
Return.
only.
skin
use.
given
and W i l c o x
are given
b y many
[9],
of
and mutual
authors,
effects.
of
impedances
of
ground
the formulae,
of
i n c l u d i n g Carson
and Kalyuzhnyi
i n the form
The v a r i a t i o n
b y some
and p r o x i m i t y
Formulae
for calculating the self
formulae
easy
for skin
No c o r r e c t i o n f o r b o t h
Formulae
frequency,
R
7
*
4.3
Inclusion
in
6.1%
Comparison
always
Error
20.5%
*•Corrected
not
X
internal
ft/km
Fig.
i n the Calculations.
0.0377
**
of
Effects
of
the
0.1422
Value
Most
Showing
0.0833
Reference
Pollaczek
the C i r c u i t
1
SKIN
ground
of
Subdivisions,
X
ft/km
**
with
of
and P r o x i m i t y
R
ft/km
No.
of
Subdivisions
4.2
the Impedance
t h e Number
and L i f s h i t s
infinite
return
series
[10,11],
[13].
and are
impedance
a r e compared
loops
i n this
with
section.
According
with
for
the
for
I
J.R.
depth
most
Model
to
Carson
of
burial.'.'.of
frequencies
to
by
calculate
various
[10],
depths
of
variation
a conductor
using
the
the
equation
impedance
burial.
of
The
is
of
ground
minimal,
(3.1).
and
This
a buried
results
the
of
is
can be
are
with
shown
impedance
calculated
verified
conductor
this
return
by
using
ground
in
return
Figure
4.4.
1.171
M69
1-167
<
•
in
IMPI
o
z
<
1
1
Figure
A
been
given
tion
of
0
DEPTH
OF
simple
Wedepohl
infinite
and
of
which,
useful
and W i l c o x
series
K a l y u z h n y i and
results
10
0-5
BURIAL(m)
V a r i a t i o n of the impedance
w i t h depth of b u r i a l
4.4
very
by
the
-0-5
-1-0
1-5
though
form
Lifshits
very
form
of
the
(equations
of
[13]
•
8
of
t-5
a buried
ground
(3.5)),
conductor
return
which
is
impedance
an
has
approxima-
solution.
also
different
derive
from
the
a
formula,
more
the
final
conventional
',.
ones,
39
are
claimed
Kalyuzhnyi
to
and
Ze
where
y
very
Lifshits
= ^
= radius
p =
P
y
-
of
Equation
which
(4.2)
is
Re
Carson's
by
=
2?r2f
electric
[17]
=
a
.
of
ground
data.
return
(Ze)
as:
constant
conductor
over
insulation
(m)
of
ground
real
10_7
of
part
(ftm)
conductor
(Re)
(m)
of:
(4.3)
Q/m
from
that
obtained
from
Carson's
equations
which
to:
TT2f
authors
current
equations
impedance
measured
(4.2)
Euler's
burial
different
equations
several
of
gives
quite
approximates
self
experimentally
<WP
= depth
Re
with
- J-
r n
buried
p = resistivity
h
the
-2—
[An
I
2TT
closely
give
0.5772157
=
..r
agree
.
10
7
(or
approximations
and
others
through
g i v e n by
(4.4)
tt/m
the
involved
ground.
Kalyuzhnyi
of
them)
with
The
have
been
analysing
very
and L i f s h i t s
i s ,
used
the
marked
for
many
conduction
deviation
therefore,
from
worthy
years
of
Carson's
of
inves tigation.
In
of
the
earth,
by
using
In
this
In
Table
the
order
the
to
impedances
subdivided
calculation,
4.3
determine
and
the
Figure
which
of
ground
the
circuits
of
representations
ground
4.6,
formula best
the
is
divided
results
into
approximates
Figure
of
Figure
five
obtained
4.5
3.1
layers
from
are
the
the
calculated
(i.e.
of
self
behaviour
62
Model
I).
subconductors.
impedance
Oo012m
(a
Figure
4.5
)
Cross s e c t i o n s of b u r i e d conductors
r e t u r n impedance c a l c u l a t i o n s
for
ground
calculations
There
is
above
method
of
but
from
more
MHz.between
4.6
at
b)
from
for
power
the
results
values
from
evaluate
the
of
Figure
the
of
the
a
slight
60 H z ,
is
the
impedance
over
of
results
when
resentation
equations,
and
compared.
and
the
Wedepohl's
Lifshits
used.
the
+
By
equation
in
12%
power
frequency
i n between
of
in
case and
neglected
approximation,
subconductors: for
from
also
in
most
a
of
fuller
Figure
3.1
a.
conductor
in
Figure
4.5
a with
ft/km
when
0.9236
Z
the
3.1
the
ground
= 0.0589
gives
an
former.
a would
+
using
the
0.9181
Therefore
the
adequate
fact
because
3.1
the
that
the
a.
To
interstices
show
only
example,
at
return
is
representation
of
of
Figure
The
less
ground
for
50%
the
ground
ft/km.
of
(over
representation
For
representation
j
Figure
Lifshits.
latter
ground
improvement
be
the
the
and
Figure
of
j
(see
Kalyuzhnyi
at
while
reactance
calculations arise
real
to
formulae
results
only
Figure
at
deviation
are
methods
values.
obtained
i s :
ground
three
calculated resistance
formula
the
latter
using
compared w i t h
given
to
first
somewhere
the
various
the
Results
the
be Z=0.0597
of
from
with
the
c a l c u l a t e d impedance
resentation
marked
this
section.
a is
are
calculated using
and Wedepohl's
subconductors
of
7%
very
approximations
cross
Figure
between
l i e
the
filled
to
the
are
the
method
obtained
in
than
Carson's
A
a were
calculated
3.1
of
frequencies.
improvement
the
other
subdivisions
influence
ground
as
using Kalyuzhnyi
c a l c u l a t e d from
range
results
between
3.1
each
reactance
frequency)
"interstices"
obtained
Carson;'s
formulae
others.
the
methods
various
ground,
Discrepancies
a l l
the
the
of
a l l
using
subdividing
widely
1.0
by
resistance values
reactance
deviations
and
the
the
The
results
I
in
the
The
deviate
Model
c l o s e agreement
equations,
deviate
using
this
3.1
latter
than
1%
return
b,
rep-
in
the
rep-
purpose.
RESISTANCE
Frequency
(Hz
)
Subdl-.
visions
REACTANCE
(ft/km)
Subdivisions
Wedepohl
Carson
Kalyuzhnyi
(ft/km)
Wede—
pohl
Carson
.. K a l y u zhnyi
2.2
.002
0.002
0.02
0.004
0.039
0.041
0.038
0.062
4.5
.005
0.005
0.05
0.009
0.077
0.081
0.075
0.119
9.0
.009
0.009
0.09
0.018
0.150
0.160
0.145
0.230
15.0
.015
0.015
0.015
0.030
0.244
0.260
0.236
0.374
30.0
.030
0.030
0.031
0.059
0.475
0.503
0.460
0.723
60.0
.060
0.059
0.062
0.118
0.924
0.979
0.914
1.39
120.0
.120
0.119
0.124
0.237
1.80
1.91
1.73
2.68
500.0
.499
0.497
0.518
0.987
7.03
7.49
6.79
10.3
IK
1.00
0.998
1.04
1.97
13.6
14.5
13.2
19.7
5K
5.05
5.05
5.20
9.87
63.0
67.6
60.7
88.3
10K
10.2
10.2
10.4
19.7
121.0
131.0
117.0
168.0
5 OK
52.2
53.1
52.5
98.7
555.0
601.0
535.0
738.0
0.1M
106.0
109.0
105.0
197.4
1064.
1155.
1026.
1389.
0.5M
556.0
611.0
530.0
987.0
4767.
5207.
4622.
6371.
l.OM
116.
1320.
1064.
1974.
9038.
9881.
8810.
11870.
Table
4.3
S e l f Impedance o f Ground R e t u r n
Path
as C a l c u l a t e d U s i n g S u b d i v i d e d
Ground,
and O t h e r
Formulae
43
Figure
"4.6
Comparison of
ground r e t u r n
calculated
loop.
self
impedances
of
a
44
Mutual
Impedance
Table
calculated
1%
in
the
for
of
Ground
4.4
the
and
two
Figure
buried
r e s i s t a n c e and
frequency
between
the
Return
Path
4.7
show
the
conductors
about
results
15%
of
in
results
of
the
of
the
mutual
impedances
Figure
4.5b.
reactance
are
obtained
at
those
obtained
from
subdivisions
and
Deviations
of
about
power
Wedepohl's
equations.
The
by
using
the
used
mutual
Carson's
frequencies
for
Despite
Carson's
impedance
overhead
used;
thus
c a l c u l a t i n g the
this
close
equations
values
line
it
equations
seems
mutual
[11,16]
Carson's
impedances
agreement
in
[11]
derived
were
c a l c u l a t e d by
the
for
are
overhead
between
results,
using
i t
subdivisions
similar
line
for
most
equations
buried
conductors
should be
remembered
conductors
located
above
and
of
may
be
[20].
that
ground.
Table
4.4
Mutual
Impedance Between
Two
Underground
Conductors
*
RESISTANCE
Frequency
Subdi-
Hz
visions
REACTANCE
(ft/km)
Carson
Wede-
Subdi-
pohl
visions
(ft/km)
Carson*
Wedepohl
2.2
0.002
0.002
0.002
0.024
0.024
0.026
4.5
0.005
0.004
0.004
0.046
0.045
0.052
9.0
0.009
0.009
0.009
0.088
0.087
0.099
15.0
0.015
0.015
0.015
0.142
0.139
0.161
30.0
0.030
0.030
0.030
0.271
0.266
0.308
60.0
0.060
0.059
0.059
0.516
0.506
0.590
120.0
0.119
0.118
0.119
0.979
0.959
1.13
500.0
0.4.99
0.490
0.497
3.63
3.55
4.25
Ik
0.999
0.978
0.996
6 . 8 2'-
6.66
8.06
5k
5.03
4.84
5.04
29.0
28.3
35.2
10k
10.1
9.60
10.1
53.6
52.4
66.0
5.0k<-
51.8
46.4
52.5
216.0
213.0
278.0
100k
105.0
90.7
108.0
386.0
385.0
509.0
500k
599.0
413.0
594.0
1376.0
1471.0
1983.
Overhead
line
equations
used.
cc-
i-
t- ID
X---Wedepohl
a
oo
100
1000
10000
FREQUENCY
C Hz
,100000 1000000
)
«nO"
+ — Subdivisions
o—Carson
(0/H)
X---Wedepohl
no
Figure 4.7
100
idoo
loooo
FREQUENCY
( Hz
iooooo
)
1006000"
I
Comparison of calculated mutual impedances
between two buried conductors.
47
4.3
Comparison
The
(1/0
of
data
AWG a l u m i n u m
formation
0.4707
the Different
i s taken
from
r e f e r e n c e [4] . T h r e e
cables with
apart.
ohms/1000'f t
listed
impedance
reduced
The c o r e
respectively
a s 515 m i l s
The
sequence
from
cored
8 inches
insulation
Results
values
matrix
of
are l a i d
resistances
the inside
cables
i n a
flat
are 0.1882
and o u t s i d e
diameters
and
of
the
respectively.
the zero
elements
distribution
neutrals)
and sheath
with
and 955 m i l s
Models
at
(0),
60 Hz
positive
(1)
and negative
i n the reference
(2)
are:
0
~0.483+j0.236
tW
•
1
-0.003+j0.001
0.0
By
and
using Model
(3.8),
symmetric
II
-0.007+j0.008
0.199+j0.096
-jO.003
and t h e ground
the sequence
ft/1000
return
impedances
impedance
ft.
0.010+j0.004_
formulae
of
equations
(3.7)
calculated are:
0
"0.483+J0.231
[ Z
012]=
1
-0.003+j0.001
0.0'
If
are
the ground
used
return
i n Model
impedance
II,
0.198+j0.089
equations
the following
(3.5)
impedance
O.OlO+jO.002
derived
matrix
is
2
0.506+J0.219
-0.002+j0.001
0.0
ft/lOOOft.
-0.007+j0.008
-jO.003
0
0
symmetric
-jO.003
by Wedepohl
and Wilcox
obtained.
1
symmetric
-0.007+j0.008
0.198+j0.083
ft/lOOOft.
0.010+j0.002
48
The
maximum
values
section
is
deviation
less
3.6
than
where
the
following
the
ground
between
3% i n
the
ground
impedances
return
the
sequence
positive
is
are
impedance
the
0 1 2
obtained,
r 0.510+j 0.225
]=L
-0.002+j0.001
The
|_0.0
maximum
none
ing
of
are
the
with
] = 1
dividing
The
calculated
for
is
3.6%
the
from
the
reference: i n
ground
return
path
derived
ground
return
comparison
Table
for
ft.
into
the
zero
sequence.
subconductors
formulae
(Model
and
I),
using
the
follow-
1
-0.007+j0.008
ft/1000
0.198+j0.900
in
this
of
the
4.5
case
i s
1.5%
comparison
The
is
Comparison
Time
(s)
I
used
purposes.
Computing
Model
being
III),
symmetric
-t-jO.003
summary
(Model
0.010+j0.002 _
2
deviation
The
(3.5)
of
ft/1000
0.198+J0.900
-0.003+j0.001
maximum
equations
model
conductor
-0.007+j0.008
analytically
-0.0
one
the
1
"0.486+j0.231
012
using
reference
symmetric
0
[ Z
only
the
calculations:
+J0.003
deviation
By
as
By
and
2
.0
2
sequence.
represented
0
[Z
impedances
10.3
ft.
O.OlO+jO.002
in
the
positive
presented
of
Various
Max
Deviation
in
sequence
Table
Models
Matrix
Size
%
1.5
101
x
101
II
0.73
3.0
39 x
39
III
0.78
3.6
40
40
x.
4.5.
magnitude.
49
4.4
Reproduction
A
[21]
on
test
a cable
measured
for
produced
in
sheath
the
phase
conducted
values
section.
are
test
The
grounded
is
values
the
lated
of
from
sheaths
of
the
British
induced
phase
Columbia
currents
currents.
impedances
both
it
two
cables
both
sheath
u s u a l handbook
are
in
to
and
bonded
test
Power
Authority
sheaths
were
results
calculate
the
are
II.
three-phase
cables
are
bonded
together
(see
Figure
4.8).
in
the
in
at
The
sheaths,
the
same
their
ends
unbalance
and
the
Table
4.6
is
measured
values
and
the
using
the
impedances
currents
re-
induced
measured.
the
methods
Hydro
These
needed
circulating currents
gives
the
on
resistances
conductor
induced
Writing
of
The
high
a parallel neutral
and
the
c a r r i e d out
sheaths
currents,
[21]
at
c a l c u l a t e d using Model
through
reference
Results
system where
currents
and
in
was
this
bank.
Test
various
The
duct
of
obtained
taken
in
current
from
predicted
calcu-
[3,24].
loop
equations
Sl,k
\
around
the
loop
formed
by
the
bonded
gives:
°
=
for
k=
Al,
0
x
=
si +
*A1
V
Due
to
the
the
mutual
X
A2
Z
I
+
+
symmetry
B l ,
X
A2,
Z
S2,k
B2,
\
C2,
<4'4>
SI,
S2,
N
(4.5)
+
B2
in
impedances
Cl,
I
S2
^1
I
+
+
ci
x
I
the
above
(4.6)
(4.7)
C2
cables
are
and
equal,
the
for
spiralling
example:
of
the
cores,
most
of
50
^
Figure
I2>70 f t
4.8
E l e c t r i c a l layout
,
of.;the
:
induced
j|-
sheath
current
test.
Figure
4.9
Circuit
diagram
of
induced
sheath
current
test.
52
Hence
Z
A1S1
Z
A1S2
Z
=
equations
0
hl
Z
B1S1
B1S2
(4.4)
I ( Z
The
equation
induced
(4.9).
=
Z
C1S2
and (4.5) reduce
-ZsiSl^l
= Z~Z;
C1S1
Z
+
( Z
A1S1+ZA1S2)
( I
A1
+
( Z
A2S1+ZA2S2)
( I
A2
+
( Z
S2N W
A1S1
+ Z
A1S2)I1
sheath
The v a l u e s
f o r two c a s e s
through
ground
resistors
There
of
subdivisions
the
latter
14%
between
the
former
are
also
mainly
cable
give
a r e improvements
produces
evident
only
be p o i n t e d
+ Z
C2
)
A2S2>I2
n
using
total
deviation
an average
and sheaths
out that
and neutral)
a r e used
there
i n the cable
were
phase
deviation
effects
of
N
) I
)
]
(
'
9
)
i n the
measurements
not
were
grounded
grounded.
[3,5,21].
method
While
and magnitude
8%.
4
using
Similar
subdivisions
results
of
calculations.
conductors
consideration.
of
case,
The improvements
when
as n i n e
under
8
f o r the ungrounded
i n the impedance
system
The
methods
i n both
a r e a s many
S1N
the sheaths
i n the second s e t of measurements.
proximity
+ Z
'
4
as c a l c u l a t e d by the
results
total
4.6.
they
i n the results
S2N
subdivisions
between
d when
+ ( Z
be c a l c u l a t e d
i n Table
t h e u s u a l handbook
the i n c l u s i o n of
conductors
sheaths
a
the c a l c u l a t e d and measured
due t o
should
from
t h e bonds
R ^ and R ^>
an average
+ I
)
(
presented
i n which
compared w i t h
B2
C1
+ I
can therefore
calculated
were
+ I
(ZA2S1
+
current
calculations are also
B1
+ I
*N
+
impedance
made
to:
are
the
It
(including
Table
MEASURED CURRENTS
4.6
Induced
Currents
i n Bonded
Sheaths
CALCULATED
(AMPS)
%
Deviations
Magnitude
*
1
Ungrc unded
1
1
x
*
si
hi:.
10.8|255°
13.0 260°
2
Mag. & Phase
_|_
*
+
*
+
11.8 261°
20
10';
23
12
10.6 270°
10.0|272°
10
5
11
6
Bonds
1
25.01 4°
53.0 191° 23.0|_0°
2
25.1 [21°
41.4 198°
17.0[0°
3
23.2[32°
44.9 198°
18.4[rj°
10.0 269°
10.8 273°
10.2 275°
8
2
10
5
4
22.0 j28°
47.4|198°
20.7[0°
10.0 269°
11.0 272°
10.5 274°
10
5
11
6
5
35.0[14°
66.9)191°
29.4 [0°
14.9|255°
16.2 263° . 15.4 265°
8
3
11
7
6
30.6|353°
65.9 191°
30.4[0°
13.5(248°
15.2 256°
13
7
16
11
12
5
14
8
9.6 j 269°
Average
Bonds
14.4|258°
% De v i a t i o n s
Grounded
1
28.4[21°
56.9 191°
22.8|_0°
13.5|262°
13.5 265°
12.9|267°
0
5
1
7
2,
28.3|21°
55.1 194°
19.9\0°
12.5|262°
13.2 263°
12.6|269°
6
1
8
3
3
29.5[21°
52.4 198°
19.9 [0 °
12.2j262°
13.0 270°
12.4|271°
7
1
10
5
4
23.0[32°
50.4 194°
21.8[0°
10.7|269°
11.5 271°
11.0|272°
7
3
8
4
5
26.4J32°
60.9 198°
25.9 [0°
13.8 273°
13.l|275°
7
2
11
6
31.7|28°
55.1 198°
18.0[0°
12.91262°
13.0j 262 °
13.7 273°
13.0|274°
5
5
1
2
9
8
7
5
5
Average
% De v i a t i o n s
i
*
Taken
from Reference
[21]
+ calculated
using subdivisions.
o
54
The
to
the
are
in
slow
used)
a l l
phase
improvement
conductors
real
advantage
for
distribution
4.5
may
Pipe
Type
cooling
mechanical
or
to
and
the
angles.
true
4.2.
This
value
This
it
of
can be
can be
said
may
(when
causes
be
for
studies
or
advantage
cables
be
due
subdivisions
further
available,
of
of
lagging
for
for
slight
systems
having
are
used.
impedances
other
method
analyzed
e.g.,
a
conductors
studies
the
only
for
calculating
of
can be
that
obtained
subdivisions
would
sometimes
liquids
or
enclosed
in
insulation
c h e m i c a l damage.
of
The
at
higher
transients
in
subconductors
which
is
handbook
sector-shaped
.
conductors.
pipes
gases
Common c a s e s
which
or
are
as
act
as
ducts
protection
found
in
o i l
for
against
and
gas
filled
installations.
Pipe
non-metallic
materials
pipes
pipes
are
also
pipes
are
highly
this
steel
if
Another
readily
are
conducting
In
phase
Cables
Cables
cable
section,
surge
types
be
4.1
calculations
switching
not
Tables
subdivisions
other
the
reactance
together
systems.
that
formulae
of
in
calculated.
this
close
occur
the
in
impedance
many
fact
angles
conclude
in
frequencies
of
illustrated
To
the
deviations
convergence
as
the
main
easily
section,
pipe
conductors
is
and
pose
may
no
assumed
pipe
plastic,
problems
treated
in
aluminum
impedance
or
steel.
Plastic
calculations
and
or
aluminum
due
to
their
constant
permeability.
Steel
far
as
their
magnetic
properties
concerned.
nonlinearity
is
not
nonlinear
the
be
as
linear
are
and
divided
having
into
taken
a
into
constant
subconductors
account
and
permeability.
and
each
are
the
whole
The
cable
subconductor
is
55
assigned
lated
from
the
permeability
according
to
laboratory
enclosed
of
pipes-and
pipe
the
currents
pipe,
pipe
the
in
in
results
addition
steel
pipe
deviations
of
to
with
pipe
the
in
assumes
the
conductors
that
the
is
4.11
i l l u s t r a t i o n of
It
curve
( i . e . ,
is
around
permeability
of
small.
may
This
Deviations
inner
of
the
in
surface
pipe.
evident
the
of
B-H
that
the
the
the
the
here
in
magnetic
an
is
the
of
because
the
the
be
-
in
s t e e l pipe
nonlinearity
of
linear
200A i n
this
in
the
to
model
current
in
of
actual
a 5
as
inch
the
Chapter
case
2
the
Figure
curve.
and
fits
saturation
s i n c e more
in
values.
portions
pipe
of
provides
nonlinear.
B-H
reactance
linear
due
is
in
percentage
model
the
the
more
4.7
cable
the
give
levels
deviations
the
on
variation
measured
the
taken
saturation
Table
unshielded
is
references
The
4.10.
calcu-
Company
current
c a l c u l a t e d and
This
of
The
values.
alongside
assumed
pipe
on
data
magnetic
[15].
Figure
then
Edison
different
most
about
may
at
is
The
of
l i n e a r whereas
around
of
and
that
deviations
be b e c a u s e
results
a l l
curve)
currents
1000),
shown
seen
are
curve
an
[14]
values
reactance.
magnetization
shows
(or
are
i t
are
The
degrees
impedance
shown
2.
Consolidated
measurements
return.
calculations
is
impedance
Chapter
various
measured
current
4.7
at
references
the
reactance
From Table
for
The
in
permeabilities
s i n g l e phase
with
in
conducted
in
and m a g n e t i c
material.
presented
steel pipes
reported
permeability
its
method
experiments
cables
the
the
of
of
the
B-H
relative
calculated
this
region
effects
flows
are
on
on
most.
the
the
inside
Table
4.7
Impedance
Degrees
Pipe
Current
(A)
Relative
Permeability
of
of
Pipe
Type
Magnetic
Measured
R
X
micro-ohms;
Cables
for
Various
Saturation
Calculated
R
|
X
micro-ohms
/ft
Errors
X
/ft
100
762.0
201
151
201
117
23%
150
980.0
223
155
223
137
12%
200
1018.0
218
154
218
140
9%
300
942.0
203
147
203
133
10%
480
784.0
180
141
180
119
15%
980
484.0
141
124
141
92
26%
3500
156.0
88
93
88
62
33%
7400
81.0
80
80
80
55
31%
in
Figure
4.11
Shape
of
magnetizing
curve
during
one
cycle
58
The
in
the
pipe
deviations
large
results
are
in
It
changes
constant
Good
the
of
points
The
pipe
of
to
the
the
the
pipe
may
by
be
not
results
i
n
only
of
is
as
of
measured
three
values.
cables
Larger
especially
when
is
concerned,
the
magnetic
constant
throughout
the
cycle
the
range
an
saturated
the
to
the
is
applied
i t
saturation.
pipe
To
the
the
from
the
produce
which
the
known
that
or
outer
is
data
as
results
(or
inner
parts.
Thus
An
layers
4.9
made.
approximates
the
not.
Table
being
current
of
portions
different
saturation.
attempt
and
gives
flux)
a
is
assigning
summary
above.
calculations,
the
is
degrees
concentric
same
assuming
permeability
outer
layers.
By
error
different
into
various
obtained
is
middle
while
to
inherent
through
the
current.
assumed
Furthermore,
than
a.c.
the
section experience
are
the
obtained
not
value
if
the
dividing
this
far
steel pipe,
current
permeabilities
is
the
most
cross
this
as
material
obtained
more
surface
for
are
calculations for
pipe.
varies.
r e s u l t s when
pipe
Impedance
alongside
4.8
instantaneous
for
pipe
Better
the
be
carry
model
different
of
may
the
Table
recalled that
the
values
in
inner
made
in
permeability
actual
the
in
sequence
a nonlinear
with
results
density
flow
of
zero
calculated reactances
should be
permeability
but
given
the
currents
of
in
especially
when
Table
the
4.9,
Table
4.8
Zero
Sequence
Cables
Pipe
Current
(A)
Relative
Permeab i l i t y
Impedance
Enclosed
in
Measurements
a Pipe
with
Pipe
on
Three
Return.
Calculated
Measured
R
(micro- Dhms/ft)
R
(micro- ohms/ft)
Errors
R
X
100
762.0
197
134
178
84
10%
37%
150
980.0
204
140
199
103
2%
26%
200
1018.0
200
139
194
107
3%
23%
300
986.0
189
133
180
100
5%
25%
500
767.0
165
124
158
86
4%
31%
970
488.0
141
106
121
60
14%
57%
3600.
154.0
76
71
70
32
•8%
55%
8000
76.0
57
57
55
25
4%
55%
Table
4.9
Impedance
as
Pipe
*
Two
of
Cables
Concentric
Current
Measured
R
X
(A)
(ufl/ft)
in
Pipes
Magnetic
of
Calculated
X
Pipes
Different
Represented
Permeabilities.
Errors
R
X
(yfi/ft)
100
201
151
203
150
1%
150
223
155
230
181
3%
200
218
154
227
192
4%
25%
300
203
147
212
185
4%
26%
480
180
141
188
160
4%
13%
980
141
124
146
114
4%
8%
.1%
:.17%
3500
88
93
89
67
1%
28%
7400
80
80
81
57
1%
29%
12%
100*
197
134
186
118
6%
150*
204
140
214
149
5%
200*
200
139
211
159
6%
14%
300*
189
133
196
153
4%
15%
500*
165
124
172
128
4%
3%
970*
141
106
129
83
9%
22%
3600*
76
71
73
36
4%
49%
8000*
57
57
56
28
2%
51%
Three
cables
in
pipe.
6%
61
pipe
i s divided
assigned
other
the
into
a lower
current
inner
layer
520.
Similar
values
Table
is
used
i s
to
reflect
assigned
and the
a
of
lower
about
4.7
layers.
i t s
these
relative
value
of
40 b e l o w
and 4 . 8
The
inner
b e i n g more
values;
for
a r e made
for
the
No
example,
given
440 and t h e o u t e r
and above
i s
saturated.
thus,
permeability
layer
is
484,
layer
the
assigned
corresponding
the other
when
permeability
results
shown
in
4.9.
deviations
can be seen
i n the
26% w h e n
constant
are
for
used
current
cables
is
the
steel
current
variations
Figure
of
to
i s assumed
to
one c a b l e
A s i m i l a r drop
from
57% t o
is
and 4 . 9
from
the pipe
also
the
as h i g h
two v a l u e s
in
22% i s
4.8
drop
8% w h e n
only
i t
should
the pipe
values
densities
no m a t t e r
4.7,
the reactance
p i p e when
could be expected
permeability
of
i n Tables
of
permeability
and the
obtained
as
for
pipe
three
pipe.
permeabilities
results
the results
permeability
980A.
i n the
from
calculated values
However,
form
980A
choices
concentric
i n assigning
l i s t e d i n Tables
As
in
two
permeability
criterion is
pipe
only
i n the
how w e l l
during
4.12a,
Figure
based
i t
any
is
4.12b.
as g i v e n
i f
out
above
layers.
done,
is
definite
the
s t i l l
form
the representation
the method
rather
layers
better
the pipe
are
assigned
such as
the
average
of
any such
linearization,
since
the l i n e a r i z e d
above
assigning
and
be i n e r r o r
of
of
arbitrary
criterion,
Furthermore,
would
of
that
is
the different
o n some
cycle
whereas
be p o i n t e d
seeks
to
the
permeability
sketch
put
i t
i n
shown
the
Figure
4.12
L i n e a r i z e d
m a g n e t i z a t i o n
curves
63
1
Chapter
The
has
been
used
accuracy
The
method
of
to
run
introduction
therefore
parallel
frequencies
test
ground
method
return
takes
be highly
test
with
the model.
of
formulae
speeds
of both
skin
to d i s t r i b u t i o n
It
values
will
circulating currents
of
It
that
subconductors
the
used.
up t h e c a l c u l a t i o n s .
systems
useful
where
i n bonded
come
effects
many
surge
sheaths
close
to
at
higher
studies.
has
the
and
conductors
especially
for switching
results
calculations
i s shown
and p r o x i m i t y
be very
a r e needed
The c a l c u l a t e d
f o r impedance
cables.
on t h e number
account
suited
impedance
case of
conductors
been
field
results.
properties
method
across
modelled
Generally
are
of
of
the impedance
i n close proximity.
The
also
calculate
where
A
studied
subdivisions
the c a l c u l a t i o n depends
This
will
of
CONCLUSIONS
5
assigned
the cross
with
better
i s also
pipe
for modelling
section.
layers
results
different
suited
Cables
assigned
are obtained
permeability
enclosed
different
when
values.
conductors
i n magnetic
values
different
with
of
layers
nonuniform
pipes
are
permeabilities.
of
pipe
material
64
LIST
[-1]
[2]
OF
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D.R.
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[4]
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[5]
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J . H . Neher, "Phase Sequence Impedance o f P i p e Type
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]
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o f Overhead
"ELEC
553 -
Lines",
-
Type
Users m a n u a l , B P A ,
o f AC Power
Advanced
Cables",
Analysis
Systems",
o f Power
IEEE
Cable",
Trans.
ibid,
June 1972.
v o l . II,
Round
General
Conductors"
Systems,
Classnotes"
of B . C . , 1978.
Armanini,
Distribution
"A Proposed
Systems",
Grounding
B . C . Hydro
and 'Bonding
Scheme
for
Underground
Report.
]
R. L u c a s a n d S . T a l u k d a r , " A d v a n c e s i n F i n i t e Element T e c h n i q u e s f o r
C a l c u l a t i n g C a b l e R e s i s t a n c e s a n d I n d u c t a n c e s " , IEEE T r a n s . o n Power
Appar- and S y s t . , v o l . PAS-97, No. 3 , May/June 1978.
]
B r a n k o D. P o p o v i c , " I n t r o d u c t o r y E n g i n e e r i n g
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"Underground
]
John G. K a s s a k i a n , "The E f f e c t s o f N o n - T r a n s p o s i t i o n a n d E a r t h R e t u r n
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IEEE Trans, v o l . PAS-95, M a r c h / A p r i l 1976, p p . 610-618.
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NELA P u b l i c a t i o n 0 5 0 ,1 9 3 1 .