Post on 06-Apr-2022
transcript
Multiconductor jSignal Propagation inDistribution Line Carrier Networks
John Dongbin Suh
Center for Communications and Signal ProcessingDepartment of Computer ScienceNorth Carolina State University
CCSP-TR-88/11~
June 1988
ABSTRACT
Suh, John D. Multiconductor Signal Propagation in Distribution Line
Carrier Networks (under the direction of J.B. O'Neal, Jr.)
A mathematical model for predicting multiconductor signal
propagation in distribution line carrier networks is formulated and
tested with empirical data. The multiconductor model accounts for
general source and load termination conditions and can be extended for
analyses of multiconductor systems of order 'n'. Current and voltage
propagation measurements (a total of 14 sets) conducted on actual
three-phase distribution lines are presented. It is shown that
certain discontinuities set up different wave patterns on each line,
which introduce electromagnetic coupling effects between phase
conductors at carrier frequencies.
Several test cases are computer-simulated to assess the
validity of the mathematical model. It is found that, overall, the
predicted results based on the multiconductor model agree with that
of the measured data, and hence, the mathematical model is valid.
The derivation of the distributed parameters of the multiconductor
model is presented and implemented in a computer simulation. It
is concluded that although the theoretically derived parameters are
adequate in predicting signal profiles, a higher degree of accuracy
could be obtained by measuring the actual parameters.
ii
ACKNO~LEDGEMENTS
The author ~ishes to thank Dr. J.B. O'Neal, Dr. Sasan Ardalan, and
Dr. Steven Vright of North Carolina State University; Kay Clinard, Lou Gale,
and other members of Carolina Power and Light's Distribution Automation
Research Unit; Ken Shuey of Yestinghouse Electric Corporation, and Jamey
Phillips for their invaluable guidance, support, and assistance in this
research.
iii
TABLE OF CONTENTS
Page
LIST OF SYMBOLS v
1. INTRODUCTION... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
2. FORMULATION OF MATHEMATICAL MODEL FOR MULTICONDUCTOR 3SIGNAL PROPAGATION
2.1 General Single-Phase Transmission Line Theory 3
2.2 Multiconductor Transmission Line Equations 7
2.3 Multiconductor Transmission Line Model 12for Three-Phase Systems
3. PER-UNIT LENGTH IMPEDANCE AND ADMITTANCE MATRICES 19
3.1 Impedance of Cylindrical Wire with Return Path 20
3.2 Self and Mutual Impedances of Parallel Wires with 24Unequal Current Distribution and Ground Return
3.3 Application of Carson's Line to the Derivation 27of the Per-Unit Length Impedance Matrix
3.4 Derivation of the Admittance Matrix 30
3.5 Effect of Earth on Capacitance 33
3.6 Experimental Determination of Multiconductor 37Line Parameters
4. NETWORK AND TEST DESCRIPTION 40
4.1 Test Set-Up and Measurements 40
4.2 Network Configurations and Boundary Conditions 43
iv
5. TEST RESULTS AND ANALYSIS 53
5.1 Experimental and Theoretical Results of Test #lb 53(Bundled Conductor)
5.2 Experimental and Theoretical Results of Test #lc 61
5.3 Effect of Capacitive and Inductive Loading on 63Propagation (Decoupled Case Studies)
5.4 Experimental and Theoretical Results of Test #ld 66
5.5 Experimental and Theoretical Results of Tests 72#2c and #2d
6. CONCLUSIONS 0 •••••••••••••••••••••••••• 84
REFEREl'lCES 87
APPENDIX A - CALCULATION OF LINE PARAMETERS: PER-UNIT ..... 92LENGTH IMPEDANCE [Z] M~D ADMITTANCE [Y] MATRICES
A.I - Calculation of [Z] for Three-Phase System 94with Neutral Yire (Vertical Geometry)
A.2 - Calculation of [Y] for Three-Phase System 105~ith Neutral Yire (Vertical Geometry)
A.3 - Per-Unit Paramters for Delta Configuration ..... 110
APPENDIX B - CALCULATION OF DISTRIBUTED PER-UNIT 112LENGTH IMPEDANCE Zp AND ADMITTANCE Yp FOR ABUNDLED CONDUCTOR
B.l - Calculation of Zp for Bundled Conductor 113(Vertical Geometry)
B.2 - Calculation of Yp for Bundled Conductor 118(Vertical Geometry)
B.3 - Propagation Constant and Characteristic 119Impedance
APPENDIX C - TABULATED AND PLOTTED MEASURED DATA 120
LIST OF SYMBOLS
V Voltage
I Current
R Resistance
L Inductance
C Capacitance
G Conductance
f Frequency
Z Impedance
Y Admittance
[] Matrix representation
y propagation constant
a Attenuation constant
a Phase constant
Z Characteristic impedanceo
Y Characteristic admittanceo
r Source reflection coefficients
rL Load reflection coefficient
H Magnetic field intensity
J Current density
B Magnetic flux density
u Permeability of free spaceo
8 Permittivity constant
r Radius
~ Flux linkage
D Self geometric mean distances
v
GMR Geometric mean radius (GMR o )s
vi
GMD Geometric mean distance
D Mutual geometric mean distancem
X Reactance
M Mutual inductance
q Charge density
E Electric field intensity
[f] Propagation matrix
K Kilo
Q Ohm
Hz Hertz
A Yavelength
v Velocity of propagationp
c Speed of light (in air)
Chapter 1 - Introduction
A great deal of progress has been made in the research and
development of distribution power line carrier (DLe) technology over
the last ten years. Distribution line carrier communications enable
utilities to implement distribution automation applications, which
include load control, remote meter reading, line sectionalization,
and fault monitoring. Distribution line carrier, which is very
different from power line carrier (PLC) over transmission lines,
utilizes two-way communications from a central point to many remote
locations in a radial tree-topology network. In such a complex feeder
network consisting of taps and multiple branches, it is difficult
to achieve uniform signal strengths.
Extensive research has been done toward the development of
distribution automation systems {1},{2} in actual feeder networks.
Over the past several years, a series of DLC studies have been made
in which testing and simulation were possible in a controlled, mini
mally complex environment. These studies were based on experimental
measurements conducted at Carolina Power and Light Company's distri
bution automation test facility. Hemminger {3} researched the effect
of distributed transformer loading on the propagation of DLC signals
in a single-phase network. Borowski {4} later extended the single
phase network to include branching, and analyzed network response to
line parameter variations.
The follo~ing investigation deals with characterizing OLC
2
signal propagation in multiconductor networks. As the number of
conductors in the system increases, electromagnetic effects of a
system of parallel conductors increase the complexity of obtaining a
solution to the classical multiconductor transmission problem. In
Chapter 2, the mathematical model for multiconductor distribution
lines will be formulated for a general system of 'n' conductors.
Steady state closed-form solutions for voltage and current as a func
tion of distance will also be presented. In Chapter 3, the per-unit
length equivalent circuit parameters of the multiconductor model will
be defined and theoretically derived for a typical distribution network
operating at carrier frequencies. A description of the propagation
measurements along several multiconductor networks will be presented
in Chapter 4, and will be analyzed in Chapter 5, where several test
cases will be compared to simulations based on the theoretical model.
Finally, an assessment of the mathematical model and program implemen
tation will be presented based on the correlation between experimental
and simulated results.
3
Chapter 2 - Formulation of Mathematical Model for KulticonductorSignal Propagation
Theoretical investigations into the propagation of electro-
magnetic waves in multiconductor transmission systems have been
carried out by various authors {7,8,13,36}. Although the theory
is developed for many applications, such as microstrip directional
couplers, shielded pair cables, and power line carrier networks, the
mathematical basis for modelling is common to any multiconductor system.
In the following sections, the mathematical model for multiconductor
signal propagation in distribution line carrier networks will be
formulated.
Section 2.1 - General Single-Phase Transmission Line Theory
The propagation of electromagnetic waves in overhead conductors
can either be characterized from a distributed parameter circuit point
of view using Kirchoff's equations, or by a more rigorous approach in
which the electromagnetic field in a multiconductor system is deter-
mined from Maxwell's equations. The latter approach is discussed by
Kuznetsov {8}, who expresses the solution to the wave equations in
terms of contour integrals. The former approach will be investigated
in great detail, since voltage and current are considered to be easily
measurable quantities, constrained to follow fundamental differential
equations based on Kirchoff's laws.
4
The general transmission line equations for voltage and
current are well-known for single-phase networks, as are the various
equivalent circuit networks (T-type, L-type, Pye, etc.). The distri
buted per-unit length equivalent circuit for an incremental length ~
is shown in Figure 2-1. From Kirchoff's circuit laws, we obtain the
following partial differential equations:
dV(X,t)/dX
ai(x,t)/dX
-Ri(x,t) - Lai(x,t)/at
-Gv(x,t) - Cav(x,t)/at
<2.1>
<2.2>
In Equations 2.1 and 2.2, also known as the general equations of
telegraphy, voltage and current are functions of two independent
variables, time (t) and distance along the line (x). The familiar
distributed parameters R,L,C, and G represent the per-unit length
resistance, inductance, capacitance, and conductance, respectively.
These distributed parameters, which appear in the above equations as
"coefficients" for single-phase lines, comprise the elements of the
distributed admittance and impedance matrices for multiconductor
systems, as ~e shall further investigate in Chapter 3.
Transformation of Equations 2.1 and 2.2 into the frequency
domain reduces the partial differential equations for voltage and
current to a system of linear ordinary differential equations as a
function of a single variable x:
+
l'
R6X L6X
G6X
6X
--- C6X
!II
5
Figure 2-1 Distributed per-unit length circuit model fortwo conductors
dV(x)/dx
dI(x)/dx
-ZI(x)
-YV(x)
<2.3>
<2.4>
6
where Z = R + jooL and Y = G + jwC. Differentiation and cross
substitution of Equations 2.3 and 2.4 yield:
ZYV(x)
YZI(x)
<2.5>
<2.6>
The solutions to the above system of linear, homogeneous differ-
ential equations are veIl known:
V(x)
I(x) (liZ )(Ae-yX - BeyX)o
<2.7>
<2.8>
where characteristic impedance 2 = (2/y)1/2 = (R+jwL)/(G+jwC)}1/2o
and propagation constant y = (2y)1/2. The constants A and Bare
determined by the imposed boundary conditions at the source and load.
It can also be shown {28} that closed form steady-state solutions
for voltage and current exist in the form:
V(x)v Z
S 0
z + Zo S
<2.9>
I(x)vs
z + Zo S
<2.10>
7
where Vs represents the source voltage, Zs the source impedance,
rs the source reflection coefficient, and rL the load reflection
coefficient. In the following section we shall see how Equations 2.1
to 2.10 for single-phase transmission lines are related to systems of
equations for multiconductor transmission lines of 'n' parallel
conductors.
Section 2.2 - Multiconductor Transmission Line Equations
Several researchers {17,24} have studied the theory of
uniform multiple-coupled transmission lines. The mathematical
model for these multiconductor systems is used extensively in the
modelling and prediction of crosstalk in various environments. The
same model will be used to predict voltage and current propagation
in distribution networks consisting of 'n' conductors. The system
under consideration consists of 'n' overhead conductors, numbered
from 1 to 'n'. The conductors are assumed to be parallel to the
surface of the earth. Vl' V2 , · .. ,Vn will represent the voltages in
each of the phase conductors. Likewise, II' 1 2 , ... , In will repre
sent phase currents. Equations 2.3 and 2.4, which describe voltage and
8
current on an elemental section of line dx, can be written in matrix
form as:
d[V(x)]/dx
d[I(x)]/dx
-[Z][I(x)]
-[Y][V(x)]
<2.11>
<2.12>
vhere [V] and [I] are vectors of dimension 'n x l' and [Z] and [Y] are
'n x n' per-unit length impedance and admittance matrices respect-
ively. Symbolically, for a system of 'n' conductors,
[V]
[I]
<2.13>
<2.14>
where T denotes transpose. Ma~rices [Z] and [Y] are arbitrarily
represented as:
211 Z12 Z1n
I 221 222 Z2n<2.15>[2] I
II
I
I Zn1 Zn2 '3nn I
I j'-
9
Y11 Y12 Ylnl
Y21 Y22 Y2n <2.16>[Y]
II Ii Yn1 Yn2 Y
nn J\-
The elements on the diagonal of [Z], namely Z11' Z22' ... , Znn
represent self- or internal impedance. They take the form Z.. =JJ
R.. +jooL .. , where R.. is the internal resistance of conductor "j",JJ JJ JJ
and Lj j its self-inductance. The off-diagonal elements Zjk (j~k),
h d i · b d t l' d t b the J' th and k t hrepresent t e Istrl ute mu ua In uc ance et~een
wires. They are expressed in complex form as Rj k + jwLj k. Similarly,
the diagonal elements Y11' Y22' ... 'Ynn represent self-admittances in
the form G.. + jwC.. , where G.. is the self-conductance and C.. theJJ JJ JJ JJ
the self-capacitance. The off-diagonal elements Yj k represent "mutual"
admittances in the form Gj k + jwCj k, where Gj k represents mutual
conductance and Cj k the line to line, or mutual capacitance. In
distribution networks where spacings bet~een conductors are signifi-
cant, the conductance component of the admittance is assumed to be
negligible. The derivation of each element in [Z] and [Y] will be
presented in Chapter 3. Intuitively, the off-diagonal, or "mutual"
terms of [Z] and [Y] exist because the lines are geometrically
positioned such that electromagnetic fields interact, thus inducing
10
voltage and current in each conductor. Differentiating equations
2.11 and 2.12 and cross-substituting yield:
2 2d [I(x)]/dx
[Z][Y][V(x)]
[Y][Z][I(x)]
<2.17>
<2.18>
Note that these multiconductor transmission line equations correspond
to Equations 2.5 and 2.6 for single-phase lines. At this time, it
is convenient to define the following 'n x n' matrices:
Propagation Matrix [f] ( [Z][y])ll2
Characteristic Admittance Matrix
Characteristic Impedance Matrix [Z ] = Inv{([Z][y])1/2}[Z]o
where "Inv" denotes the inverse of a matrix. The corresponding
multiconductor solutions to Equations 2.17 and 2.18 are analogous to
the single-phase solutions (Equations 2.7 and 2.8), and are obtained
by solving a system of '2n x 2n' linear, homogeneous second-order
differential equations:
[V(x) ]
[I(x)]
<2.19>
<2.20>
11
where 'n x n' matrices [A] and [B] are determined from a set of '2n'
boundary conditions. In solving the equations for a system of 'n'
parallel conductors, there are generally 'n' roots (eigenvalues) that
correspond to 'n' "modes" of propagation. This approach involves the
diagonalization of the matrix product [YZ] in order to isolate or
effectively decouple the voltage and current equations.
In {13}, Paul presents a method which utilizes chain
matrix parameters in constructing solutions to the classical multi
wire transmission line equations. In {14}, several matrix identities
are given by Paul, along with a set of matrix equations which incor
porate terminal constraints for the total solution of line currents.
Similarly, a highly systematic technique for solving systems of
multiconductor equations which utilizes Green's matrix is formulated
by Gruner {39}. Gruner's method is valid for arbitrarily ter
minated networks and can be applied in various situations, which
include voltage or current excitation applied at any point along the
network. Alternatively, a derivation by Riddle {33} arrives at
multiconductor closed-form solutions for voltage and current as a
function of line length. These closed-form solutions, which exist in
matrix form for a multiconductor system, are very similar to Equations
2.9 and 2.10 for single-phase systems, and are convenient for com
puter simulation and algorithm development. The aforementioned
methods to the solution of the multiconductor transmission line
equations are all empirically verified and assessed by computer
12
simulations in {33}. The method developed by Riddle is shown to be
more computationally efficient and versatile. Consequently, the
measured results (to be discussed in Chapter 5) are simulated using
the closed-form multiconductor solutions developed by Riddle.
Section 2.3 - Hulticonductor Transmission Line Modelfor Three-Phase Systems
~e will no~ consider a multiconductor system composed of three
homogeneous lines above a neutral plane. The per-unit length equiva-
lent circuit model for an elementary length dx is sho~n in Figure 2-2;
The mutual coupling elements in [Y] are connected in shunt from phase
to phase. Similarly, the self-capacitance terms located on the
diagonal are represented by a phase to neutral shunt connection.
Distributed resistances per-unit length are connected in series with
the line. The arro~s between conductor self-impedances denote the
mutual inductive coupling between phases, which is analogous to ideal
transformer coupling. Assuming that all conductors are homogeneous
implies that [Y] and [Z] are symmetric matrices. Hence, Yjk=Ykj and
Zjk=Zkj. From Figure 2-2 we utilize Kirchoff's relations to
produce a system of six differential equations:
dV (x)/dx1
<2.21>
<2.22>
13
Ij (x~ Ij(~+h),.~
I
ZjJ dx. ++
\.
V/yJ Y,odx \ V/:t:+dx)JJI
ref ~k(h- ", " " '" " -,
I
IIVk(x) I Ykkdx Vk(x+dx)
II
IZkk dx ++
Il- "4'(~) I k (x+d~)
Figure 2-2 Per-unit length multiconductor transmission line model(two conductors)
14
<2.23>
<2.24>
<2.25>
<2.26>
The above system of equations can be expressed in the matrix form
of Equations 2.11 and 2.12 as follows:
V1(X)l(,...
Z13lI 1(X)l
I
211 212I
d/dx V2 ( x ) 221? 223 12( x ) <2.27>~22
V3
( x ) 231 232 233 13 ( x ) J
" iYll+Y12+Y13 '" ( "II (x ) I -Y12 -Y13 \IVl (X) ,
II
d/dx 12
( x ) ' = - I -Y21 121+Y22+Y23 -Y23 V2( x )I <2.28>I lV3( x )13 ( =< ) 1 I -Y31 -Y32 Y31+Y32+Y33
/ "
From the system of equations above, vhich ~ere determined by the per-
unit length multiconductor model of Figure 2-2, it is evident that as
the number of conductors increases, the general solutions become more
computationally involved. In addition, nonuniformity of lines, and
15
the incorporation of terminal constraints further complicate the
solution process. Thus, we assume that the system is composed of
uniform conductors, and that wave propagation is constrained to TEM,
or "quasi-TEM" mode. A fundamental property of TEM (transverse electro-
magnetic) waves is that the components of the electric field intensity
vector E and magnetic intensity vector H only exist perpendicular
to the direction of propagation along the line. This implies that
E and H are zero, where x denotes the distance along the line.x x
In a strict sense, since losses are eminent in any real transmission
network, the longitudinal components E and H are not zero, andx x
hence, we cannot assume pure TEM wave propagation. The waves actually
exhibit a combination of TE and TM modes of propagation. These
"hybrid" waves are referred to as "quasi-TEM" waves. Since the
longitudinal electric and magnetic field components are assumed
to be considerably smaller than the transverse components, the so-
called "quasi-TEM" wave can be approximated by the TEM wave. For
the remainder of the discussion, we will assume "quasi-TEM" wave
propagation.
For a system of 'n' equations, '2n' boundary conditions must
be incorporated to solve for the constants ('nxl' vectors) [A] and
[B) in equations 2.19 and 2.20. Note that these matrices are
equivalent to the constants associated with forvard and backvard
travelling waves for single-phase transmission lines. For our
specific system (as modeled in Figure 2-2), a total of six boundary
conditions are needed for the solution.
16
Ye will consider for simplicity, a linear, reciprocal network
in which a source voltage is applied at the sending end of only one
line, the other two lines terminated in an arbitrary impedance between
phase and neutral at the source, as shown in Figure 2-3. Similarly at
the load, arbitrary impedances are also connected between phase and
neutral. Consequently, the source and load impedan~es of the system
can be represented as 3x3 diagonal matrices. It is often convenient
to represent source and load impedances as respective admittances in
shunt with a Norton equivalent current source in order to accomodate
the open-circuit load, ~hich is quite ·common in distribution networks
operating under normal conditions. Thus, the source and load termina-
tions can be represented by diagonal admittance matrices as follows:
a
a
a
a
o
a
o
a
a
a
o lo I
YL3 J
<2.29>
<2.30>
Note that the terminations between phase and neutral produce diagonal
17
Figure 2-3 Three-phase transmission line with source and loadterminations
18
source and load matrices. Additional line to line terminations would
create off-diagonal terms (The analysis for incorporating these
termination conditions is discussed in detail in {l} and (12) ).
Yith boundary conditions known at the source and load, along
with the per-unit length impedance and admittance parameters, the
closed-form expressions for voltage and current as a function of dis
tance can be solved.
The solution to the multiconductor transmission problem is
important in characterizing forms of crosstalk, which exist in a
multitude of applications, including distribution line carrier. The
coupling effects, ~hich must be analyzed from an electromagnetic point
of view, inherently reside in the per-unit length circuit parameters,
which are covered in the next chapter.
19
Chapter 3 - Per-unit Length Impedance and Admittance Matrices
The distributed parameters of the mathematical model present
ed in Chapter 2 will be defined by considering individual elements
of the impedance matrix [Z] and the admittance matrix [Y]. The terms
of [Z], namely the self and mutual resistances and inductances, can
be derived by examining the flux linkages both internal and external
to the conductor. Equations describing self and mutual impedances
are based upon a modification of Carson's line equations for wave
propagation in parallel overhead wires with ground return. The- 1
actual calculations become complicated at higher frequencies when
currents tend to redistribute themselves toward the outer surface of
the conductor, thus resulting in an increased resistance per-unit
length and a decreased inductance per-unit length.
The admittance matrix is composed of self and mutual con
ductance and capacitance terms. The capacitance terms can be
derived by considering the physical conductor geometry in reference
to an e~uipotential earth surface. The conductance per-unit length,
on the other hand, is affected by factors that may not be control
lable or measurable.
The derivation of net~ork parameters [Z] and [Y] will be
outlined in this chapter. Other references {11,30,31,32) contain
more complete and rigorous analyses of these parameters. Calcula
tions for [Z] and [Y] based on actual physical test network
characteristics are presented in Appendix A. These calculated net-
work per-unit length parameters are used in computer simulations of
several test cases, which will be discussed later in Chapter 5. An
experimental method for obtaining actual network parameters from
open-circuit and short-circuit input impedance measurements will
also be presented in this chapter.
Although the derivations of [Z] and [Y] are considered
separately by effectively isolating respective magnetic and electric
fields, it is evident that the equations for voltage and current
accommodate both field effects. Hence, the term transverse electro-
magnetic (TEM), or "quasi-TEM" (as discussed earlier) applies in
the formulation of per-unit length distributed parameters.
Section 3.1 - Impedance of Cylindrical Vire with Return Path
From Ampere's law for static magnetic fields,
,. re H -d I = : J. ds I , <3.1>
which states that the line integral of a static magnetic field
intensity around a closed path must equal the total current enclosed
by the path. For a typical segment of cylindrical ~ire, as sho~n in
Figure 3-1,
closed path
ds
;> II;'~
I~J I1( ..( ,
\\ ~
\\",
Figure 3-1 Ampere's Law for a cylindrical conductor
21
1 <3.2>
<3.3>
22
H<p 1/2 nr <3.4>
The magnetic flux density B around this path is expressed as:
B u Ho <3.5>
where u is the permeability of free space (uo 0
-74nxlO Him) •
Thus, B -72xlO Ilr <3.6>
Assuming a medium of constant permeability, the equations for
inductance of a return circuit consisting of two parallel wires can be
derived. The total inductance of the circuit is found by dividing the
sum of the internal and external flux linkages by current I. Yoodruff
{S} expresses the total number of flux linkages per meter length about
one wire as:
~tot-710 1{2ln(D/r)+u/2u)o
<3.7>
where D denotes the distance between phase and neutral wire and r the
radius of the conductors (assuming both are homogeneous). Thus, the
inductance per-meter (Him) of one vire is:
L lO-7(2ln(D/r) + u/2u )o
<3.8>
23
Woodruff extends the theory to a parallel system of 'n' homogeneous
conductors, in which mutual geometric mean distances (GMD) and geo-
metric mean radii (GMR) are utilized. As a result, the total number
of linkages about a conductor with self GMD D and mutual GMD Ds m
with respect to neutral current is:
~tot 2xlO-7I{ln(D /D)} linkages per meterm s <3.9>
Dividing by I and converting to units of miles, we obtain the
following expressions for inductance and reactance:
O.3219(ln(D /D)} mH/milem s
2.020xl0-3j{ln(D /D)} ohms/milem s
<3.10>
<3.11>
From these equations and Carson's line equations, we can derive line
parameters (see Appendix B for derivation of Z and Y ) for ap p
homogeneous three phase system in which phase conductors are injected
with the same current, and this effectively reduces to a single
"bundled conductor".
24
Section 3.2 - Self and Mutual Impedances of Parallel Vires withUnequal Current Distribution and Ground Return
Ye will now consider a group of parallel, non-zero, current
carrying conductors, in which each wire experiences an induced voltage
due to flux linkages between current-carrying conductors.
Two segments of parallel wires (denoted by a-a' and b-b') are
shown in Figure 3-2. This particular circuit model will be used to
describe self and mutual inductance terms of the per-unit length
impedance matrix [2]. The circuit model is analogous to a one-turn
air core transformer equivalent. From field theory, if an applied
potential V creates a current I in the direction shown, a mag-aa a
netic flux ~ba linking coil 'b' due to the current in 'aT will be
established. Lenz's law states that a counterflux ~ab will oppose
~ba' thus creating an induced current in the direction b-b'. Thus,
a mutual impedance term establishes the effective induced voltage in
the opposing wire, and the circuit equations may be written as:
Va V 'a
v 'b
<3.12>
<3.13>
Severalwhere Zaa = Raa + jwLaa , and Zab = Ra b + jwLa b , etc.
references {31,32} show the computations for parallel cylindrical
wires. This involves the summation of partial self-inductance terms
due to internal and external flux linkages. For transmission lines
~~ + '10(4.' - --- ~T
:l~ClI-0.- «a ~
+ 0 t
Va. Vr;.,1
- .i- o -.-L
to..b : t htLI I
liLt- ---~I. -- ----------T-- tI
lH Ib Ib~
j,'
+ G I\IV' ) 0 +Vb Vb'---L. -.L -
Figure 3-2 Mutual coupling between two parallel wires
25
26
where length of line's' is much larger than radius r, the inductance per
unit length is:
Lis -7 -710 /2 + 2x10 {In(2s/r)-1} <3.14>
Since the GMR for cylindrical wires is Ds
therefore express self-inductance as:
O.779r {5}, ~e can
L -72x10 {In(2s/D )-l}s <3.15>
Likewise, mutual inductance M is determined from geometric mean
distance (D ) and is defined as:m
M -72xlO {In(2s/D )-l}m <3.16>
Although the above formulas for self and mutual inductances imply that
they are functions of line length s, we shall see that these terms
"cancel out" when equivalent expressions for self and mutual impedances
are derived by Carson's line equations.
27
Section 3.3 - Application of Carson's Line to the Derivation ofthe Per-Unit Length Impedance Matrix
The basis for describing wave propagation in overhead conduc-
tors with earth return was presented by J.R. Carson in 1926. Various
authors (11,30} have used Carson's line with earth return in trans-
mission line applications, such as zero-sequence impedance calculations
for fault analysis. Here, we will derive self and mutual impedance
terms using a rather heuristic approach in which earth return is used
in the circuit model. This involves the utilization of Carson's line
with earth return, as shown in Figure 3-3. An overhead wire of unit
length (denoted by length a-a') carries a conductor current I , anda
returns through the earth through a ficticious "ground conductor"
beneath the surface of the earth (denoted by length g-g'). Similar to
the method of images, which is commonly used in the computation of
sequence capacitances, the earth is assumed to extend infinitely with
uniform resistivity. The distance between the overhead conductor and
the ficticious "ground conductor" is denoted by D . This distanceag
is a function of earth resistivity p, and is adjusted so that the
calculated inductance is equal to that measured by test (30}. From
equations 3.12 and 3.13, we can represent Carson's line (Figure 3-3)
as:
28
I za. a aa I
.-.,.. a
+
rva
D REFag
)7
V !g= -I a J+
---... ~I Fictitious earth3 return conductor
I,. 1 UNIT ·1· .I
Figure 3-3 Carson's line with earth return
zag
[Va - va'l= 'Zaa
lVg-Vg'j
: -, II' -I
a J
<3.17>
29
Note that voltages V , V " V , and V ' are all referenced to ground.a a g g
Thus, we know that Vg
0, and V ' - V ' = o.a gSubtracting the two
equations enables us to solve for V :a
Va
(z + Z - 2z )1aa gg ag a Z Iaa a
where Zaa z + Z - 2zaa gg ag Z denotes the "total" selfaa
impedance of conductor "a" vi th earth return accounted for, whereas ..
lower case z denotes the self-impedance of conductor "a" withoutaa
earth return. Z can be regarded as the total self impedance, sinceaa
it contains an earth resistance term r. From equations 3.15 andg
3.16, we can express each component of total self-impedance Z as:aa
zaa
zgg
zag
r a + jOOk{ln(2s/Dsa)-1} Q/unit length
r + jwk{ln(2s/D )-1} Q/unit lengthg sg
jwk{ln(2s/D )-l} Q/unit lengthag
<3.18>
<3.19>
<3.20>
where D and D denote self GMD's of conductors "a" andsa sg "(J""o ,
respectively. Combining terms from Equations 3.18, 3.19. and 3.20 and
substituting into Equation 3.17, we obtain:
Zaa (R + R ) + jwkln(D /D )age sa <3.21>
30
where D is commonly defined as D = D 2/D {1l}. The parametere e ag sg
De is dependent upon both earth resistivity p and frequency!, and
is defined by:
De2160(p/j)1/2 <3.22>
An identical methodology is followed for deriving a system of three
phase conductors ~ith or vithout ground wires. The derivation is shown
in Appendix A. Given the physical geometries and conductor specifi-
cations, it is possible to theoretically calculate per-unit length
impedance parameters. Factors such as skin effect can be estimated by
a method also shown in Appendix A and also in {22}.
Section 3.4 - Derivation of the Admittance Matrix
Just as magnetic field effects are considered for studying
inductance, the distribution of the electric fields determine the
capacitance of a system of parallel conductors. The shunt admittance
matrix [Y], as mentioned earlier, consists of conductance and
capacitive reactance terms. However, the conductance term is usually
omitted because of its negligible contribution in most applications.
31
Unknown and often uncontrollable factors such as changes in atmos
pheric conditions, dirt, and corona contribute to leakage current
between conductors, which often make conductance impossible to
measure. Thus, we assume negligible conductance contribution at
at distribution voltages and consider only capacitance terms.
Capacitance between conductors is defined as charge per unit
of potential difference. It is dependent upon the size and spacing
of the conductors relative to each other and to an earth conducting
plane. Intuitively, we can visualize current, or the movement of
charge, to increase and decrease with the instantaneous value of the
alternating voltage impressed on the system. This is evident in
standing waves, where charging current flows even in the presence
of an open-circuit load, as we shall later investigate.
Recall from field theory that the potential difference
between tva points P1 and Pz external to a linear charge density q
(see Figure 3-4) is equal to the integral of the potential gradient E:
J E-dx J (q/2n£x)-dx <3.23>
Thus, by superposition, for an n-wire system carrying charge
densities qa' qb,···,qn' located above a ground plane, the
difference in potential between any two wires will be the sum of
component potentials due to each of n charged wires:
+q
-............
/f
II
I
/ //~ /
//
'",..( "
/'
32
Figure 3-4 Path of integration between two pointsexternal to a linear charge density
v .aJ
(1/2n8)(qa 1n{Daj/r} + qbln{Dbj/Dba} +
+ ••• + q In{D ./D })n nJ na
<3.24>
33
Section 3.5 - Effect of Earth on Capacitance
The presence of earth as a conducting medium must be account-
ed for when calculating capacitance. The assumption that the earth
is a perfect conductor of infinite extent in a horizontal plane will
enable us to understand the effects of a conducting earth on
capacitance calculations.
Consider a parallel two conductor system with earth return
as shown in Figure 3-5. The physical location of the conductors is
defined with respect to a coordinate system in which the earth plane
is used as the horizontal reference axis and the axis of symmetry of
the pole structure as a vertical reference. In charging the conductor,
the earth surface and conductor plane can be regarded as equipotential
surfaces, since the earth has a charge equal in magnitude to that of
the conductor but opposite in sign. Assuming the earth is of uniform
resistivity and infinite, its surface can be replaced by a ficticious
conductor of the same size and shape as the overhead conductor at a
di~tance equal to that of the overhead conductor to earth. This en-
tails that if the earth is removed and a charge equal and opposite to
that of the overhead conductor is placed on this ficticious conductor,
34
polestructure
conductor j
/
II
/!!
)II
d ..1J
conductor i
earth
I
IIIII
~image of
conductor i
\D.. \1J
\\
\
I
II
I
II
II
1image of
conductor j
1~
j
j~,-----=-~----
Figure 3-5 Effec~ of earth on capacitance - Method of images
35
then the plane midway between the two occupies the same position as
the equipotential surface. This ficticious conductor, having charge
equal and opposite to that of the overhead conductor is called the
image conductor.
Thus, since calculations involve only lengths between con-
ductors and their respective image conductors, the admittance matrix
is dependent only upon the physical geometry of the conductors
relative to earth.
From equation 3.22, a system consisting of four overhead
conductors can be equivalently expressed in matrix form as:
V B Bab B B
\;' qa 1a aa ac an !
Vb 1/2Jt£ Bba B
bb Bbc BbnI
:: I<3.25>
V B Bcb B Bcdc ca cc
Vi
B Bnb B Bqn Jn na nc nn
where the elements of matrix [B] are determined by the geometry of the
conductors as follows from Figure 3.5:
B..1J
1n(D . . /d .. }IJ 1J <3.26>
d d i t b . th d' th dij IS ance etween 1 an J con uctor for (i~j)
radius of i t h conductor for (i:j)
D.. distance between the jth conductor and the image of the ith1J
36
conductor.
Equation 3.25 can also be expressed in a form similar to equation 3.23:
vhere
[V] = 1/2n£[B] [1']
~ = [qa qb qc qn]T
<3.27>
We can arbitrarily define a charge coefficient matrix [P] as:
[P] = (1/2Jt€)[B]
where [P] = [C]-I, since q
[V] = [P] ['f]
cV and
<3.28>
<3.29>
To obtain the total per-unit length admittance matrix [Y], we apply the
following relations: From Ohm's law, the current vector [I] is:
[ I ] [Y] [V] <3.30>
Current is also defined as the derivative of charge vith respect to
time. Thus,
[ I ] d['f]/dt j -r '¥] <3.31>
From 3.29 and 3.31 ~e obtain the relation:
[ I ]
Thus, from 3.30:
jw[C] [V] <3.32>
37
[Y] [1] [V]-l . p-1JW-1
jw(2nEB ) <3.33>
The complete calculation of the per-unit length admittance matrix for
the actual test network is given in Appendix A for a four conductor
system (Three phase wires, one neutral) with earth return.
Section 3.6 - Experimental Determination of Hulticonductor LineParameters
A measurement technique for determining the per-unit length
parameters of a multiconductor network is presented in {21}. It is
formulated in terms of measurable short and open-circuit line
impedances at a particular frequency. From transmission line theory
for single-phase lines, the short-circuit input impedance Z andsc
open-circuit input impedance Z can be expressed in terms of attenoc
uation constant a, phase constant a, characteristic impedance Z , ando
line length 1 as:
zsc
zoc
Z t anht « + jS)lo
Z co tht « + jf3)lo
<3.34>
<3.35>
Multiplying equations 3.34 and 3.35 and solving for Z we obtain:a
38
Zo
{Z Z }1/2sc oc <3.36>
The expression for propagation constant in terms of Z and Z is:sc oc
y = ex + jS = (arctanh{Z IZ }1/2)/1sc oc <3.37>
Similarly, for a multiconductor system consisting of{N' conductors
(excluding ground wire), the resulting expressions for the short-
circuit and open-circuit input impedance matrices are:
[tanh(fl)][Z ]o
Inv[tanh(rl)]-[Z]o
Solving for Z we obtain:o
<3.38>
<3.39>
[Z ]o
{[ Z ]Inv[Z ]}-1/2 [2 ]sc oc sc
<3.40>
The propagation matrix [f] is expressed as:
[ f]1/')
{arctanh([Z ]Inv[Z ]) ~}/lsc oc
<3.41>
39
Having obtained [Z ] and [f] from equations 3.40 and 3.41, we cano
solve for [Z] and [Y] by the following relations {14):
[Z]
[ Y]
[f][Z]o
Inv[Z ]·[f]a
<3.42>
<3.43>
Thus, from the knowledge of the input impedance matrices for open-
circuit and short-circuit load conditions, the multiconductor line
parameters [Z] and [Y] can be obtained.
A technique for measuring these input impedance matrices
is presented in {21}, where ratios of voltage to current are measured
by effectively "isolating" self and mutual impedance and admittance
terms at a chosen frequency. The reader is referred to {21} for a
detailed explanation of this measurement procedure and the results
for a four-conductor line.
This experimental method of calculating line parameters could
not be implemented in our tests because the network was not conducive
to the measurement of open-circuit and short-circuit input impedances.
Thus, the validity of the mathematical model can only be determined
by the theoretical calculations presented in the previous sections.
40
Chapter 4 - Network and Test Description
In order to gain an understanding of carrier signal propa
gation on multiconductor distribution lines, several tests ~ere
performed on actual de-energized distribution networks at Carolina
Power and Light's Distribution Automation Test Facility. The test
facility offers a controlled environment in which propagation
measurements of voltage and current as a function of distance can
be performed. The 23 kV test facility, which was constructed to
Carolina Power and Light's distribution engineering standards, is
composed of spans of single-phase and three-phase sections of line.
These spans can be configured into various lengths of three-phase
and single-phase "netTHorks" by controlling oil break switches located
at various "switching poles". A more detailed description of the
test facility is presented in {6}.
Section 4.1 - Test Set-Up and Measurements
Propagation measurements were performed by injecting a 25 kHz
carrier signal at the sending end of a multiconductor network and
measuring voltage and current magnitudes at approximately equidistant
intervals along the network. The 25 kHz frequency was also used in
previous OLe experiments {3,~} in order to "visualize" nodes and
antinodes in standing ~ave patterns. These signal nulls occur at
quarter-wavelengths, ~hich correspond to approximately 9000 feet, or
41
1.77 miles at 25 kHz. Due to the physical limitations of the network
(approximately four miles each of three-phase and single-phase sections
of line), about one-half of a wavelength can be plotted for either
the three-phase or single-phase spans, which should yield sufficient
information for standing vave analysis.
Although the test facility accommodates both single-phase
and three-phase sections of overhead and underground distribution
lines, only the overhead conductors were utilized. These conductors
are classified as #2 AVG aluminum, and are spaced according to distri
bution standards (Refer to Figures A-2 and A-3 as shown in Appendix
A). The neutral conductor follows an intermittent grounding scheme,
where grounding occurs at each pole (located about 300 feet apart).
Figure 4-1 shows the experimental set-up for carrier injection
at the sending end of the network, where a function generator (HP
3311A) in series with a power amplifier (HP 467A) is used to generate
the 25 kHz sinusoidal carrier signal. The signal is coupled to the
phase (denoted by A,B,C) and neutral (N) conductors by twisted pair
16 gauge wires, which are clamped on to each of the overhead
conductors. A connection box (banana-type connectors) vas built to
select either single-phase injection on conductor A, or three-phase
injection on all phases A,B, and C. The peak-to-peak source voltage
used for the propagations tests was SO volts- At the receiving end,
16 guage twisted pair wire was also used for any loads which were to
be attached from phase to neutral. No loads were attached from phase
HP 3311A FunctionGenerator
HP 467A PoverAmplifier
R
R lOkQ
42
FrequencyCounter (25 KHz)
/
TO PHASE B
Figure 4-1
ConnectionBox
Equipment test set-up for carrier injection
43
to phase.
The actual voltage and current measurements were performed in
a bucket truck using a battery-operated portable dual-channel oscillo
scope (Tektronix 305 DHM). The voltage probe was modified by attaching
large alligator-type clips to signal and ground leads of a coaxial
cable, enabling voltage measurement from phase to phase and phase to
neutral, simply by clamping on to the wires. The current measurements
were made using a combination of a Fluke current transformer (clamp-on
with 1000:1 turns ratio) and a Tektronix current probe (model P6021).
In order to compensate for the 1000:1 decrease in current, the output
current of the Fluke was increased by placing 500 turns (type 40 A~G
wire) of the Fluke's secondary to the primary side of the Tektronix
current probe. Thus, the calibrated net transformer ratio was approx
imately 1.75:1 at 25 kHz. Currents were measured on phases A,B, and
C only, since Hemminger {3} showed that almost no current flowed on
the neutral conductor more than two pole spans (about 600 feet) from
the source.
Section 4.2 - Network Configurations and Boundary Conditions
Multiconductor propagation measurements of voltage and cur
rent were performed on three different networks. These net~orks are
shovn in Figures 4-2, 4-3 and 4-4. "Network #1", as shown in Figure
4-2, consists of a homogeneous span of parallel conductors A,B,C,
and N. The actual physical geometry of the conductors in this network
~_._-------- 3.59 miles--------~
Ia O~--~--------------------..,O
b o--------~...--..---..-;---....-----=o
C O,..,~~-...I:.:III:I:lI----...------------_rj
n(")c~----------------------...o
44
source
Figure 4-2
load
TABLE 4-1 Boundary conditions for tests conducted onNetwork #1
45
# OF PHASES SOURCE LOADTEST # INJECTED CONDITIONS CONDITIONS
ZSA SQ ZLA OPEN1
la 2SB OPEN ZLB OPEN(PHASE A)
ZSC OPEN ZLC OPEN
ZSA = 5Q ZLA OPEN
Ib 3 2SB
5Q ZLB OPEN
ZSC 5Q ZLC OPEN
ZSA SQ ZLA OPEN1
lc ZSB lOKQ ZLB +jZol(PHASE A)
ZSC lOKQ ZLC -jZ02
ZSA 52 ZLA OPEN
Id 3 ZSB SQ ZLB +jZol
ZSC SQ ZLC -jZ02
201 132 + j367 ohms '7 -j395 ohmsLJ02
46
corresponds to Figure 4-2, where the parallel conductors assume a
vertical geometry with phase A on top, B in the middle, and C on the
bottom, the closest to neutral conductor N. The total length of the
network is approximately 3.59 miles, or 18,962 feet. In a sense, the
network is symmetrical in that all conductors (A,B,C,N) span the same
total length, (i.e. there are no discontinuities that result from
unequal line lengths). Thus, this particular network is easily
modeled as one continuous, homogeneous section of line. In actuality,
this is not the case due to variations in conductor geometry result-
ing from transpositions, differing pole heights, etc. However, for
simplicity, we shall model this "symmetric" network as one continuous
section of line, characterized by the same set of distributed per-
unit values throughout.
A series of tests was made on Network #1 by applying sev-
eral combinations of boundary conditions at the source and load.
At the source, current was injected on either: (i) phase A only
(single phase injection), or (ii) on all phases A,B, and C (three-
phase injection). On the receiving end, load conditions were applied
as follows: (i) all phases A,B,C open, or (ii) phase A open, phase B
terminated in +jZ , and phase C terminated in -jZ (loads connect-o 0
ed between phase and neutral). No phase to phase source or load
terminations were applied. The combinations of two source and two
load conditions resulted in a series of four independent tests
conducted on Network #1. They are listed in Table 4-1, where ZLA'
47
for example denotes the load impedance connected between phase A and
neutral. Likewise, ZSA denotes the source impedance connection between
A and N. The term "open" implies an open-circuited load.
Network #2, as shovn in Figure 4-3, is very similar to Network
#1 with the exception of phase A, which extends about 1.18 miles
further, spanning a total length of 4.77 miles. The lengths of Band
C are unchanged at 3.59 miles. The extension of phase A causes the
network to be asymmetric. Hence, a discontinuity exists in the net
work, for which two sets of line parameters must be employed to the
multiconductor solution equations presented in Chapter 2. Some
interesting coupling phenomenon result from this discontinuity, as
we shall see in the next chapter. The source and load conditions
for this network and the corresponding test cases are summarized in
Table 4-2. Note that the load conditions change only for phase A.
Network #3, shown in Figure 4-4, adds another discontinuity
by effectively reducing the length of phase conductor C by one-half.
Here, we are interested in the effect of discontinuities (which result
from an open-circuit) on the propagation in each of the phase conduc
tors. The three-phase netvork is reduced to a two-phase network after
propagating 1.82 miles, and is then further reduced to a single-phase
network after traveling 3.59 miles. Because the test facility is
composed entirely of single and three-phase distribution lines, a
"true" tl,rJo-phase system W'as not physically configurable. However, it
was p0ssible to simulate a two-phase system by "opening" the span of
line from c~ to c,. t-J.S dena ted by the do t ted line in Figure 4-4, the
48
4.77 miles
!,
"3.59 miles
b o~-------------------o
I
Ic o~--------------------o,
n 0-----------------------------.......0i .. source load
I
~I
Figure 4-3 Network #2
TABLE 4-2 Boundary conditions for tests conducted onNetwork #2
49
# OF PHASES SOURCE LOADTEST # INJECTED CONDITIONS CONDITIONS
ZSA = SQ ZLA SHORT1
2a ZSB OPEN ZLB OPEN(PHASE A)
ZSC OPEN ZLC OPEN
ZSA = SQ ZLA OPEN1
2b ZSB OPEN ZLB OPEN(PHASE A)
ZSC OPEN ZLC OPEN
ZSA SQ ZLA SHORT
2c 3 ZSB 5Q ZLB OPEN
ZSC SQ ZLC OPEN
ZSA 5Q ZLA OPEN
2d 3 ZSB 52 ZLB OPEN
ZSC SQ ZLC OPEN
50
-------- 4.77 miles
I
~3.59 miles~._------_.
b O~-------------------O
~ 1.82 miles ~I
C c...--------~o--
n c,·~-------------~-------~---.--.-(O!
load--~
Figure 4-4 NetYlork #3
TABLE 4-3 Boundary conditions for tests conducted onNetwork #3
51
# OF PHASES SOURCE LOADTEST # INJECTED CONDITIONS CONDITIONS
ZSA SQ ZLA SHORT1
3a ZSB lOKQ ZLB OPEN(PHASE A)
ZSC lOKQ ZLC OPEN
ZSA = 5Q ZLA = OPEN1
3b ZSB lOKQ ZLB OPEN(PHASE A)
ZSC lOKQ . ZLC OPEN
ZSA = SQ ZLA 4101
3c ZSB lOKQ ZLB OPEN(PHASE A)
ZSC lOKQ ZLC OPEN
ZSA = 5Q ZLA SHORT
3d ')
ZSB SQ ZLB OPEN..J
ZSC SQ ZLC OPEN
ZSA SQ ZLA OPEN
3e j
ZSB SQ '7 OPEN...) L1LB
ZSC SQ ZLC OPEN
ZSA SQ ZLA 410Q
3f 3 ZSB SQ ZLB OPEN
ZSC SQ ZLC OPEN
52
wire physically exists with both ends open, and can therefore be
considered as an equipotential neutral conductor. Table 4-3 sum
marizes the various source and load boundary conditions that were
applied to Network #3. A total of six tests were conducted. Note
again that loads ZLB and ZLC at their receiving end remains open in
all test cases. The 10 kilohm source impedance was used to simulate
the impedance of a substation transformer {3}.
The results of all propagation tests are tabulated and
plotted in Appendix C. Because of the extensive amount of experi
mental data, a few tests, particularly those which were conducive
to simulation, will be analyzed and commented upon in greater detail
to follow.
53
Chapter 5 - Test Results and Analysis
The experimental and theoretical results of several test
cases are analyzed in the sections to follow. Because of the ex-
tensive amount of data from the various tests described in the last
chapter, only a few of the more significant results will be discussed.
Computer simulation results will be compared to empirical data to
check the validity of the mathematical model.
Because the solution process to the multiconductor problem
is so mathematically complex, it lends very little insight to the
actual physical occurrences along the distribution network. Conse-
quently, an analytical approach will be taken which involves the
application of transmission line fundamentals. The Smith chart, for
example, provides an accurate solution and a physical interpretation
of what happens on the line.
Section 5.1 - Experimental and Theoretical Results of Test lIb(Bundled Conductor)
As described in Table 4-1, test case #lb was conducted on
a symmetric netvork, ~here each conductor spans a distance of approx-
imately 3.59 miles. For this particular test, three phases (A,B,C)
were injected with a 25 kHz sinusoid. At the load, all phases were
terminated in an open circuit. The standing wave patterns for voltage
and current versus distance from the source are shown in Figure 5-1.
54
A total of ten locations were chosen as measurement points. A spline
function was used in plotting the subsequent standing wave patterns
for each phase. Note that the voltage minimum and current maximum
both occur approximately at the midpoint of the network, a quarter
wavelength from the load. This is because the network length is very
close to an electrical half-wavelength at 25 kHz.
For a lossless line, a half-wavelength (\12) at 25 kHz
corresponds to a distance of 3.73 miles. From previous propagation
tests conducted by Hemminger {3}, the velocity of propagation of an
unloaded distribution line was measured to be 95% of the speed of
light in air, which indicates that the line is virtually loss less and
conducive to TEM, or quasi TEM modes of propagation. Thus, an electri
cal half-vavelength adjusted for 95% speed of propagation is actually
about 3.54 miles, which is very close to the total netvork length of
3.59 miles. The symmetry of the standing wave patterns indicates that
the network is indeed very close to a half-wavelength.
Figure 5-1 shows the voltage standing wave pattern for phases
A, B, and C. Because of the uniformity of the network (all phases
being of equal length and all conductors uniform), we would anti
cipate that all phases would exhibit identical standing ~ave patterns.
In essence, this particular three-phase system can be considered as a
single-phase bundled conductor, in which the three phases constitute
the bundle. As discussed in Chapter 3, a system of parallel wires
carrying unequal currents will mutually induce voltage on neighboring
wires due to nonzero flux linkage terms. A flux ~ab' links phase
PHASE TO fo'[UTRAl VOltAGES (T£ST,es)
SIMUlATED PHASE TO NEUTRAL VOLTAGES (TEST liB)... V~lt.:.~ (va
'0
to
.0
,oL 10,S
ao
10
~~
..........
I"
...
It .•
.,. .
.... l." I." .... I.... ..M ,... 1.&1
o.04 0 40 0 ' .0 1. 20 I. 60 2 . 00 J . It I . " I. 19 J. Jt
DUrAlIIICI n. SOuUI (MIL'S)
Olst·JnC. fr'Olfo ~~ (",tlea)
PHASE CuRMNTS (TEST'I8)
'ea • -..----I • ---- ..., .u
....
r:'".'
;/
.'!,-
i
\\
\
\
\.~\\ \...... \~ ...-\
.110
.'70
.,Iot
..Jt
SIWULATED PHASE CURREKTS(TEST liB)C\S·('~t lA).'M
.,...
....
6 ......J
1....-,
.....-.
I~ .I""--lS' ~... ·I
-,~
'.- t-+' 0 ..···0I ~,_.-A
o
", -A- ' ...
""~-'''-..,.'/,,.>: D-<,~_..
,,/ / '0,.'/,.,D»<>
./a"
"
:~y'
70
'0
60
10
40
20
'0
10
10
0.00 0.40 0.10 r. 20 1.60 Z. 00 2. J, 2. 7f J. l' 3. ,.
DISUNCI "'OM SOURel (HILlS)
.... .... ..at .... .... ..... •.•• J.lI 1.W
[Iht·:...:~ fr.jfA ~.:. (",1l~&)
lJ1V1
rtlU~e 5-1 I Me••ur.d volt... and cu~r.nt v•• di.tanc. (t•• t lib) rilUre 5-2 I Theoretical volta,. and current va. d18tance (t•• t 11b)
56
'~ via current in phase \a~ producing an induced voltage across the
mutual inductance term Zab. In the case of a bundled conductor, or
any balanced three-phase circuit ~here IIJ can be considered as the
superposition of equal currents I , I b, and I , the flux linkagesa c
between phases will essentially be zero because the magnetic field
intensity H is zero. Thus, no mutual induction takes place. This is
evident in Figure 5-1, where all phase voltages are equal. The phase
to phase voltages were also measured to be zero, which justifies the
absence of mutual coupling effects on a uniform network.
The phase currents, as expected, are not exactly equal at all
points along the network. This can be attributed to the physical
geometry of the conductors. Although the network is considered to
be symmetric, other factors, such as unequal spacings between phases,
variations in conductor heights above ground (due to terrain), and
skin effect can cause unequal current division between homogeneous
conductors. Skin effect results in a decrease in current density
toward the center of the conductor. This inequality in current
density is caused by a longitudinal element near the center of the
conductor being surrounded by more magnetic lines of force, hence
reducing the net driving emf at the center element. Thus, virtually
all of the current is concentrated near the surface of the conductor.
In addition, the netvork experiences changes in conductor geometry
in the form of "tvists". These tvists involve the transposition of
phase conductors from a vertical to delta configuration. The afore-
57
mentioned nonuniformities have some effect on voltage and current
distribution, and is the probable cause for the slightly different
standing wave patterns between the phase conductors shown in Figure
5-1.
Since this particular network can be treated as a single
bundled conductor, certain transmission line parameters such as
characteristic impedance and propagation constant can be evaluated
from empirical data. For single phase lines, the characteristic
impedance is obtained by the following relation:
zo
(2 Z )1/2sc oc <5.1>
where Z represents the short-circuit input impedance, and 2sc oc
the open-circuit input impedance. From Figure 5-1, the impedances
for short-circuit and open-circuit load conditions can be obtained
by assuming that the netvork length is equal to an electrical half-
wavelength. This assumption is valid due to the symmetry of the
standing wave patterns, as discussed before. Since voltage and cur-
rent minimas and ~aximas repeat every half-wavelength, the input
impedance for this symmetrical network can be represented by:
Iv II Ioc oc
12 I 375 ohmso
55 volts/12.3 rna
7 volts/222.2 rna
4.47 kohms
31.5 ohms
58
The open circuit input impedance is obtained at the end of the line,
where V = V ,and current I = I. = I + Ib
+ I. Like-oc max oc mIn a c
wise, the input impedance looks like a short circuit at the midpoint
of the network where V = V . , and I = Isc mIn sc max Note that
the currents for both short-circuit and open-circuit loads are the
superposition of individual currents flowing in each of the phase
conductors. Hence, we see that the input impedance is dependent upon
line length. Because this particular network is one-half wavelength
long, the input impedance always "looks" like the load impedance. If,
however, the network length were ~ quarter-wavelength, then the input
impedance for an open-circuit load would look like a short.
The empirical value for characteristic impedance 2 ofo
375 ohms is consistent with that derived theoretically for a bundled
conductor. The theoretical value for Z of 372 ohms is derivedo
in Appendix B, and is based on methods described in Chapter 3 for
computing per-unit length matrices [Z] and [Y]. From a previous
experiment conducted on an unloaded single phase network {3}, 20
was measured to be in the neighborhood of 450 ohms. The decrease in
Z for a bundled conductor is expected since the equivalent geometo
ric mean radius is greater than that of a single conductor. Hence,
the inductance is decreased with the addition of conductors, while
the capacitance to ground is increased, which results in a net de-
crease in the magnitude of the characteristic impedance.
Since the relative positions of voltage minima and current
59
maxima for each phase conductor all occur at the same location
with respect to the load (midpoint of the network), we may deduce
that the propagation constant (y = a + jS) for each conductor
is the same. Furthermore, the propagation matrix r is diagonal.
Recall from Chapter 2 that the propagation matrix contains elements
(eigenvalues) that define the modes of propagation. It is evident
from Figure 5-1 that the imaginary components <a , a, a ) of thea ~b c
propagation constant are equal.
A simulation program based on the mathematical model presented
in Chapter 2 was written by M. Riddle {33). Several test cases were
simulated for comparison vith empirical results. Figure 5-2 shows the
theoretical standing wave patterns generated for the symmetric network
denoted by test #lb. The per-unit length parameters, namely the [Z]
and [Y] matrices were derived by the methodology described in Chapter
3. These parameters are derived in Appendix A for the test network
used in making the propagation measurements. In comparing the measured
data (Figure 5-1) to that of the theoretical (Figure 5-2), we see
noticeable differences, the most prominent being the lack of symmetry
in Figure 5-2. Another discrepancy between the two plots exists in
the phase voltages being unequal in Figure 5-2, ~hereas the measured
voltage standing wave patterns in Figure 5-1 are identical.
There are several possible explanations for these apparent
differences. The first is that the theoretically derived per-unit
length parameters do not accurately represent the actual test network
parameters. In our derivation, we assume a uniform network. In
60
reality, this is not the case. Many approximations were made due to
uncertainties in the network parameters. For example, the physical
geometry of the conductors changes throughout the netVlork from "verti
cal" to "delta" configurations, as mentioned previously. These
conductor t~ists, which cannot be regarded as a true transposition
in a strict sense, have an effect similar to that of a complete
transposition cycle in that the rotating of conductors effectively
reduces, or "cancels out" mutual impedance effects. In the derivation
of the per-unit length impedance matrix, these twists were not taken
into account. Other uncertainties, such as conductor heights are
inherent in varying terrain levels. In addition, non-uniform conductor
spacings, and earth resistivity also have an effect on signal propaga
tion. These "non-uniformities" in the network could affect the
calculated parameters [Z] and [Y], but their sensitivity to these
non-uniformities is not yet knoVln. However, based on several "trial
and error" variations of line length, the experimental standing wave
forms could be matched. In particular. the network seemed especially
sensitive to line lengths corresponding to multiples of quarter-
wavelengths, vhich is the case in test #lb.
Another possible source of error could be attributed to lack
of precision in the numerical solution process. Since the computer
simulation involves matrix functions (multiplication, diagonalization,
inversions, etc.) vhich are used iteratively, a very slight error in
precision at the outset could invariably be magnified. thus "blowring
61
up" the solution. If a network is sensitive to certain parameters,
such as line length in this case, a slight precision error could have
a significant effect in prediction accuracy.
Section 5.2 - Experimental and Theoretical Results of Test #lc
We shall now investigate loading effects on a network ~ith
equal line lengths, excited by injection on a single conductor (Test
#lc). The net~ork description for test #lc is described in Section
4.2, where we recall that at the receiving end, phase A is left open,
while phase B is terminated in an inductive load (+jZ ), and phaseo
C terminated in a capacitive load (-jZ). At the sending end, onlyo
phase A is injected, while phases Band C are terminated by a lOkQ
resistor to ground (Three-phase injection on this network is analyzed
in the next section). Measured and theoretical voltage and current
profiles are shown in Figures 5-3 and 5-4, respectively. From Figure
5-3, we see the same "symmetric" voltage and current standing wave
pattern on phase A as in test #lb (Figure 5-1). The uninjected phases
(B and C) give indication of coupling as shown by the induced voltage
and current patterns. The smaller magnitudes of voltage and current
along the uninjected phases indicate that this coupling is weak.
Thus, the loads on phases Band C have very little effect or signifi-
cancel We viII later see that in test #ld, where all phases are
injected, the loads on phases Band C have a significant mutual
SIUUUTED PIIASE TO NEUTRAL VOLTAGES (TE~' ,IC)PHASE TONEUTRAl VOlTACES (nSf' 'e) ,t.'V-=,lt~ (Vt
.:~;---=----::-==::-==~.:-===':=---=:-~-=~ ~ -.~-=--.-:---=~~iii iii , , r
0.00 0 .•0 0.'0 1.20 1.'0 1.00 2.1t 2.19 1.1' I."
DI IT ANCI '''OM tOUl'C[ (MillS t
J ...
'.... --_ ..'a - --_.-
, -..
.... .... I." I.... I." I.... l."
.r>
.~".:.' .......<:..'.: -'~:':':"""'-=:'::':-:':-:'"
'I 1 I I I I' ,•
~...
n .•
4'.'
".0
01 stO"Ce f,..O#A .~c. (al1.-.,)
"..
SHIULATED PHASE CURREJfTS (TEST ,Ie).HOCUt"r-er.t CA)
....
.ns
.aso,
~i 1 1+I' _;1. 1 ~JZ'I--- J.S! .... --.j I
~ ~ _ T
,k"'---+
/
'M- +-+'a.. 0··..·0
'ea - b-._o~
/
/
Pt~ CURM:NTS(TaT "C)
110
110
100
'0
so
10
.0
10yoL1, 20
1-0.us
DISUHCI '''0'' SOWtC! (MllfS)
i , 1 r ~ T -~ - ,- ~ - Til
0.00 O. 40 0 . 10 1 . 20 1. '0 2.00 2. 3f Z . 7t J. if 1. "[11.t-:.-..:. frOM .~ce I.t l~.'
...........__.._..-..-.~=..:-~.----:.:.. -:.-...._..... '.il .... ,... .... I.'. J." I.".4M
.......,....-,
'0..,.-'....;.
. 0 -:-:-:-:"_: 0" 0, .a' - - ..0·-.... <,
,.' , \",' / .', U
.r~~'··- " ."> / ".... '0 ' , ...- -'
20
70
~o
60
I'»
10
10
0"N
rl~re ~-) I "'••ul'ed voltal. and current v•• diat.nee (t.at lie) rleur • 5-4 J Theoretical volt.,. and current V8. di8tance (t.at 11e)
63
effect on propagation.
As before, the theoretical plots in Figure 5-4 are slightly
different than the measured in that the line lengths for the simulated
plots appear to be longer. This same effect can be noted in the
previously discussed simulation plots in Figure 5-2. One observation
that can be made in both simulated and measured plots is that the non-
driven phase conductors (B,e) exhibit some coupling, but this coupling
is considered to be "wreak".
Section 5.3 - Effects of Capacitive and Inductive Loading onPropagation (Decoupled Case Studies)
To get an intuitive feel for loading effects on multiconductor
signal propagation, we shall apply the Smith chart, which is shown in
Figure 5-5. A great deal of information can be extracted from the
Smith chart such as VSVR, voltage reflection coefficient, and input
impedance. For this particular test case, it will be useful to
consider the location of minimum and maximum impedance ~ith respect
to the load.
Consider the load attached to phase B (ZLB
normalized impedance is denoted by lower case zLB:
+jZ /Z = +jl.0o 0
+jZ). Theo
For simplicity, ~e will assume that the reactive component of the load
64
impedance is exactly equal to the characteristic impedance Z (Theo
actual measured value for the inductive load was 132 + j367 ohms).
Since a full rev·olu t ion around the Smi th char t corresponds to one-half
wavelength (Al2), a half-wavelength line with an open-circuit load
such as conductor A corresponds to a point located on the real axis
where the normalized resistance component (R/Z ) approaches positivea
infinity. The end point for this open-circuit load on phase A is
denoted by "a" in Figure 5-5. The inductive load has a normalized
impedance of zLB = 1.0 and is located at point "b". This corresponds
to one and a quarter revolutions around the Smith chart, or .625A.
On the other hand, a capacitive load with a normalized capacitive
reactance component equal to 1.0 is located at point "c". Note that
as we move around the Smith chart from the load to the generator in
a clockwise direction, we intersect two possible values for which
the normalized impedance is real. The maximum value is on the positive
real axis and has the same value as the VSVR. The minimum value is
on the negative real axis and corresponds to normalized impedance
values less than 1. For the inductive load (point b), as we move
clockwise toward the generator, we reach a voltage maximum after
tra'leling O.125A, or approximately 0.85 miles. After traveling
another quarter ~avelength we reach the voltage minimum.
For the capacitive load (point c), a voltage minimum is first
reached after traveling O.125A and a successive maximum after another
half-wavelength. These waveforms are plotted concurrently in Figure
65
IMPEDANCE OR AOMITTANCE COOROINATES
',' ':.'i I '-6 ',~' I
'. '~ .e I,.,.•'. I
........Ly IC.&\.O NA6MCTI..
"'" .....~-.. ..r\. Q . . , .I. .. .
~ ~, ., '1 '. ...
I, ,~"
,~ ': .' "
.,I 4' "t.·'c·
Figure 5-5 Smith chart - effects of loading on line length
66
5-6 in order to visualize the "shifting effects" of different loads.
\.Ie see that the capacitive load "shifts" the unloaded line to the right,
while the inductive load "shifts" it to the left.
These standing wave patterns shown in Figure 5-6 were generated
using theoretical R,L,C, and G parameters that were obtained from the
actual test network. The parameters were calculated from conductor
geometries, as described in Chapter 3. These "theoretical decoupled"
plots serve as a basis for understanding signal propagation in multicon
ductor networks, where coupling effects alter propagation in each of the
phases. Since these waveforms in Figure 5-6 do not exhibit coupling
effects, the positions of the signal minimas and maximas are a function
of conductor geometry. Hence, we will refer to these standing wave
patterns shoYln in Figure 5-6 as "decoupled".
Section 5.4 - Experimental and Theoretical Results of Test ild
In Figure 5-7, the actual measured voltage and current
standing wave patterns are shown for this test case. At first glance,
the experimental results appear to be notably different than the
theoretical decoupled patterns descr~bed earlier in Figure 5-6. ~e
would not expect these graphs to be the same, since the actual
measurements were performed on multiple "coupled" distribution lines.
Hovever, there are some notable similarities bet~een the decoupled
simulation plots and the actual data. First, we see that the relative
I
iJ'.I~-- 15' .a•• ----.....~
I .
67
'I'
THEORETICAL DECOlJPLED VOLTAGES (TEST lID)ve l t~l~ (V)
to\) ..
'10.0
10.0
"~o.o
~o.o.:
"
.-~ .
..0.0
"0.1
,
'\
.......«"
0+-_-+-_-+-_-...---+--......-_+---........--......---+o ."04 .800 1.2t 1.~ 2.00 2.4t 2.'0 1.20 3.6t
~.o
to.O
10.0
DTstonee fr~ SOYrce (miles)
'.... --THEORETICAL DECOUPLED CURRENTS (TEST 11 D) 'bit· --_ ...
.l2O!C'-..t" r '!'f'".t
.1$3
.115 ... ",
lA) v • -.-----CJI
.110.:
.r:"----- ..':.", ", ....~,._--
""...,i'
,,
,".'' ....-
Qtst~~ fr~ s~ce (miles)
rieur. 5-6 Deeoupled volt.._ and current VI. distanc. (test tid)
68
voltage minima for phases Band C seem to match the theoretical plots.
Phase C experiences a voltage minimum (and current maximum) at approxi
mately 2.8 miles from the source, or about 0.8 miles from the load.
This is consistent with our Smith chart analyses for determining mini
mum points in relation to their distance from the load, as described
in the previous section. By a similar comparison, phase B incurs a
voltage minimum at approximately one mile from the source, which is
also consistent with standing wave theory for a decoupled system.
The differences, however, are quite noticeable when comparing
maxima and their positions relatrve to minima and the load. For
instance, the measured voltage maximum for phase C has shifted to the
right to a point approximately 1.6 miles from the source. The distance
from the voltage maximum to voltage minimum (defined as a quarter
wavelength for a decoupled distribution line) has decreased from a
distance of about 1.8 miles to a distance of 1.2 miles due to this
shift. This indicates that the propagation constant has changed
due to the interaction of the other phases. Since our voltage and
current distributions are unequal betveen conductors as we move along
the network, mutual coupling effects begin to take place. This phy
sical interaction is supported by the mathematical model described by
equations 2.15 and 2.16, which suggests that each individual line
voltage is composed of the superposition of all component voltages,
vhich include induced voltages from other lines. Intuitively, if
we consider the entire system in a multidimensional sense ~here
69
reflection coefficients are actually reflection matrices, then we
can say that mutual reflections of forward and backward traveling
waves exist, as we shall further investigate in test case #2d. Fur-
thermore, one particular "mode" of propagation for a phase is actually
the sum of all modes represented by the system.
From Figure 5-7, it is evident that phase C is the dominant
mode in that its peak values of voltage and current (dictated by the
input impedance at the point of injection) are substantially higher
than phases Band C (This is discussed in detail in Section 5-5).
The most prominent change in the shape of the voltage standing wave
pattern occurs in phase A. Theoretically, the pattern should exhibit
symmetry about the midpoint of the network, with the endpoints at
about the same voltage. However, coupling effects from phases Band
C have shifted the voltage minimum of phase A to the right by about
0.6 miles. This effect can also be seen on the voltage standing wave
pattern of phase B, in that its voltage minimum, which normally ~ould
occur at a distance 2.8 miles from the source if decoupled, is less
pronounced and occurs nearer to the load. The voltage levels at the
endpoints of phase A, which ~ere the same for the symmetrical test
case #lb, are different due to these mutual coupling effects from
the other parallel conductors. It appears that the dominant propaga
tion mode exhibited by phase C reinforces the voltage levels on both
phases A and B as ~e move closer to the load.
The theoretical voI tage and current waveforms for test #ld
are shown in Figure 5-8. The per-unit length parameters used in
PHAS( JU N(lJntAL VOLfAG(\ (I(Sf "0)SIMUlATED PHASE TO MEUTRAL VOLTAGES (TEST liD)
Volt~.,. (V)m ..1'0
' ... a
, .M
, .----..-.
./
,I
.I
.>:--:/ ~ \.''/ ..- -_..- , ..,.;
.:........ . ~_ \
IN
Itt
III
.St
.n
': .'u _1__ . L • I I I 1 I
DI.t0nc4 fre- 8OU"'C. (.11•• )
H.'.
n .•
".,
SIMUlATED PIUSI CURRENTS (TEST 110)C\Sr~t "d.,..
.jl.
":'
! ~Jl.·I', I
1---1S' ~... .1I •
T
'•• +-+'.- 0···..0' •• 6-._.-6
I'i/
Ii,
i,,,
ya.. /_.n_ oO· i,
;,"\ ,
'6'"
'*M CUMINTI(TaT '10)
--........ .-. "
.It., '"' ,
.I' \
\~. \/ \
' ,' \
J \
\-/ .. 0 '.,. .SO11.- • •
0" , Ii' , i , i i
0.00 0.40 ,... LID 1." 1.00 I." J. " 1.1' •. tt
DUTAtlCI , .... aeu-CI ("ILl')
100
150V
•L,• 100
o,00 O. 40 I .•0 1 . 20 1. .0 I. 00 2 . ., 2. U J. l' J . "
DIU.IICI rlOM IIUIU (MillS)
-."
_,__._f#
.., .
iI
.'. -~.__ -..
I 1 I I ,.... .... • .•, I.N .... a.~ .... I.)' J.to
.......,..
" • I 1 I. ..
Olat·:.-r:o 'rOttl ~c. 1",11•••
'.
\
//
.US
.l4"
.IM
.In
.IN
..,.
,.~....,
--0
Q
/
, , / '''---~.'.''''''~ ,
.I"" '-.,.
/---" i" I""'a. I
""\... /\ /-.... \ /'a 0._ \
. ..--........\ .................
400
'00
JOO
100
100
tOO
JOO
.a
-......Ja
rip... '-7 I .....ur.d volt... and current va. 'tilataDc. (teat ltd) 'Iaur. 5-1 I Theoretical volt... and curr.nt v•• dlatance (t•• t lId)
71
generating the plots are derived in Appendix A. In comparing the
measured data versus the theoretical, for the most part, the plots
shov good correlation, especially in the relative locations and
magnitudes of local minima and maxima. The shifting of the voltage
minimum in phase A (Figure 5-8) from the midpoint (1.8 miles from
the source) of the network to a point approximately 2.4 miles from
the source indicates that the program is capable of simulating the
coupling effects which cause apparent "shifts" and irregularities
in the waveforms. The minor differences betveen Figures 5-7 and
5-8 can be attributed to estimations and assumptions (due ·to netvork
uncertainties) used in calculating [Z] and [Y], vhich were discussed
earlier in Section 5.1. The accuracy of prediction should increase
if actual per-unit parameters can be obtained. However, the theo
retically derived parameters for the test network seem to be suf
ficient for simulation purposes.
72
Section 5.5 - Experimental and Theoretical Results ofTests #2c and ~2d
Tests #2c and #2d (see section 4.2) were performed on
an asymmetric network where the line length of phase A was longer
than phases Band C. The purpose of these tests was to study the
effects of network discontinuities resulting from different line
lengths on signal propagation. The two extremes for load conditions,
namely the short-circuit and open-circuit load, were applied at the
receiving end of phase A.
Figure 5-9 shows the theoretical short-circuit load patterns
on a single phase line. The voltage is zero at the load and at points
every half-wavelength from the load, while the current is a maximum at
the load and every half-wavelength thereof. The line length of phase
A is approximately 4.77 miles, which corresponds to 0.673\ at 25kHz
(assuming v = .95c, where c is the speed of light in air) {3}.p
Thus from Figure 5-9, we would expect the input impedance at the send-
ing end to look like a pure inductor at approximately .67\ from the
load. The measured voltage and current standing wave patterns are
shown in Figure 5-10. As expected, the voltage level on phase A is
close to zero at the load, while the current is at a maximum. The
voltage and current standing wave patterns on phases Band C (equal
in length) exhibit symmetry, similar to that of test #lb. Thus, all
phases conform to theory which can be applied to single phase trans-
~., 7~
------------'"""T
51T 91i21T
7rr 31T 51T ! ~ 0 ~d radians1T2 4 4 2 4 2 4
450 0 4050 3600 315:) 2700 2250 1800 90e 45(. 0 Degrees
5A 9;\ 7':\ 3" SA A 3A A A0A
4 8 8 4 8 2 8 4 8
73
Figure 5-9
+j
----1t--
z
\~ -j
I1 1 1
T T
~j~
~II
Short-circuit load patterns for a single phasetransmission line
I3.5' .... ---i-- 1.18 ..i.~
74
50.1K., ,1I
C'1
40 1\
---t;.
" .. -6'" c
l\
~ ~'1 ,/ ",,-1 \
\e ~
30 ~\
\ //1" /r' \"V \. \
0 \ \ ///
I
L I J
T ~ I ,~s ::..20 l I
II,
/\
\ /"\\ ./
.' ?10 1 I, A'"1 \,J....-
1 \
0- ~
0.00 0.40 0.79 1.19 1.S9 1.99 2.38 2.7S 3.18 3.58 3.97 4.37 4.77
OISTANCE FROM SOURCE (MILES)
90 ~
80
70
60
50lit
AAO
30I
2~ ~,f:.
~ /
~o ~
PHASE CURRENTS (TESTN2C)
/
/I
VLn • +-+
Vbn • O·····{J
Ven • 6---6
/~/
0.80 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.'8 3.97 .&.37 A.77
DISTAHCE rROM SOURCE (~ILES)
Figure 5-10 Measured voltage and current vs. distance (test t:c)
75
mission lines terminated in an open (phases Band C ) and short
circuit (phase A). Note also that the first phase A voltage maximum
(approximately one-quarter wavelength from the load) coincides with
the voltage maxima of phases Band C, located about 3.6 miles from
the source. Thus, the voltage and current standing wave patterns
appear to "track" one another for all three conductors. This, how-
ever does not occur when phase A is terminated in an open-circuit,
as in the case of test #2d.
The theoretical voltage and current standing wave patterns
for an open-circuit load along a single-phase transmission line is
shown in Figure 5-11. The open-circuit load, like the short-circuit
load acts as a circuit element (inductor, capacitor, or resonant
circuit) which varies with line length. For a lossless line, the
current at the load is zero, while the voltage is at a maximum, as
shown in Figure 5-11. At a network length of 4.77 miles, or approx-
imately .67A, the input impedance looks like a pure capacitor. The
resulting theoretical standing wave pattern is simulated in Figure
5-12. This pattern was generated by using calculated per-unit para-
meters of the actual network used in the test. From Figure 5-12,
a quarter wavelength equals about 1.8 miles (dlv - dlv. =max mIn
4.77 - 2.98).
25 kHz
This agrees with Hemminger's {3) conclusion that at
velocity of propagation vp
), t = 7.084 miles, A /4ac act
O.95c.
1.77 miles<5.2>
I v I
76
I v I
III
5" 919'2"
7r. 3;,' 5" 31' ! !2 4
1r 0 ~d radIans4 2 4 4 2 44500 40S e
360~ 3150 2700 2250 1800 1350 90° 45° 0 DegreesSX 9\
A 7>- 3A SA X 3A ). x4 8 8 4 . 8 2 8 4 8
0 Wavelengths
1
\1
\1
I,
+j
z
Figure 5-11 Open-circuit load patterns for a single phasetransmission line
I-I
/'/ ,
r~;~ "r--.-------....--..
G ~ 3,5''';
DECOUPLED PHASE AVOLTAGE (TEST #2D)
77
lO.O
10.0
\
\
I'I
//
//
/I
//
/
,j
DECOliPLED PHASE ACURRENT (TEST #2D)
.1751
I
:::]\\iI
r, ~~.·2l.
~.~:~·2~III
I').
.~!17 .7~
/~
/ -./ "
, \/ \
/ '
/ \I \
I \/ '.
/I
/
I I I
\
\\
\\
\
\
\
\
\\
\\
Clist·.·,:~ f(,:,m s,:'l.lr.:~ ("',:l~s)
Figure 5-12 Simulated phase A voltage and current vs. distance(test #2d)
78
where Aa c t is the actual wavelength adjusted for the 95% propa
gation factor.
Now consider test #2d, and the measured voltage and current
patterns shown in Figure 5-13. Note the distorted standing wave
patterns for phases Band C. Normally, we would expect to see a
symmetrical pattern as in tests #lb and #2c. This "distortion" is
the result of a reflected component of voltage or current from phase
A. This mutual coupling effect is more noticeable in this particular
test case because the phase angle relationships between waveforms are
not the same as in test case lb (Figure 5-1), where waveforms are
identical due to network uniformity, and test case #2c (Figure 5-10),
where the length of conductor A enables all waveforms to coincide.
The measured current pattern of phase A is altered from the theoreti
cal pattern shown in Figure 5-11 in that the minimum values are not
the same. The presence of current on the other conductors has the
effect of raising the value of the relative minimum located 1.6 miles
from the source.
From Figure 5-13, phase A can be considered the "dominant"
mode by its larger peak magnitudes of voltage and current. The
magnitudes of these peaks are largely determined by the input im
pedance at the point of injection. From Figures 5-9 and 5-11, which
show short-circuit and open-circuit load patterns as a function of
line length, we see that the input impedances can be represented by
equi~/alent circuit elements which vary as a function of line length.
..PtiAS( TOt~UI"Al \t'OllAiClS('(S' I~D)
.>:
SnWLAT~U I'HASE TO H~lITRAL VOLTAG~~ (T~~11~()'
I~O v.:.. 1 t..::.~~ tV)
10\
fQ
........\0
,...
/
' ... h __
, -_.' _..•ca
).~'.1•a.•~.1'I.tlI.M
'.. --
,". t(nl:. fr-t;>ta s.<x.rc.. (", 11•• )
•4~ ....
... .". 1.1' I.. I." '.D '.'1 1.11 J."-t '.M
SU'UUTED PHASE CURRENTS (TEST 12D)
'\.
"':;.;:;~.::::.;::.,.~
Cun' ter,t (A)
[Itat·:.-",. fr-Ola &'.A.i'C. (",11 •••
'I I I I I I I I•
.~
•
.1~O
.jU
.14'
.'"
J'l.'
to.'
e...o
4 ....-'
.......
I
I
II
1.18 ••• -1I
I3,5' .a. ---.!.-
I
I
~
e. ----. __ ....-
'•• t-+' ... o-...Q9•• 4--4
'*S( CUM04J1 (lOr '10)
, 0'-_
_ a ",,-~~.- ':...~-\' " ........~/ U~_ I
'o-:::-..QI
,---.../.A',/.....-Q~\
s-:" \'._......... <,
,,/ \ ----- ....... _.........'. .~. . --0- - -o-~
,
• ''r.....--ri~.....,,~.....,Ir-..... T"ir¥Y'''''''-..."..........."..............,'r-"'..-,',..............r'-'-...-r,......-T,
.0
.0
10
10
10
v•L .0,•
.... I... O. ,. I. 11 •.•t 1." J. It a." •. II I. II •. " •. n •. n_UtallCl , ... MuIU (MllII)
0.00 0.'0 '.J' •. ., I." •. " 1.'1 2." I." I." I." '.17 •. ,'
.lIf_. , ..... _. (MILII' I~,
I"...HO...IN1...N
I"U..1..
A "toJOtoto'010.0100', , i , i J , , , , , , ,
.11U~. '-1] • Me..u~" volt... an4 current v•••1.tanc. (te.t '24) 'JlUre '-14 I Theoretical volt... and current v•• dl.tanc. (t•• t 12d)
80
For example, the peak voltage for a half-wavelength open-circuited
line ~ould be less than that of a quarter-~avelength line, simply
because the point of injection for a quarter-wavelength line corres
ponds to a low impedance point (represented by a series L,C). Thus,
since the input impedance of phase A in test #2d is lower than that
of phases Band C, the voltage and current maximums reach higher
values, as shown in Figure 5-13. These maxima and minima are also
affected by coupling in the presence of nonuniform current distribu
tion in multiconductor systems. Not only are the magnitudes affected,
but also their positions relative to the load. These shifts in signal
nulls can be explained in terms of the propagation matrix and charac
teristic impedance matrix.
Instead of analyzing one line or one mode at a time, multiple
coupled transmission line phenomenon must be approached by consider
ing the entire system. The single-phase transmission line theory was
used to give us an intuitive feel of propagation in the absence of
coupling. This served as a basis for comparison vith measured results
and provided insight to the degree of coupling in various network con
figurations. Multiple coupled theory must be considered in terms of
vectors of forward and back~ard traveling vaves, and reflection
matrices. In general, equations for single-phase transmission theory
are directly applicable to multiple coupled lines. For example, if
a set of lines is terminated in a netvork ~here the load impedances
match the respectiTfe characteristic impeaances, then the vector of
81
incident voltages will experience no reflection and maximum power
transfer is established. However, in terms of propagation, one must
visualize reflections in a multidimensional sense. Recall that for
a three-phase system (with ground wire), the propagation matrix [f]
is 3x3 in dimension, as is the characteristic impedance matrix [2 ].o
If we assume a uniform set of lines with load impedances equal to
their respective characteristic impedances, we can say that the lines
are "self matched". To have no reflections in a multidimensional
system, the lines not only have to be self-matched but mutually
mat~hed as veIl. The mutual elements of the characteristic impedance
matrix are represented by the off-diagonal terms. Supposing that the
lines are self-matched but not mutually matched, then no self-
reflection will occur, but a mutual reflection will {24}.
Consider again the waveforms for test case #2d in Figure 5-13,
where xl denotes the transition point from a three-phase system to
a single-phase system. As the vectors of incident voltage and current
propagate from source to load, they effectively "see" a different
characteristic impedance when making the transition at distance xl
Likewise, the propagation matrix also changes at this point. As the
vector of potentials encounter a reflection matrix at xl' a compo
nent of phase A due to coupling on phases Band C gets reflected
back with waves Band C, vhile another component of phase A travels
to the load and gets reflected. The distortion on phases Band C is
a final result of continually varying portions of component voltages
traveling at different speeds. Because the lines are neither self nor
82
mutually matched, self and mutual reflections occur. In effect,
these self and mutually reflected components of for~ard and backward
traveling waves "reinforce" or "cancel" according to their respective
phase relationships, and often produce "distorted" patterns. Evidence
of these reflections and their effect on propagation constant can be
seen by noting the different lengths between voltage maxima and minima
of phase A in Figure 5-13. The distance from the load to the first
minimum as we travel towards the source is approximately 4.77 - 3.1
1.67 miles. From this point to the next relative maximimum spans a
distance of only 3.1 - 1.87
to the next minimum is 1.87
1.23 miles. Similarly, the distance
0.6 = 1.27 miles. This is consistent
vith our previous discussion of differing propagation and character-
istic matrices for single-phase and three-phase spans of line.
The theoretical results of test #2d are shown in Figure 5-14.
The per-unit length parameters are the same as before (Appendix A).
Since the simulation program {33} does not presently accommodate
discontinuities (branching, taps, etc.), the input impedance of phase
A for the extended portion (about 1.2 miles) was calculated from the
folloving equation:
z.In<5.3>
vhere fL
= -1 for an open circuit load. The line length of 3.89
miles was used to match the measured plot in Figure 5-13. In com-
83
paring theoretical to experimental plots of voltage and current versus
distance, we see good correlation. Again, the calculated per-unit
length values may be the cause for the slight differences in magnitudes
and positions of signal nulls.
84
Chapter 6 - Conclusions
A general mathematical model for multiconductor transmission
lines has been presented and specifically implemented for character
izing signal profiles along distribution lines used for carrier
networks. Propagation measurements were conducted on various distri
bution lines, and several experimental test cases were computer
simulated in order to assess the validity of the theoretical
multiconductor model. As a result, the following general conclusions
can be made:
(1) Based on the computer-simulated results, overall, the multi
conductor mathematical model appears to predict, with reason
able accuracy, voltage and current profiles along unbranched,
arbitrarily terminated three-phase conductors. In particular,
the model is very accurate in predicting signal profiles of
test cases that exhibit "strong" degrees of coupling (ild,#2d).
In cases where coupling is not as prominent (ilb,#lc), the
model appears to lack precision when comparing ~ith experimental
data. It is conjectured that these discrepancies can be attri
buted, in part, to precision error in the numerical solution
process (vhich involves iterations of matrix inversions, multi
plications, diagonalizations, etc.), and also to the sensitivity
of network response at multiples of quarter-~avelengths.
85
(2) The accuracy of prediction is subject to uncertainties in the
network parameters. These uncertainties exist in the theore
tical derivations of the per-unit parameters [Z] and [Y].
(3) The presence of standing waves is verified for three-phase
systems. As in single-phase systems, load impedances act
as circuit elements that vary with line length. In addition
to the effect of loading, relative positions of signal
nulls can "shift" due to interactions from other current
carrying conductors. The extent of these interactions, or
"cross-coupling" effects is dependent upon relative phase
angle relationships between signals on each conductor, which
thereby produce signal cancellation when "out of phase",
and signal reinforcement when "in phase".
(4) The propagation velocity of "weakly coupled" systems, parti
cularly those containing "floating" or nondriven phase
ductors, is approximately the same as that of an unloaded
single-phase distribution line reported by Hemminger (O.95c)
at 25 kHz {3}.
(5) A uniform three-phase system consisting of homogeneous
conductors of equal length can be considered as a single
phase "bundle conduc tor" vhen the same boundary cond i t ions
are imposed on each of the phase conductors.
86
(6) The appearance of st~nding wave patterns can be altered at
network discontinuities by the interaction of propagation
"modes", vhich, in effect, change the distances between
relative minima and maxima.
Although the multiconductor model has been formulated and
empirically verified for a few specific test cases, further research
should be directed toward improving prediction accuracy and extending
the theory to accommodat~.more complex, tree-structured systems.
One way of possibly improving prediction accuracy is to obtain
experimentally the per-unit length parameters [Z] and [Y] for the
test net~ork. In addition, a sensitivity study of the various
net~ork parameters would be useful in establishing bounds on the
computability of the network. Another topic of research that may
provide useful information is the effect of distributed loading
(transformers and capacitor banks) on three-phase DLC systems.
Because actual OLC systems present a variety of topological
uncertainties in modelling, it may be of interest to research
statistical methods to reduce the complexity and uncertainty of the
model.
87
REFERENCES
1. "RF Model of the Distribution System as a CommunicationChannel," General Electric Co., contract No. 955647,Phase II, Vol. 2, Task 4, July 28,1982.
2. "Field Demonstrations of Communication Systems forDistribution Automation," Electric Power Research Institute,EL-1860, Vol. 1, Project 850, Final report, March 1982.
3. Hemminger, Rodney C., "The Effect of Distribution Transformerson Carrier Signal Propagation for a Power Distribution Lineat Power Line Carrier Frequencies," Master's Thesis, NorthCarolina State University, 1985.
4. Borowski, Daniel, "Simulated Signal and Noise Profiles inDistribution Line Carrier Networks," Master of ScienceThesis, North Carolina State University, 1986.
5. Yoodruff, L.F., Principles of Electric Power Transmission,New York: John Wiley and Sons) Inc., 2nd Edition, 1956.
6. "The Distribution Automation 23 kV Test Facility, An Introduction," Carolina Power and Light Company IntroductoryReport, Raleigh, NC , May 16,1984.
7. Carson, J.R. and Hoyt, R.S., "Propagation of PeriodicCurrents Over a System of Parallel \.Tires," Bell SystemTechnical Journal, vol 6, 1927.
8. Kuznetsov, P.l. and Stratonovich, R.L., The Propagation ofElectromagnetic Vaves in Multiconductor Transmission Lines,The MacMillan Co., New York, 1964.
9. Magnusson, p.e., Transmission Lines and Vave Propagation,Second Edition, Allyn and Bacon Inc., 1965.
10. Carson, J.R., "\.lave propagation in Overhead TJires with GroundReturn," Bell System Technical Journal, Vol. 5, 1926.
11. Anderson, P., Analysis of Faulted Power Systems, The IowaState University Press, Ames, Iowa, 1973.
12. Gardiol, Fred E., Lossy Transmission Lines, Artech House,Inc., 1987.
13. Paul, C.R., "On Uniform Multimode Transmission Lines," IEEETransactions on Microwave Theory Tech., vol. MIT-21, August 1973.
14. Paul, C.R., "Useful Matrix Chain Parameter Identities for theAnalysis of Multiconductor Transmission Lines," IEEE Trans.Microwave Theory Tech., vol. MTT-23, September 1975.
15. Paul, C.R. and Feather, A.E., "Computation of the Transmission Line Inductance and Capacitance Matrices from theGeneralized Capacitance Matrix," IEEE Transactions onElectromagnetic Compatibility, vol. EMC-18, Nov. 1976.
16. Paul, C.R., "On the Superposition of Inductive and Capacitive Coupling in Crosstalk-Prediction Models", IEEETransactions on Electromagnetic Compatibility, vo~MC24, no. 3, August 1982.
17. Paul, C.R., "Computation of Crosstalk in a MulticonductorTransmission Line," IEEE Transactions on ElectromagneticCompatibility, vol. EMC-23, no.4, November 1981.
18. Gale, Louis J., "Distribution Automation at CarolinaPower and Light Company," Proceedings of the AmericanPower Conference, April 28, 1987.
19. Anderson, E.M., Electric Transmission Line Fundamentals,Reston Publishing Company, Inc., 1985.
20. Paul, C.R. and Nasar, S.A., Introduction to El~ctromagnetic
Fields, McGraw-Hill, Inc., 1982.
21. Agrawal, A.K., Lee, K.M., Scott, L.D., Fowles, H.M.,"Experimental Characterization of Multiconductor Transmission Lines in the Frequency Domain," IEEE Transactionson Electromagnetic Compatibility, vol. EMC-21, no.l,February 1979.
22. Galloway, R.H., Shorrocks, V.B., and Yedepohl, L.M.,"Calculation of Electrical Parameters for Short and LongPolyphase Transmission Lines," Proceedings IEE, vol. 111,no. 12, December 1964.
23. Davalibi, F., "Ground Fault Current Distribution BetweenSoil and Neutral Conductors," IEEE Transactions on Po~er
Apparatus and Systems, vol. PAS-99, no.2, March/April 1980.
24. Murray-Lasso, M.A., "Unified Matrix Theory of Lumped andDistributed Directional Couplers," The Bell System TechnicalJournal, January 1968.
25. Veldhuis, J., "Computer-aided Research on Multiwire Telephone Cables," Phillips Tech., vol. 40, no.4, 1982.
26. Members of the Technical Staff, Bell Telephone Laboratories,Transmission Systems for Communications, fifth edition,Bell Telephone Laboratories, Inc., 1982.
88
89
27. Central Station Engineers of the Westinghouse ElectricCorporation, Electrical Transmission and DistributionReference Book, Westinghouse Electric Corporation, 1964.
28. Dworsky, L.N., Modern Transmission Line Theory and Applications, New York: John ~iley and Sons, 1979.
29. O'Neal, J.B.Jr., Hayden, L.C., "Important Performance Criteriafor Distribution Line Carrier System," IEEE Trans. on PowerApparatus and Systems, July, 1982.
30. Yagner, C.F., and Evans, R.D., Symmetrical C~mponents, firstedition, McGraw-Hill Book Co., New York, 1933.
31. Calabrese, G.O., Symmetrical Components Applied to ElectricPower Net~orks, The Ronald Press Co., New York, 1959.
32. Stevenson, William D., Elements of Power System Analysis,third edition, McGraw-Hill Inc., 1975.
33. Riddle, Michael L., "Modeling Multiple Conductor TransmissionLines," Master's Thesis, North Carolina State University,1988.
34. Amoura, Fathi K., "Distribution Power Line Carrier AnalysisBus Impedance Approach," Phd. Dissertation, North CarolinaState University, 1986.
35. ~edepohl, L.M., "~ave Propagation in Nonhomogeneous MultiConductor Systems Using Concept of Natural Modes," ProceedingslEE, vol. 113, No.4, April 1966.
36. Vedepohl, L. M., "Application of Matrix Methods to the Solutionof Travelling ~ave Phenomena in Polyphase Systems," ProceedingslEE, vol. 110, No. 12, December 1963.
37. Bickford, J.R., Mullineux. N., Reed, J.R., Computation of PowerSystem Transients, Peter Peregrinus Ltd., 1976.
38. Brown, H.E., Solution of Large Networks by Matrix Hethods, John~iley and Sons, Inc., 1975.
39. Gruner, L.M., "Multiconductor Transmission Lines and the Green'sMatrix," IEEE Transactions on Microwave Theory and Techniques.vol. 22, 1974.
40. Alexander, Susan, and Ardalan, Sasan H., "Computer Modeling andAnalysis of Transmission Line Networks," Center for Communications and Signal Processing Technical Report, No. CCSP-TR-87/4,North Carolina State University, Raleigh, NC, March 1987.
90
41. Spencer, Richard, and Gale, Louis, "Standing Wave Study on PowerDistribution Feeders," Carolina Power and Light Company, ReportNo.1, January 1983.
42. Cheng, David K., Field and Wave Electromagnetics, AddisonYesley Publishing Co., 1983.
43. Hayt, V.H., Engineering Electromagnetics, New York: McGraw-Hill,fourth edition, 1981
APPENDICES
91
APPENDIX A
CALCULATION OF LINE PARAMETERS : PER-UNIT LENGTH
IMPEDANCE [Z] AND ADMITTANCE [Y] MATRICES
92
SYMBOLS USED IN DERIVATION OF [2]
93
z (p=q)pq
z (p eq )pq
Ra
Xa
D (p;tq)pq
De
kz
Dsa
f
Self-impedance of conductor "p"
Mutual impedance between conductors "p" and "q"
Internal resistance of conductor "a"
Internal reactance of conductor "a"
Geometric mean distance between conductors "p" and "q"
Equivalent depth of earth return (formulated by Carson)
Constant used for units of Q/mile
Self-geometric mean distance (GMR) of conductor "an
frequency
Subscript "a" Refers to phase A
Subscript "b" Refers to phase B
Subscript "c" Refers to phase C
Subscript "n" Refers to neutral conduc.tor N
Subscript "g" Refers to ficticious ground conductor G
94
Section A.l - Calculation of Per-Unit Length Impedance Matrix [Z]for Three-phase System with Neutral ~ire
For a three-phase system with neutral wire, as shown in Figure
A-l, Carson's line equation can be written as
rVaal
r- V I r '"', (1V Z zab Z z Za a' I aa ac an ag ! I a
I jI Vbb,! Vb - Vb'
! Zbg II
bi zba zbb zbc zbn II I
I,
V - j V - V TI cc' -, C c' z zcb z z :cg
I
.1.I ca cc en c
I
I <A. 1>Vnn' l::
Vn , z znb z z I Ina nc nnZng I n
l V - V z Zgb z Z lI ggg' g' ga gc gn gg)
The neutral T,t/ire, labeled as "n", is connected at each end to a common
ground point of zero potential. Since the neutral wire is in parallel
with the ficticious ground conductor "g", the return current \Jill
divide between the two paths:
-(I + I )n g<A.2>
I g-(I + I b + I + I )a c n <A.3>
Using the earth current relation in Equation A.3, we may subtract out
v , from each of the other equations to obtain the follo\Jing pargg
tioned system of eauations:
95
la. la.a.I--.
a 0 '\,. ~
lb Zdb
b0~ Zbb
~..;
I7
Ie - '1'Zee
~
C "'"\
\
ZenT'
Z~ \.l.r.--..
+I
Vn~O ZQ~
Zb5 I...., "" -, I ZC3 , " " , ,"" '-"" , '\.
Yg: 0~
I ljj...
Figure A-I Carson's line for three-phase system with neutral wire
96
r
zan 1r
V Z Zab z I Ia aa ac I a
Vb zba zbb zbc z I I I bIbn II <A.4>
V I Z Z Z Z I Ic I ca eb cc en j c
~::o J---------------------
Jz z z z Ina nb nc nn n
As described in Chapter 3,
zpq z - zpq pg z + Z
gq gg <A.S>
where p,q represent combinations of {a,b,c,n}. From Equations 3.18-
3.21, assuming conductors a,b,c, and n are homogeneous, ~e can vrite:
Zaa zcc Znn
R + R + jwk In(D /0 )age s <A.6>
where D represents the self GMD of each conductor (0 =D b=D =D ).s sa s sc sn
Equation A.6 describes the self-impedances of phases a,b,c, and n.
If all conductors are homogeneous, note that the self-impedances are
all the same. Likewise, the mutual impedance terms are expressed as:
R + jwk In(D /D b)g e a <A.7>
<A.8>
Zac<A.9>
97
In general, for p#q, the equation for mutual impedance becomes
zpq
R + jook 1n(0 /D )g e pq
<A.I0>
where D represents the distance between conductors p and q.pq
For p=q, the general equation for self-impedance takes the form:
zpq
R + R + jook In(D /D )q g e sq
<A.l1>
Note that the general equations for self and mutual impedances (A.l0
and A.1l) do not account for skin effect. Several references discuss
skin effect in great detail {S},{32}, arriving at closed form expres-
sions in terms of Bessel functions. A simpler solution is presented
in (22} and {37}, where the conductor internal impedance terms are
calculated at sufficiently high frequencies (above 5 kHz). This
solution assumes that the surface magnetic field intensity H iso
tangential at all points to the surface of the conductor and propor-
tional to the current density J at the surface. For the completeo
derivation, the reader is referred to (22}. The formula for conductor
internal impedance is:
z..11
1/22 . 25 (p (;)u )
rII(n+2)(2)1/2
R..11
x..11
<A.12>
98
where:
2.25 = constant due to stranding
r = radius of outer strand
n = number of outer strands
p resistivity of wire
u = permeability of conductor
00 2nj
From {22}, the impedance matrix [Z] consists of a total of five
components:
(1) self-resistance term from the conductor
(2) self-reactance term from the conductor
(3) resistance term from the earth return path
(4) reactance term from the earth return path
(5) reactance term from the physical geometry of the conductors
Equation A.12 describes (1) and (2) with skin effect taken into
account. (3) and (4) are determined from implementing Carson's line
for a multiconductor system as shown earlier, and (5) is determined
from effective GMR's and GMD's of the conductor geometry.
Thus~ ~ith all five components considered, the general equations for
respective self and mutual line impedances are modified from Equations
A.I0 and A.l1 as follovs:
99
For p=q,
zpq
",here Rq
For p~q,
zpq
R + R + jook (In(O /D )) + jXqq g e sq
X .q
R + jwk {In(D /D )}g e pq
<A.!3>
<A.14>
Furthermore, the 4x4 system of equations in Equation A.4 can be re-
duced to dimension 3x3 using a method known as Kron's reduction {38}.
The resulting system of equations is shown below:
r ( ~ ( ""\! 1 1
I val zaa' Zab' Zac' I I II
Zbc' I
a
Vb Zba' Zbb' I b <A. IS>
V I Z Zcb' Zcc' J Icj ca' c
\Jhere Zpq'
The terms in Equations A.13 and A.14 for a vertical conductor geo-
metry are calculated as fo110"'s:
The value for conductor internal impedance (with skin effect) is
obtained from equation A.12:
100
2.25 ( )1/2pwuR Rb R R X X
bX Xa c n a c n -;~(~:2)(2)I72
where: r 0.0487 inches = -31.237x10 meters
p = 3xlO-8 Q·m (for aluminum)
u = 4nxlO-7 Him
n 7 (# of outer strands for AWG #2)
· Ra R
c x = 5.6324 Q/milen
The formula for earth resistance R was derived empirically by J.R.g
Carson {31}. Note that R is a function of frequency. At 25 kHz,g
-3 .R = 1.588xlO f = 39.7 Q/mlle .g
The equivalent depth of return D , also empirically derived by Carson,e
is a function of earth resistivity p and frequency f. As mentioned
in Chapter 3, it is common to define D as :e
D D 2/D (unit length)2/(unit length)e ag sg
(Carson's derivation of this parameter is discussed in great detail
in Vagner and Evan's (30)). The empirical formula is given below:
vhe r e : earth resistivity
f = frequency = 25 KHz
101
· D 136.61 fte
Figure A-2 shows a vertical conductor geometry for a typical
distribution line. The conductor spacings between phase conductors
are equal, and all conductors are classified as type A~G #2. For this
type conductor, the GMR (D ) is 0.146 inches {11}.sa
From tables {11) , D Ds b D D 0.292/2 in 0.012167 ftsa sc sn
From Figure A-2,
D 9 ft Dbn 6.5 ft D 4 ftan en
Dab 2.5 ft Dbc 2.5 ft D 5 ftac
The self and mutual impedances elements of the [Z] matrix are now
computed based on the conductor geometry shown in Figure A-2:
w = 2nf = SO,OOOn
k -7 m/mile) -3z = 2xlO Him (1609.344 0.3219xlO H/mile
Z Zbb Z Z R + R - j tok In(D /D ) + jXaa cc nn a g e sa a
Z Zbb Z Z 45.33 + j477.2 Q/mileaa ec nn
Z R + jwk In(D /D ) 39.7 + j137.53 Q/milean g e an
Zbn R + jwk In(D /Db ) 39.7 + j153.98 Q/mileg e n
Z R + j tok In(D /0 ) 39.7 + j178.53 Q/mileen g e en
1/t0 A-I"30 =- 2.5
11'0 8+30
1' == 2.5" I
"10 C+I
I' 4'48 =
I
30
102
Figure A-2 Conductor configuration for vertical geometry
103
Zab Zba R + jwk In(D ID b) 39.7 + j202.30 Q/mileg e a
Zbc Zcb R + jwk In(D /Db ) 39.7 + j202.30 Q/mileg e c
Z Z R + j tok In (D ID ) 39.7 + j167.25 Q/mileac ca g e ac
From Equation A.15 (Kron's reduction formula), we may reduce the 4x4
impedance matrix to an equivalent 3x3 impedance matrix by eliminating
the phase to neutral impedances Z ,Z. , and Zan ~n en
Zaa' Z (Zan2
/ Znn) 26.07 + j439.04 Q/mileaa
Zbb' Zbb - (Zb 2/2 ) 24.31 + j428.82 Q/milen nn
Zce 'Z - (2
2/ Znn) 21.87 + j411.48 Q/mileee en
Zab' Zab - (2 Z )/2 19.53 + j159.31 Q/milean bn nn
Zbc' Zbc - (2 Z )/2 17.40 + j145.87 Q/milebn cn nn
Zac' Z - (2 Z )/2 18.17 + jIl7.0S Q/mileac an en nn
26.07 + j439.04 19.53 4- j159.31 18.17 + jll7.0S
[Z] 19.53 + jlS9.31 24.31 + j428.82 17.40 + j145.87 Q/mile
18.17 + jl17.0S 17.40 -+- j145.87 21.87 + j411.48 I"
SYMBOLS USED IN DERIVATION OF [YJ
D D' t b t ' th d d · f ,th d' , IS ance e ween J con uctor an Image 0 1 con uctor1J
d D' t b t ' th d' th d (1' .J,J' ), , IS ance e ween 1 an J con uctors r1J
r ad i us of I' th d f"con uctor or l=J
104
y, .I)
Y..1J
C
G
Self-admittance of conductor "i" for (i=j)
"Mutual" admittance term between conductors "i" and "j"(usually considered as a line to line capacitance)
Capacitance
Conductance
£ = Permittivity constant
[B] Matrix which defines charge distribution due to zeometry
[P] Charge coefficient matrix [P] = (1/2Ttt)[B]
B. , In {D. . / d. , )1) 1J I)
P,. (1/2n£) In{D, .v d . .J1) 1J 1J
-12£ = 8.854xlO F/m
k = 2n£ = 89,525 nF/miley
105
Section A.2 - Calculation of Per-Unit Length Admittance Matrix [Y]For Three-phase System with Neutral ~ire (Vertical Geometry)
From Figure A-2, the method of images is used to determine the
folowing distances:
D 78 ft Dbb 73 ft D 68 ft D 60 ftaa cc nn
Dab Dba 75.5 ft 0 D 69.0 ftan na
D 0 73.0 ft Dbn Dnb 66.5 ftac ca
Dbe Deb 70.5 ft D D 64.0 ften nc
d dbb d d 0.012167 ftaa cc nn
dab d. 2.5 ft d d 9.0 ftba an na
dbc deb 2.5 ft dbn = d
nb 6.5 ft
d d 5.0 ft d d 4.0 ftac ea en nc
From Equation 3.34,
From Equations 3.26 and 3.27, Ne obtain the folloving form for
the inverse of [B]:
- fIn(D Id ) In(Dab/d ab) In(D Id ) In(D Id )aa aa ae ae an an
[Br1 In(Dba/d ba) In(Dbb/d bb ) In(Dbc/d bc) In(Dbn/d bn)
In(D Id ) In(Deb/d cb) In(D Id ) In(D Id )ca ca cc ec en en
lln(D /d ) In(Dnb/dnb) In(D Id ) In(D Id )na na nc nc nn nn
From Equation 3.30, we know that
[V] = [P] [til]
where [P] = (1/2n€)[B]. Thus, in partitioned matrix form,
'I -- I
,
V I P Pabp I p :
qaa aa ae1
an
Vb Pba Pbb Pbe I Pbn qb
= I
l::-Jp Peb
p I P qeca eeI
en
--------------------1-------P Pnb P 1 p qnna nc nn
Since V = 0, the system can be reduced to three equations byn
eliminating the fourth row and column. This is accomplished by
solving the last equation and substituting back (ll) to obtain:
[V] = [P']['t'] = (1/2Tt£)[B']['Y]
106
where P' ..1J
P .. - (P. P .)/P1J In nj nn
Similarly, B' ..1J
B.. - (B. B .)/BIJ In nj nn
107
As a sample calculation, element B' is calculated as follows:aa
B'aa In(Daa/daa) - (In(Dan/dan)ln(Dna/dna)}/ln(Dnn/dnn)
~ B'aa = 8.27785
In a similar fashion, the remaining terms are calculated and the re-
suIting [B'l is numerically equal to:
[B' ]
/'
I 8.27785
2.85082
2.01688
2.85082
8.06359
2.58111
2.10688l
2.58111J
7.72454
-599.3 E-6
Since [Y] j tok [B,]-1, where 00= 2njy
1.983 E-3 -317.4E-61
[Y] j -599.3 E-6
-317.4 E-6
2.134 E-3
-556.6 E-6
-556.6 E-6I
2.089 E-3)
mhos/mile
The above matrix represents the admittance matrix. However, from
Equation 2.28, the actual admittance matrix derived from the mode~
is different to account for line to line capacitance. Specifically,
the diagonal elements are of the form:
108
Y11 Y11 + Y12 + Y13
Y22 Y21 + Y22 + Y23
Y33 Y31 + Y32 + Y33
Thus, the actual per-unit length admittance matrix [Y] t used inac
the simulations is derived from the mathematical model described
in Chapter 2 :
j
2.899 E-3
-599.3 E-6
-317.4 E-6
-599.3 E-6
3.290 E-3
-556.6 E-6
-317.4 E-6
-556.6 E-6 Mhos/mile
2.963 E-3
Table A-l Sample Input Data File for Simulation (Test #lb)
109
Parameter Value
Line Length, L 3·59 13 miles
- -Input 50.0
Voltage} V. 50.0 V
50.0- -
Source ~ 0.2 0.0 0.0 -Admittance,
0.0 0.2 0.0 ey.
0.0 0.2 _.. 0.0
Load ~ 0.0 0.0 0.0 -
Admittance, 0.0 0.0 0.0 U
YL .. 0.0 0.0 0.0 _
Per Unit ,- 20.1 + j439.0 19·5 + j 159.3 1~.2+i117.1-
Impedance, 19.5 + j 159.3 24·3 + j428.8 17.4 + j 145.9 (2
ZL.. 18.2 + j 117.1 17.4+j145.9 21.9+j411.5_
- -3 -6 .6lPer Unit j2.gxI0 -j599.3xlO -j 317.4x 10Admittance, -6 -3
-i556.6xIO·6
(j-j 599.3x10 j 3.3x10Y -6
-j556.6xIO-6
j3.0XIO·3j-j317.4xIO-
Section A.3 - Per-Unit Parameters for Delta Configuration
Figure A-3 shows the conductor configuration for a delta type
geometry. From the methodology presented in Sections A.I and
A.2 in Appendix A, [Z] and [Y] are as follows:
110
r 23.6 + j424.3 17.3 + j148.5 17.3 + j 148.5 11 17 . 3
I
4- j414.2 16.6 + j85.SI
ohms/mile[Z] + j148.5 22.2 I
I III
l17. 3 + j148.5 16.6 + j85.5 22.2 + j 414 ..2.J
r j3.27 e-3 -560.2 e-6 -573.1 e-61I
[Y] I -560.2 e-3 j3.10 e-3 -419.8 e-6 mhos/mile
l-573.1 e-6 -419.8 e-6 j 3 .15 e-3
111
A
/II
27.15
8/II I
27.75 :; Z.31
~-C
--- -------f -II I
20.25 : 1.69
II
m.....------1 II
I
I
II 416\8 :
Figure A-3 Conductor configuration for delta type-geometry
APPENDIX B
CALCULATION OF DISTRIBUTED PER-UNIT LENGTH IMPEDANCE ZP
AND ADMITTANCE Y FOR A BUNDLED CONDUCTORp
112
113
Section B.l - Calculation of Per-Unit length Impedance Z for aBundled Conductor (Vertical Geometry) p
For a symmetric network in which all conductors are uniform,
as in test case #lb, the conductors A, B, and C can be treated as
a single "bundled conduc tor" . For a three-phase sys tern, as shown
in Figure B-1, phases A, B, and C can be equivalently represented
by a single bundled conductor located at a geometric mean distance
of 6.162 feet from the neutral conductor.
The three-phase system with neutral wire and earth return
described earlier in Appendix A is now reduced to a single-phase
system consisting of composite conductor "a" and neutral conductor
"n", as shown in Figure B-2. The total current in composite conductor
"a" is denoted by I. For clari ty, the conductors in the bundlea
are labeled as 1, 2, and 3 (instead of A,B,C). Under the assumption
that the current divides equally among the individual conductors,
we can say:
Ia <B.l>
If R denotes the resistance of anyone conductor (1,2,3) in the bundle,
then:
R R /3 , Xa
X /3a <B.2>
and the self-impedance seen by current Tis:-La
/~
t.-J\ I~--/ I
II
6.1 fJ 2
,30
114
Figure B-1 "Bundled conductor" geometry via equivalent GMD
115
-,\
I \
Composite conductor "a"
C------Z141
- ........-- -
""./
I 1 O-'~~JII8II'\/Y\---.--------...-------_..u....f{
( 2~...:::;..~r\i_----DIlIzaII--.:lIl~----=------_<\r
{\ 3 ~..::::....~;.....-...-----..---=-----~~--o."-- -
Figure B-2 Carson's line for a bundled conductor
r"7LJaa R /3 + R + jwk In(D /0 ) + jX 13 Q/mileage aa a <B.3>
116
where Daa is the modified geometric mean radius (GMR) of the bundle
defined by:
oaa <B.4>
Similarly, D denotes the equivalenteq
geometric mean distance between the composite conductor "a" and neutral
conductor "n".
oeq(D D D )1/3
In 2n 3n <B.S>
From Equation A.I0, the mutual impedance between "a" and neutral is:
ZanR + jook In(D 10 ) Q/mileg e eq
<B.6>
From Equation A.Il, the self-impedance of the neutral wire is:
ZnnR + R + jwk In(D /D ) + jX Q/mile
n g e sn n<B.7>
Using Kron's reduction, as in Equation A.IS, the per-unit length
impedance for the bundled-conductor system can be expressed as:
Zaa'z
pr'7<-I
aa(Z 2/ Z ) Q/mile
an nn<B.8>
As before, the self resistance and reactance of each individual
conductor is:
Thus, the total self-impedance of composite conductor "a" is
From Appendix A, the following values remain unchanged:
D 136.61 fte
R 39.7 Q/mileg
k -3 H/mileO.3219xlOz
w = 2ft!
D = 0.146 in 0.0121667 ftsn
From Equations B.4 and B.S,
D {(0.146)3(30)4(60)2}1/9 = 5.93 inches = 0.4942 ftaa
D {(l08)(78)(48)}1/3 = 73.95 inches = 6.162 fteq
From Equations B.6, and B.7
117
Z 1.877 + 39.7 + j2n(25000)(.3219 E-3) In(136.61/.4442) + jl.877aa
Z 41.58 + j 291. S4 Q/mileaa
Z 39.7 + j2n(25000)(.3219 E-3) In(136.61/6.162)an
Z 39.7 + j156.68 Q/milean
118
Znn 5.6324 + 39.7 + j2n(25000)(.3219 E-3) In(136.611.012167) + j5.6324
Znn 45.33 + j477.2 Q/mile
Thus, from Equation B.8
Zp (41.58 + j291.54) - (39.7 + j156.68)2/(45.3324 + j477.2)
Z 20.276 + j241.38p
Section B.2 - Calculation of Per-unit Length Admittance Y fora Bundled Conductor (Vertical Geometry)p
As before, B.. = In{D . . /d . . ). The distances between1J 1J 1J
conductors and their respective images, as discussed in Appendix A.2
are obtained from the physical geometry of the conductors (Refer to
Figure B-2).
Daa72.324 ft d dan na 6.162 ft
Dan Dna66.126 ft d d 0.0121667 ftaa nn
D 60 ftnn
USIng the above equivalent distances for a bundled conductor, ~e obtain:
[B]In(72.32410.012167)
In(66.162/6.162)
In(66.162/6.162)
In(60/0.012167)
[
8.5034(1/68.265)
-2.373
. -1Since [Y] = Jook [B] ,Y = jookBy aa aa
-2.373 l8.6902 J
119
Yaa-3Y = jook(8.S034/68.265) = 1.7517xlO mhos/mile
p
Section B.3 - Propagation Constant and Characteristic Impedance
From Z and Y , the propagation constant y and characteristicp p
impedance Z can be calculated as follows:o
y = (Z Y )1/2 = {(20.276 + j241.38)(1.7517 e_3)}1/2p p
y = 0.02728 + jO.6508
zo
{Z /y }1/2p p 371.5376 - j15.577 Q 371.86 -2.4 Q
APPENDIX C
TABULATED AND PLOTTED MEASURED DATA
120
A
~
c !• Ct
I
Ii \. 3.~9 tr.~, ·1.J-
I
121
Table C-1 : Pleasured data tor test Ila
, I I I I I I I I IK.A5ur •••nt I I 1 I 2 I 3 I 5 I 6 I ., I & I 9 I 10 I
I ,1 I I I I 1 I II , ,
DistAnce trom I I I I I I i I I 1
(ailes), 0 I o. 4 , 0.1 1.2 I 1 .6 I 2 .0 I 2 •• I 2. a I 3.2 ( 3 .6 IsourceI I I I I I t I I ,
\
I I I I I 1 I I I IVan (V) I 48 I .6 I 3& I 25 I 10 I a . a 123 .5 I 36 I 46 I 50 1
! I I I I I I I ! I II I I I I I I I I
X 'Ibn (V) 115.5 1 16 I 16 116.5 t 16 I 16 I 16 I 16 I 16 116.5 I! I ! I I J I I ! I II I I I I I I I I I
X Vcn (V) 115.5 I lS 115.5 j15.5 I 16 115.5 115.5 115.5 I 16 \16.5 IJ I ! I I I I , I I II
I I I I I I I I ! I'lab (V) I SO I tti 34 I 26 I 16 115.S 116.5 I 4J I ., I SO I
I I I I I I I I I II I I I I I I I I I
Vbc: (V) I S • 5 1 1.0 5.6 I 5 I 2.4 I 2.2 t 4 • a 1 6 • 4 I 7.21 7.6 II
I 1, I I I I ,
1!
I I I I I I 1 IVar: 'V) 48 I S2 38 I 30 I lS I lS I 30 r .~ I 52 I SS I
I I I I I I I , II I I I I I I I t,
: .. faA) 7 I 42 1 7 8 . a I 105 I 119 1122.51106.8180.5 \43.8 Il.75 II I I I t t I I I Ii I I I I I I I I I
It> (aA) 3 • S I 14 124 .5 129.8 13 a .5 13& • S I 3S 124 .5 112.2 11.75 I, I I I ! I 1 ! 1 II I I I I I I I I ,
:r: (_A) 1. a 119 .2 13", • 5 I 42 1 49 147.2 140.2 I 25 I 14 I 0 tI ! ! I I I ! I i I
x Poss:.ble error In measurement due to 60 cycle couplingfr0m nearby hlgh '/01 ~ a ge transm~ssion Ilne
122
PHASE T0 NEuT?AL VOLTAGES (TEST #1A)
'"
10 j
J"'r-i~----r---r-.,---,--~~----r-""'--"----",------r--
v 30aLT5 20
0.00 0.40 0.80 1.20 1.60 2.00 2.39 2.79 3.19 3.59
DISTANCE F'ROM SOURCE (MILES)
V . t-+PHASE Ct..;RRENTS (TEST #1A)
an
Vbn • 0······0
130 -ofv . 6-·_·~en
120
110
100
90 //
80
/m 70
A 60
/50 _ ,_._~._._._.-.'t:r- ....
40 l ."G J .. _-_ ......._..Q .... _...........~~ .../
."
30 / .,,~ 1""'\
....-... \oj a'.,:1 ,"
j ",
20 "......k-
,- ..... 8' .'A.\10 ..... "'
~ ' ......... \
O~'.....,~
I
0.00 0.40 0.80 1.20 1.60 2.00 2.39 2.79 3.19 3.59
DISTANCE "'ROM SOURCE (MILES)
123
......--- 3.59 ,..;ltS ----......,
~E :
! ~__C ~1·
~! !
C7 t· I
TAbl. C-2 : M.asured dAtA tor test tlb
I I i I I I I I IKe&sur ••• nt I I 2 I 3 I 4 I 5 I 6 I 7 I I 9 I 10
I I t I I I 1 I IDistance fro. I I 1 I I I I I I
0 I o.4 I o.! I 1.2 1 1.6 I 2.0 I 2 • 4 I 2.8 I 3 .2 I 3.6source (aile.)I I I I 1 I I I !I I I 1 I I I 1 I
Van (V) 4' 1 41 I 34 I 25 I 14 I i 123.5 I 4O I 51 I 55I I I ! I I t I I,
I I I 1 I I I I IVbn (V) 4& I 41 133.5 127.S 113.& I 7.2 123.5 139 .5 I 51 I 54
! I I I J I I ! 1I I I I I I I I I
'len (V) 45 I 41 I 34 I 28 I 14 I 7 123.5 I 39 1 50 I S4I
,I I I ! ! , II !
I I I I I I , I IVab (V) 1 I 0.4 1 o.3 I 0.4 1 0.4 1 1 1 0.6 I 0.6 I o. 8 1 0.3
I I I I I , I I !I I
I I I I 1 I I I IVbc (V) 1 I 0.3 I O.S J 0.6 1 O.i I 0.6 I 0.6 I 0.6 I 0.5 I 1
I ! I I I ! i I II I I I I I 1 1 I
Vac (V) 0.5 I 0 I 0.6 , 0.5 I 0.5 1 0.8 I 0.6 I o. 4 I 1 .1 1 1 • 2I ! I I I ! I I 1 ti t I I I 1 I I I 1
I. C.A) 110.5 I 21 1 42 I 56 166.5 173.5 /66 .5 157.! I 28 I 7! I I I I I I 1 I II I I I I 1 I I I I
Ib (aA) 110.5 , 21 138.5 ISo.a 161.2 161.2 161.2 I SO • a 126.2 I 1.8f I I ! t , I I ! !I
I I I I , I I I I 1Ie (a,A) 110 . S 122.8 14~.S 161.2 1'3.5 110.5 lil.I 151.S 131.5 I 3.S
I I I I , I I J I It
124
60
150
40
V0L 30TS
20
10 jj~
04I
0.00 0.40 0.80 1.20 1.60 2.00 2.39 2.79 3.19 3.59
_.-~ .... ....
.c:~,
.......(J.- ..
,/I!. 0 0
~,/
,/
" 0/. "
. \,
Ji' .. ' " .....\\I-It .' \\
/',/0' \\1 \,\
I \~I
~~\1
\ \
""\
"-"lI--
~
0.40 0.80 1.20 1.60 2.00 2.39 2.79 3.19 3.59
DISTANCE F"RQM SOURCE (MILES)
90 ]
80
70
60
50mA
40
30
20
10 I
oJi
0.00
DISTANCE FROM SOURCE (MILES)
PH.:.SE CUPRENTS (TEST# 18)"'bn· C·· ..··O
Ven • 6-·_·-6
A
3. 59 ~"
I
-jl. T-·f ~
125
Table C-3 : MeAsured dAta tor test 'lc
I I I I I IM•• sure•• nt t 2 I I 5 I 6 I 7 I 8 9 f 10
I I I t I !Olstance tro. 1 I I I I I
(ailes)0 o. 4 I o. 8 I 1 • 2 1 .6 I 2.0 I 2.4 , 2 • S 3 .2 I 1.6
sourc.I I 1 I I !I I I 1 I I
Van (V) 50 48 I 40 1 2S 9.2 I 9 1 24 I 39 48 I 50
i I I I I ,I
I I I I I II
Vbn (V) 1.2 0.6 I 1 .1 I 0.8 1.0 J 1 .2 I 1. 2 I 1 .4 1.6 I 1 . 2I I 1 1 I I
I
I I I I I IVcn (V) 1 • 5 3 .2 I 3 I 2. S 2 • a I 3.2 I 3. 2 I 3 .2 3 .4 I 3 .0
I I I I I I II I I 1 i I I
Vab (V) S2 I 4! 138 .5 124.5 10 I •• 6 123.5 I 38 47 I 51I f I I 1 t II
I I I I I 1 IVb.: (V) 3 • 4 , 1.6
,3 .2 I 3 3 I 3 J 3 .1 I 3 • 2 3 .0 I 3.2I
I ! I I , I II I i i I J I
v&C (V) SO I 48 I 40 I 25 10 I 8.5 124 .5 I 39 47 t 51I ! I ! 1 , I I;I I i I I I I I I
Ia ( aA) 7 I 35 1'3.5 196 .2 1110.21 112 tlO3.217S.2 13a . 5 I 3 .5I I I I I I I II I I I I i I i I
It'> <sA) 0 11 .75 \17.5 I 21 124 .5 11.75 124 . S J 19 • 2 !12.2 I 3.5I I ! I I , I ,
iI ,
I I I I I I 1Ie (aA) 0 11.7S 110.5 115.7 I 0 119.2 117.5 0 I 7 I ).5
I I I I I I I !
126
F~4 :: c: TO: lEu :-F .:. '- \/0L: A.GE3 (TEST #1C)
0.00 0.40 0.80 1.20 1.6~ ~ 2.00 2.39 2.79 3.19 3.59
DISTANCE tROM SOURCE (MILES)
PHASE CURRENTS (TEST #1C) Van +-+
mA
120 i110 ~
100
90
......_.0c "",,,,, -'-A
,,~
1.20 1.60
Vbn• 0 ......0
Ven • 6-·_·-6
\
" ...... -.,0········
~~
G/
"/
" "0·. \/
" ~.~.~/ , "."." ·-·....19''0' ... ...... - -
T2.00 2.39 2.79 3.19 3.59
DISTANCE rROM SOURCE (MILES)
A
B
I I ~j z,,
-/'-rI'1 3.59 "'; Ie ~ ~
I J-:-
-:"
127
Tabl- C-4 : Measured dAtA for tast t 1 d
I I I I I 1 IMeAsur •••nt I I I I 5 6 I 7 I a I 9 I 10
I I I I t 1 IDistance teo. I i I I i I 1
(-ll."a o. 4 I o.! I 1 .2 I 1 .6 2.0 I 2 .4 1 2 • S 1 3 .2 1 3 .6source
I ! I I t I II I I l I i 1
VAn (V) 44 47 1 46 1 42 I 37 26 I 21 I 50 1 78 I 102I I I I J I 1/ / I I I I I
Vbn (V) 43 31 /12.8 I 9 .4 128.5 41 I SS I 67 1 '7 I 42I I I , , I I!
I I I I I I IVen ( V ) 46 102 I 162 1 204 I 224 196 I 110 /14 .6 I 106 I 200
I I I I ! ! [I I I 1 1 I I
Vab (V) 3. S 21 136 .5 1 49 I 5S 48 136.5 11a . 5 I 1 I 20I I I I 1 I II I I I I I I
Voc (V) I 18 87 I 156 I 208 244 216 I 154 I 6 ) 130 .5 I 120I I I I I I I! I I I I I I
'lAC (V) 114 • 4 68 I 118 I 162 las t 176 I 120 I 46 130.5 1 100! ! I I I ! I I!
I I I I I 1 t I I IIa (aA) I 21 131 . 5 14i.2 159.5 I 14 196.2 192.7 )73 .5 138.5 I 5.2, I , I I I I I I I
! I
I I I I I I I I I IIb (aA) /12.2 I 203 I 192 I 147 194 .5 13a . 5 I 98 I 15~ I 192 I 3S
! 1 ! I I ! I t ! II
/ I I I I /I I II
Ic (-A) I 567 I 532 I 399 I 262 110 I 490 I 630 I 700 I '65 I 560I I I I I I I I I
128
?H;.,:E TI; I~E'.)Tr·;'L \/\=)LT~/.-,~'3 \ It.3T /f 1[.)
250
- -t':t- _
200 ,is
A t:;' \
/ ,II ,
I~ ,I
150 J,
IY I \
0 I \I
100 jIL ,
IT I ,I
5 j A, ,-/I '.
sJI ,
I
"I , .~:........ ~
I
0.. /'
\/' I
II
,~ I
~./
I
10 .. I
\ I
0 ...................0 ....... \.6,/
04i
0.00 0.40 0.80 1.20 1.60 2.00 2.39 2.79 3.19 3.59
DISTANCE FROM SOURCE (MILES)V . +-+an
PHASE CURRENTS (TEST #' D) Vbn • 0 ..····0V . 6-,-·-6en
700 ~ ., ~_.- ..............
j~ ....
600 l._.- ~,,~ "
"/I
~
j 'A.. I
I
:::J~,
I,I
I~ ;1 !m
1 !A ,I
"300
1\
,I.:- I,
I] \ i200 1 c ::; I
ICG. I
100 i ,~
~
0.00 0.40 0.80 1.20 1.60 2.00 2.39 2.79 3.19 3.59
DISTANCE FROM SOURCE (MILES)
$1I
II i
I I-=-
1 !.S9 Ini. ~ IIS .... j -;-
129
Table C-s Me~sur.d dat~ fjr test 12~
I I 1 I I I I I I IM.asurement t I 1 I I 6 I 7 1 1 1 10 I 11 I 12 I 13
I 1 ! I I I ! I ! ! !!
Distance fro. I I I I I I I I I I I(ailea) I O... 10.79 11.19 1.59 11.99 12.31 12.78 13.18 13.59 13.97 14.37 14.7'lource
I I I I I 1 I I I I II
I I I 1 I I 1 I I IVan (V) I 47 39.5 I 27 I 9.2 7.8 22.5 1 30 134.5 I 34 130.5 I 22 112.6 1 o.4
I ! I I I , I I I II I I I I I I I I
X Vbn (V) f 16 16 I 16 115.5 15.5 15 115.5 I 15 I 15 16 1 I I! I I I I I ! I II ! I 1 ! I I I I
X Vcn (V) I 15 15.5 115.5 I 15 15 15 I 15 t 15 115.5 15.5 I I II I ! ! , ! I I II I I I I I I
Vab tV) I 42 I 36 I 27 I 15 16 15 I 30 1 3. 40 36 1 1I I I I I I II I I I I I
Vbc: (V) 1 7.6 1 7.8 I 6 ... I 2. , 3.2 1 4.8 4.1 I II ! ! I I I 1
I I I I I I IVac: (V) I 47 1 43 1 33 1 16 16 21 33.5 I 40 42 40 I I
I I I I I
~I iI !
I I I I I I It. (aA) 131.5 17 1 . a I 98 1 115 121 175 . 2 59.5 I 28 5.2 35 56 I 70 171.a
I I I I ! I ! I !I I I I I t I I I
Ib (aA) I 1 .8 I a .5 I 14 /17.5 28 138.5 3 e .5 I 2& /12.2 0 I I, , 1 I ! I I ! II I 1 I I I I I I
Ie ( ..... ) I 3 .5 112 . 3 I 2 ~ .5 133 .2 31.5 42 136 .7 124 .5 112.2 1 II I I 1 I I I I I
x Possible ~rror in measurement due to 60 cycle CQUpllngfrom nearby hlgh voltage transmission line
\
\:
\,+-
I i \ iiI
130
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.183.183.58 3.97 4.37 4.77
DISTANCE rROM SOURCE (MILES)
mA
130 1
~:: ~100 1
90
80
70
60
, I I '
PHASE CURRENTS (TEST #2A)
ii" i
//
;¥
Van· +-+'lbn· D·····D
Ven • 6-,-·-6
, ,
0.00 0.40 J.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.31 4.77
DISTANCE ~ROM SOURCE (MILES)
131
I
II
--'Ols",...j3.59 ~,.
~ 0----[~-
~I....--.--,
':'able C-6 Measured ~ata ~':): test i2b
i j I I I I I I
Measure.eat • I I I I I I I 10 I • 1 12 13
I I I I I l I !Distance froJl I I I I i 1 1 I I
0.4 0.79 11.19 11.59 11.99 12.3 a 12.7S I J .18 13.59 13.9i 4.37 4.77source (.iles)
I I I I I I , ,i
, !
I 1 I I I I IVln (V) 43 16 17.5 I ~5 I 6S I j'j 1 50 I 21 110.6 136.5 I 61 75 78
I I I ! I I I II I I I I I I I
X Vbn (V) 23 16 15.5 115.5 I 15 115. S I 15 I is I 15 I 16 II I I I I I II I I I I I
X 'len (V) :'8 15.5 15 I 15 15.5 I 15 115 . 5 I 15 I 15 I 15I I I I I II I I I I I
'lab (v) 17 15.5 16 I lS 48 I SO I 40 I 22 115.5 I 16I I I ! I I II I i I I I I
V'bc PI) S I 4.2 I I s.a I S.4 I 2 .6 I 2.2 I 2 I 2 ...I I I I I II I 1 1 I I
"lac ('I) 22 I 17 121 .5 14 55 I 54 .1 122.5 I 15 I 16! I i 1 I !I I I I i I
Ia (aA) 161 I 17a I liS 141 112 194 .5 140 11il.SI .12 I 154 ~:1 61.3 5.2I , I I I II I I I 1 I I
:b ( ah) 5 . 3 112.2 119 .2 26.2 28 12: .7 124 .5 11i.5 110 .5 II ! i I , I II
I I I I I I I I 1:c (2Al 3.5 8 .8 115.7 124.5 I 26 . 2 119 . : i17.5 115.7 18 . is I
I ! I I ! I I!
X Possible error ln ~easurement due to 60 cycle couplingfrom nearby h1.gh vol-cage t:-anSml.SS1On line
132
... •••• - , ~ J [ 1,_' • ~~ ,', L " ~ L ~:. :'l ~ ._ J :I ~ t, .\
I
t/
I/
I
\
I:t
./
/I
/
\
\~ / \
""0 -1 \ I +,'L. ~_ \ f. \ /
~ - -~ ~_JQ - - - e-·_·-.-g ~._ .... ~-- -g-..--.-e-----u-~ - -2
10~! ' ii' ,. \~'. I i I I , I I i I
80 iIl
:: jl1
v 50 1o JL 1
~T • \5 ~o ~ ,
I \
~ \30 ~
j
0,00 0.40 0.79 1.19 1.591.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE FROM SOURCE (MILES)
"&It. t-+
\+
\
\
+
\~
'Ibn· O··· ..{]
Ven • C:r-._.~190 ~
: 80 : /~
4-
170 ~-,
160 ~ -,1 \
150 "1 '+
14'0 : \
\
: 3 'J ~
120 ~\-1 ' ~ ....• I...J I
\
""0 1m 1 'oJ 1A 9c) ~
80 ~
70 160 ;5G ~
40 ~3 ,J ~ ... ~ _ - - .~ _20 - 2_""'-10~ ~ c:.
g - ---a- I II
- ·0- ~'.:."~ '- -6 - - - .§
\
0.:0 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.:8 3.:3 3.97 ~.37 4.77
JrS7ANCE ~ROM SOUR:E (M!~~S)
133
T~bl. C-7 Measured -:!ata for: test t2e
I I I I I I I I I I II
M.a5urea.nt • I 1 1 I 1 I 7 I I 10 I 11 I 12 1 13
I !r
I 1 I I I I I I
Distance fro. I I I I 1 I I I I I(ai1.s) 1 0.4 10.79 1.19 11.59 11.99 12.38 2.78 13 .18 13 •S9 13.97 1~·37 14.77
sourceI ! I ! I I I I ! I1 I I I I I I I I I
Van (V) I 48 40 12a.S 12 I 6 • 4 ! 22 I 31 36 I J8 I 33 I 24 1 14 1 o .3I I I I I I I I 1 !I I I I I I I I I 1
Vbn (V) I 48 44 I 36 22 I III .2 23 33 1 40 I 40 I I II I ! I I I I I II I I i 1 I I I 1
Ven (V) I 48 4S I 37 23.5 I ! . e 110.2 22.5 )3.5 1 41 I 42 I I II I I I I I ! I I II I I I I 1 I I I I
Vab (V) I I 4.6 I I 11 112. 4 I 11 7.2 2 ... I 2.4 I , I I 1I I I I I I , I I I I I, !
I I I I I I I I I I I IVbc: (?) I 1. S 1 0.5 1 1.1 I 1.4 I 1 .6 I 1.2 0.4 I 0.5 I 1.3 1 2.2 I I J
I I I I I , ! I I , , II
t
I I I I I ! I I I I I IV4C: (V) I 0.5 I I 112 .4 I 14 112 . 4 1 I 2.2 I I 9 .5 I I 1
I 1 ! I ! I I I I ! I I II !
I I I I I I I I I I I I II~ (&4\) 131. S I 56 I 70 1 77 I' 3 . 5 129 .8 117.5 I 5.2 124 . S I 42 161 .2 I '7' I SO. 5
I I I 1 I I 1 ! I ! I II I I I I I I I I I I I
Ib (aA) 110.5 I 21 140.2 154.2 161 .2 I S6 I 49 136 .7 119 .2 1 1. '7 I 1,1 I 1 I ! I I I ! I II
I I I I I I ; I I I I IIe (.A) 110.5 12:: .7 145.5 I 63 I ~ 3 . S I '70 161.2 145.5 126 .2 I 1 .7 I I
I I I I I I I ! ! !I !
134
+
, I
50~" ,
l' -. -0'J \, \
40
1 "\ \6\,
v 30 io 1
~ J
s 20 i1~
~
10 1
o~I.or-r-'"-"~i"T"I"""""'~i""T"""!""I""" ir-T'''''I~i """I,,..,,-r'"T'"i-,,...,,'r-T/..,.I'T"/.....' ""'1j-r,'T"'_I,...."j-r/"T"I-,,...,,'l""'Tj""i"T'"i"'-,'''''''-''j"T'"i-r-.,..,....,...,.......r'"I'"""P'~ ---.-~"'T'"-r-~
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.9/-4.37 4.77
DISTANCE FROM SOURCE (MILES)
Van z t--+
PH.A.SE CURRENTS (TEST#::) Vbn:l: C ..····O
Vcn· 6-·_·-6
0.40 0.79 1.19
O:STANCE FROM SOURCE (MILES)
90 J
80170
601I
SO jmA
40
130
I1,
20 i1 /
10 fJI1
o ~I i'
0.00
.C
I:JI
.I.I.
II
/+
/
,///
fI
0' /" //
'I'
;'
4.77
135
TAble e-a Measurlild d~tA tor: test i2d
I I i I I I i I IMeasurement • I I I I I I I I 10 II I 12 r 13
I I , I I I I ! I I
Olstanc.- fro. I 1 I I i I I I i I II o. .( 0.19 11.19 11.59 1 .9 9 12.38 \2.71 13 .1 a 13.59 13.97 14.37 14.77
sourc. (miles)! I I t , , I I I I!
I i I I I 1 I I 1'l~n (V) 43 1 16 17.5 I 46 I 6& 70 1 SO 21.5 \ ~ 0.6 I 37 I 63 I 76 1 80
I ! ! I I I I I II I 1 1 I I I I
Vbn (V) 46 I 34 20 \ I 15 23.5 1 19 12 I 6 . 2 I I \
I I I I I i I II I I I , j \ \
~lc:n (V) 46 I 36 24 I 8 .6 I e. 5 19 118.5 15.2 !10 .8 7.5 I I II ! I ! I I ! !I
I I I I I I IVab (V) 2.4 119.5 34 47 I 52 47 I 32 10.0 111.2 31 I I I
I I I I I I II i I I I I I
Vbe (V) 1.2 I 2.2 .( .5 6.1 I 6.8 4. a I 0.5 3. a I 7.6 10 .4 I I II I 1 I I I II I
I I I 1 I I I.Vac (V) 121 .5 38.5 S3 I 59 Sl I 32 7.2 11 a .5 41 I I I
I I I I I I I II
I I I I I I I IIa (aA) 154 I 175 168 I 133 192. a I 115 I 157 182 I : a9 lSi I 124 I 63 I 7
I ! I ! I I ! I !I j I I I I I I I
Ib (aA) I 21 40.2 147.2 I 56 1 1 2 . 2 112.2 7 I 3.5 :'.7 I I II I I ! ! I ! ! I! t
I I I I I I I I I!c (2A) 122. a 43 • ., 159.5 I 6J I) 1 . 5 129.5 17.5 110 • S 1 . ., I I I
I I I I I I I I ! II !
136
80
70
c- ",
'I i
\,\
\\\
\\\\-
L-'-'- iil_ _ \.,-.~\
Q .•\ ..... I
.. ?, ---Q"'0" u
, I ".' .1' Iii ,i. I I . I ':
+,I
,/
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.183.58 3.974.374.77
DISTANCE ~ROM SOURCE (MILES)
PHASE CURRENTS (TEST #2D)
'a _ -,-'0 ' ...
\\
\
\.
Van· +-+
Vbn• 0""'-0
Ven • 6-·_·-6
\\
\\+
+
I'
0 ..··· ....~ .... - - ~-.' 0 0 ~
o
i I Ii I Ii
~··· ....0 \
c
. 'I0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
JISTANCE PROM SOURCE (Mr~ES)
r-1.82"'-"lI 3.59 'hI.
II
I--_..~ '-l.l8."~
137
Tabl. C-~ Measu:-ed dat:a for test '3.
I I I I ! I i ! I IXe&5ur •••nt • I I I I I I 7 I I 10 I 11 1 12 I 13I
I I I ! I I ! I ! I IDistance fro. I I I I I I I I I I I
(aJ,1.5) 0 , 0.4 10.79 11.19 11.59 11.99 12.38 12.7& 3.18 13.59 \3.97 14.37 14 . "sourceI I I I ! ! I I I I II I 1 I I I I I 1 I 1
Van (V) 48 I 40 1 27 I 10 I 123.5 130.5 135.5 3& 1 32 1 2S I 13 I 0.6I I
, I I ! I I I I I,I 1 I I I I I I 1 I I 1
Vbn (V) I 6.& I I a.s I '.5 I 5.6 I I I 7 11.5 I 15 1 1 1I I ! I I I f I I I I I II I I I I i I I 1 I I I
Ven (V) , 5.4 I 3 .5 I 1 I I I I I I I II I ! ! 1 I I , I ! !I I I I ! I I 1 I I I
Vab (V) I 51 I 46 34 116.5 I 2.5 121.5 I 33 41 I 44 1 44 I II I I I I I I I I !I I I I I I I I I
Vbe (V) I 12 I 11 10 7 t I 1 1 I II I I t I I
,II I !
I I 1 i I I i I"I.e (V) , 44 I 37 25 10 I I I I I I
I I ! I I I I!
I I I I I I IIa (aA) 13a. 5 173.5 98 IlS I 115 173 .5 152.5 3S 12.2 I 35 154. :2 70 75.2
I I ! I ! I I !I I I I I I I
Ib ( aA) I 3 .5 11" . S 131.5 42 I 42 IS" • 2 49 24 .5 1'.5 I 1.7 II I I I I ! !I I I I I I
Ie (.A) I 1 .7 I I 1. , I 3 . 5 I II I I I
138
Iii I ii' i J i"
\\
\\.
iii
\
i ' .;. I Ii iii' '
n
n....
I/
!+\
50 i.,40j "\
l1
30 1
~ 1
~ 20 ~ \
! \l \
10 1 0' """o,,,oQ, __", \R.······· ··5.\~
j---6·- _ _ ~ ".
"0_ "o - -t:::--"iii I i ~ ii' , iii Ii' iii' I' I
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.91 4.37 4.77
DISTANCE FROM SOURCE (MILES)
PHASE CURRENTS (TEST #3A)
120 .r<.* +
//
\110 /100 /
90 \80
70 \m 60A a······· ,
Van· t-+
Vbn • 0 ..····0
Ven • 6-·_·-6
u·· ·· 0
50
40 J;1
30 -:
2°110 .
- - -e - - - ~_.- - .J.,- - - ~o I I ! ' Til Ii i I
"\
\:J
i I
.. a ..... ·· ..C,I'.; t '
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE FROM SOUR:E (M:LES)
I I
~LU ..;----.JI II. :.:1 "".
I
II
I---......,...·-1.18.~-,
139
'!'aole C-lO :1aasiJ:'''~ data tor test .3b
I I I I 1 I I,
I,
Me.surement • I I I 7 I I 1 10 1 11 12 I 13I I I I I I I I
i :Olstance troll I I ; I I I I I
(ailes) I 0 o•4 10.79 11.19 11 .59 11 .99 2.38 12. '8 13.18 13 .59 13.97 4 .3 i 14."';7source
I ! i 1 I I I II I I 1 I I I I
Van (V) I 4~ lS 21 I 50 I 69 72 53 I 23 I 10 I 39 I 60 72 I a4I I I I I , I!
I I I I I I IVbn (V) 116.5 10 10 I 16 I 22 20 112.5 I 25 I 36 I
I ! I I , I! ,:I I I I I I I
7cn (V) I 12 3 .5 I 10 I ' c: I I I I... J
[ I I I I I II I I I I I I
'lab (V) 31.5 I 20 I 21 I 36 I 4& 55 47 I 34 I 21 18.5 I I! ! I I I I I II I I i \ I I I
Vbc (V) 10 I I 7.5 I 6.6 I 7.2 I , I II ! I I I I ! I!
I I I I I I I I I'-lAC ('1) 32 I 14 Il7.5 I 40 I 55 I I I I I
! I I ! ! I I ! I
I I I I I I I I IIa (mA) 175 I 192 I 192 I 161 I 112 I 91 140 I 164 I 178 112. 2 124 164.7 \10.5
1 I I I I I I I I!
I I I I I I I I I:b (aA) 1. 7 110 .5 I 21 I 28 I : ~ . 5 59 • 5 54 140 . 2 122.7 I 1 • 7 I I, I t
, I I I !I I I i I
:c ~:lA) 1. 7 3.5 I 1 • 7 3 .5 3 .5 I I I II I I I I
140
70/~
+ ,
60
"i j iii, iii iii'
\ /\ /
~ ,I
\ /\ c /
\ /\ /
o \.JD ....·····
i i j iii iii i, iii iI ' I Ii
,.0···..·· :J
. ,~
o
vQ 50 I
~ 40 f'5 30 J \
! \20~ \p \ ,~ ..~~
10 J '''' ... o. ...... cI ....~. ,;'
Q ~ - _._ 6
'" iii ii' ii' iii 'I" i' i ii,
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE FROM SOURCE (MILES)
PHASE CURPENTS (TESi #38) Van· +-+
+\\.
200 I
190180170160150140130 ~
120 ~110 ~
~ 100 ~90 180 j78 .60 ~
~~1 0 G h O c20 l .
~ o·10 .h---~---~---~--'--c.°1 I
\/ \
I \/ \
of I
/ \
\/
\-.........,;f
I
\c \
:J. i!
G III
G /
,I i
f-\
\\
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE FROM SOURCE (MILES)
I i
~~~1 , II ' I. I
rLU""'1 . I3.S~ ~" II Ll8.,,-,
':'
141
'ri1bl. C-ll M.i1sul:'e~ data tor test t3c:
I i I I I I I IMeasur ••• nt • 1 I 1 1 I I I 10 I 11 12 I 13
I I I ! I I I ! i I
Distance frOID I I I I I I 1 , I Io• 4 0.i9 1.19 11.59 11.99 12.38 12.78 13.18 13 • S9 13.97 14.37 4.77
source (ai1es)I I I I I 1 I I
I I I I I I I iVan (V) 48 38 26. S 20 1 26 132 • 5 1 33 1 32 I 33 I 31 129 .5 I 28 27
I I 1 I I I I II 1 I I I I
Vbn (V) 10 10.5 10 I 10 I 6.5 1 I 10 I 16 21 I .I 1 I , I I
I 1 I I I 1 IVcn (V) I 3 .5 I 1 I I I . -I
I I I I , I !I !
I 1 I I 1 1 1 1Vab (V) so I 4S 34 1 21 11i.S 127.5 I 36 1 42 1 44 ~3 1
I I I I ~ I 1 II
I 1 I I I I I IVbc (V) 12 1 12 11 I 1 I I 1 I I
I I I I I I I !:I I I i I I I
Vile: (., ) 42 I 35 25 11i.5 I 21 I I 1 II I I ! 1 I J
1 I I i I I I II. (.A) 70 194 .5 11~ I 122 I 117 I 75 70 159.S 15 J • S 63 I 63 63 164.7
I I I I I I 1I I I I I I I
Ib (.A) ) .5 117 . 5 21 143 .7 I 42 I 5s . 5 56 13a . 5 21 1 • ., II I I I I I~ I I I I I
Ic: '.,\) 1 . 7 I I 1 . 7 I 3.5 I 1 II I ! I
142
I i" i i" i I
o
i i
.0
o
j , I ' Ii I i I "
········0··
" i' "
1i
"'0 ~£ 1
110k...' .., ..·0 ... ··iJ······o······c
p '" <; .e: _,6
i ~.... ...,o ~ ..... - -!r'-
II "I' iil"II" i "ill.ii!
50 ~ ..
1 \\~ \
40 ., \
1 \1 \
30 ~ \\V i \o 1 '-\LTS
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE tROM SOURCE (M~L£S)
?~:'SE CURRENTS (TEST #3C) Van· t-+
Von· 0 .. ···-0
Vc:n· 6-,-·-6
; 'i
co.·· "~
13°1' ~'\\~:: / \
j / \100 1 ,,/ \
::V/::1
140 1
30 j20 i c··· G
10k _--6o t""' - - - ~ _.- - ~ - _._.e..-
If i
mA
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.71
DISTANCE FROM SOURCE (MILES)
""'tooc-----o
I
I
[8G
". e
I
I, :---- 3.5' "':. ---......,.- I.J!"u.--:
143
Table :-12 :~ ~ ! S '.1 :- '= ~ ddta for :est *3d
I I I I I I l~.asurement f i I I I 7 I I 10 :1 I 12 1 13, ! I , I , I ,
! I
Olstance from 1 I I I I I I I !(1I11es) I o•4 10.79 11 .19 I: .59 11 .99 12.38 \2.78 I 3 .1 a 3.59 ! 3 .97 '4 .37 14 .77
sourceI I I I I ! ! I1 I I ! \ I I I
Van (V) 39 I 22 I 16 I 7.2 3 .2 :. 0 13 I 16 I 1 7 13.3 10 I 5.6 II I I 1 I I II I
I I I I I I IVbn (V) 39 123 .5 117.5 I 10 -t .8 I 10 I 14 I 16 16 I
t I I I ! I I II I I I I I ! I I
V=n ('1 ) I 36 ~64 I 4~2 I 536 544 I I I I II I I I I I ! I,
I I I I I I ! I"'"a0 (V) \ 2 . 5 6 • 4 I 5.! I 4. ! 4 • 8 " .8 \ 4 .4 I I ... a 5 .6 I I
I I I I I / ! II
I I I I I I I IVbe (V) 129 .5 264 440 I 544 560 I \ I I I
1
I 1 I I I II I i I I
'J~c (V) 30 260 432 544 552 I I I I I! ! ! I I
!
I I I I I:& (2A) 252 234 210 154 I a" . 5 21 Il7.5 I I ';.2 14 II s .2 1 .23 t 3 a . S
! I ! I ,I!
I I I I iIb (aA) 441 413 343 227 9a ~a 126 . 2 119.2 112 . : 1. , I I
! I I I I I,I I I I I I i I
1= (mAl IleSS 11750 11312 735 :ao I I I II , I I I!
144
...... " '.. I J ',.f II'... • ~. '" r ~ _.... . '". "J
.' .
, I, I'
. · ..0
i ,ii'ilillij'''' i
!'"1JODI :-,'
100 ,
~Cl~__""""'--......-..------r-,-':; 00 0 '0 0.7' I It I " I 19 , JI , '. ) It ] ,. J U • " • n
40 ~p\l '\1 .~
20~ Q~< ..] ".
""-,G.
1 ~ . .. 0
o~"rj"T"f"'1""",..., ,.....,'f"""lj-""'Ti"T''''''"T''''''''1T"""iir-t,-r''''-'-j''''l"'1""".,...i,.....,f"""ll,-r'"'TI-r."T'""'1""",.....,r-:i.-ri"'Ti-ri"T"j""l"""',...,r"',il'""'T'""T"'"P"'T'"'T""'T'""r-T"""l'"""T"""f"'~T-r-I~""-'-~"""""'~"""""~
60vaLTS
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE FROM SOURCE (MILES)
PHASE CURRENTS (TE3i #3J)
v . +-~1900 ~ an
1800~
Vbn = 0······01700 , V . 6-·_·-61600 , en
1500 ,,1400 \
1300 l.\
1200 \
"1100 \
1000\
m ,A 90a
\
\
800 1 \
s700 \
6001 \1
500 bCj
400 : s300 s:200 l100 1
i.l ~io ~i 'I i
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.78 3.18 3.58 3.97 ~.37 4.77
DISTANCE F"RQM SOURCE (MILES)
145
•, 8
o
-162m;.~
----~~,,,,i.----'.I="". ~
'-.coo-----~
':'able C-13
~17 91
770 30!
I175 192.7
!
Measurement •
Distance fro.source (ailes)
Van (0")
Vbn (V)
v e n (oJ)
'lab (V)
Vbe: (\')
"lae: ('/)
Ie faA)
r e (sA)
38
39
35
II 30II1 31IIt 308I
II 441II1:'830
I
II 0.4II110.5I
21
260
10.6
264
II 264I
iI 2a7III 420II11750
I
I10.79I
I110.4
432
18
432
432
252
350
i::295I
I11.19!II 26III 1.2!II 523ItI 2SIII 536,
520
I11 .59III 35I
II 52!II127.5II1 52!!II 512
II
11.99I
39
13
27
63
a . i
I12.38III 28III 10I
II 11I
tI a7 • 5!
I12.751
II 11I/I 5.5I
I1 6.6I
II 981
IIS.2I
I13.18II1 6.4II1 2.4,
II 6. aI
I199.7III 3.5I
II 10I1I) .59I1120 • 5I1I 5.2I
17
11
I13 .97III 26I
152.5
II 12I1/4.3"7III 34I
28
II :'3II14 .77III 48I
146
80
, .....( C 'J '" J'~"" ""'-'a.c.L j •.• _ .... ,~
'001'col
.QO Io I• toO,.
Ii'
o
'~
····O.~..······0
.....-G 0
o
~:l~'__-,-..-r---.- ---.._~~ _
o OQ 0 .~ C 7t I 19 \ " I " l.ll J " , a J " J " , " • 11
OU,..C& tal. 10\1-'1 1.1~ISJ
0····'1""'1""1/"11111 If I
vaLTS
0.00 0.40 0.79 1.19 1.59 1.99 2.3S 2.78 3.18 3.58 3.97 4.37 4.77
DISTANCE FROM SOURCE (MILES)
PHASE CURRENTS (TE3T #2E)
Van· t-+
Vbn• 0···· ..0
Vcn· 6-·_·-6
0· ......·..···0 ........··D ........·..·G·· .. _r " I' ! i' , ii i
\\
\
\
.1
, I
1900 -b."1800 i1700 1 l..,
1600 j \1500 \1400
1300112001
1100 im 1000 1A 900 1
800 -1700 j
600 iseo ~
400 f ..·····0300 i
200 J1
100 jO_~...,...,.....,...,...,-.-r"""""~"""'"T"''''''''''~T''''''"'''1r-'1'"''''1'''-'-~'''''''''''''r'T''"'-''''''''~r''''T''T'''''''''r-''"'''rT"T'T'''''-rT'....--o:----~----,.-
0.00 0.40 0.79 1.19 1.59 1.99 2.38 2.733.18 3.58 3.97 .37 4.77
DISTANCE ~ROM SOURCE (MILES)
147
I---- ,3.5' ""i. ----t.16.~._i
rfJ "o-g----.,~
I - I.8l "';. ---i(~-:»
'!'abl. C-14 ~eaSt;:-e= :l:a for test t3f
: I I I IKeasureDlent • I I I 7 I I 10 ; 11 I 12 :'3
• J.. I I I I I\ !
:;)lstanc:e fro. I i i I i I I I I I I
(a11.s) I 004 10. i9 11.19 11 .59 i 1 .99 12.38 12.78 13 01 a 3 .59 13.97 14 .3 i 14oj;source
I I I I I I I ! !i I I I I I I I
Van (V) 38 19 ! 14 110. Ii i 13 :5 I is I 15 I 15 14 .6 111.5 111.5 I 14; I , I I I I II I I I I I I I
'ibn (V) 39 122.5 115 . 5 I a 04 I I 10 112.4 113 .5 13 II I I ! I I I
!
I I i I I Iv c n (V) 3: 256 42'; I 52e i 536 I I I
I ! 1 I II I I I I
Vab (V) 205 804 10 110 0a 110 . 5 I 5.6 I 3.2 I 9 • 4I I 1 II I 1 I
Vbc: (V) 23 260 ~2~ 52! 536 I I II I !
,I
1 I I
Vac (V) 30 256 42.- 528 525 I I I II I I !
: I I I II. (sA) 245 227 199 147 ii 129 • '7 1 3S r 3S 3S 131 • 5 I : ( 05 11 S .:2 35
I ! I I !!
I I I 1::b (sA) 434 ~O, 343 224 I s 2 • 2 ! : 2 . , 1:2 2. 7 117.5 a 0 i 3 • S
I I I!
I I i j II z (cA) !1855 \1630 I 1):' 2 :52 308 ! I
I I ! ,,