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 ! ,,