Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Communication frequency response of high voltage power lines Naredo V., José Luis A. 1987

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


831-UBC_1987_A7 N37.pdf [ 4.05MB ]
JSON: 831-1.0097155.json
JSON-LD: 831-1.0097155-ld.json
RDF/XML (Pretty): 831-1.0097155-rdf.xml
RDF/JSON: 831-1.0097155-rdf.json
Turtle: 831-1.0097155-turtle.txt
N-Triples: 831-1.0097155-rdf-ntriples.txt
Original Record: 831-1.0097155-source.json
Full Text

Full Text

COMMUNICATION F R E Q U E N C Y RESPONSE OF HIGH V O L T A G E POWER LINES by  JOSE LUIS A. NAREDO V. B. OF ELEC. ENG., UNIVERSIDAD  ANAHUAC, MEXICO D. F.,  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING  We  accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA APRIL  1987  © JOSE LUIS A. NAREDO V.,  1987  1984  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  DEPARTMENT OF ELECTRICAL ENGINEERING The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: APRIL  1987  A B S T R A C T  Several  methods  for calculating  the  electrical  phase  and modal parameters of  overhead transmission lines are described in this thesis; then, a graphical method for evaluating communication frequency response of delta transmission lines -based on the guidelines given by W. H. Senn [12,13,14]- is developed. The graphical method, combined with the parameters calculation methods, obviates  the need of  large mainframe computers for the analysis of power line carrier (PLC) systems.  A new technique for assessing coupling alternatives, based on Senn's method, is developed.  The technique is applied to generate coupling recommendations; it is  found that many of the current recommendations  given elsewhere [21]  reliable.  Finally, future work to be done in this field is proposed.  u  are not  TABLE OF CONTENTS ABSTRACT  ii  TABLE OF CONTENTS LIST OF FIGURES  iii .  v  LIST OF TABLES  vii  ACKNOWLEDGEMENTS Chapter  viii  1. INTRODUCTION  1  Chapter 2. ELECTRICAL CHARACTERISTICS OF OVERHEAD TRANSMISSION LINES 4 2.1. Electrical Parameters of Transmission Lines 5 2.2. Earth Impedance Calculation 6 2.3. Impedance Correction Due to Grounded Ground Wires 10 2.4. Example of Electrical Parameters Calculation 13 Chapter 3. MODAL ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 17 3.1. Modal Solution of the Propagation Equations 18 3.1.1. Equation for voltage 18 3.1.2. Equation for the current 21 3.1.3. Nonhomogeneous transmission systems representation. 22 3.2. Numerical Computation of Power Line Eigenvalues and Eigenvectors 23 3.3. Modal Parameters of Delta Transmission Line Configuration. .. 25 3.3.1. Example of eigenvalue/eigenvector calculation in a delta line 27 3.4. REMARKS 28 Chapter 4. GRAPHICAL METHOD FOR PREDICTING FREQUENCY RESPONSE OF DELTA LINES 30 4.1. Reflectionless Wave Propagation 31 4.2. Supplementary Losses in Delta Transmission Lines 34 4.2.1. Homogeneous lines 36 4.2.2. Transposed lines 37 4.2.3. Insertion loss calculation : 41 4.2.4. Modal cancellation poles 43 4.3. Senn's Method for Evaluating Insertion Losses 45 4.3.1. Example 48 4.4. REMARKS 54 Chapter 5. COUPLING RECOMMENDATIONS 5.1. Proposed Method for Assessing Coupling Alternatives iii  59 62  5.2. Coupling Considerations for Common Line Cases 5.2.1. Untransposed lines 5.2.2. Single transposed lines 5.2.3. Lines with two transpositions 5.2.4. Phase to phase coupling 5.2.5. Lines with three transpositions 5.3. Nonconventional Couplings  66 68 71 73 73 75 77  Chapter 6. CONCLUSIONS 6.1. Future Research  84 85  APPENDIX  9  2  REFERENCES  8  9  iv  LIST OF FIGURES. FIGURE 2.1)Conductors above a perfect conducting ground 7 FIGURE 2.2) Complex depth of penetration 9 FIGURE 2.3) Example of a 500 kV delta line 13 FIGURE 4.1) Example of carrier coupling on a nontransposed line 39 FIGURE 4.2) Example of coupling on a single transposed line 39 FIGURE 4.3) Transmission line transposition layouts most commonly found in practice, a) Untransposed line, b) One transposition, c) Two transpositions, d) Three transpositions unequal spacing 42 FIGURE 4.4) Plot of the function A = 201oglo|(X2-6X-3)/8| a) Three dimensional graph, b) Contour representation 46 FIGURE 4.5) a) Graph of Aa-AG for different frequencies and different earth resistivities, b) line data for the graph 4.5a 49 FIGURE 4.6) Example , 50 FIGURE 4.7) Example, a) Modal curve, b) Theoretical minimum attenuation. ...51 FIGURE 4.8) Contour maps with modal curve superposed. a) Coupling Ct=(l,0,0)/C = (0,0,l). a) Couplings Ct=(0,l,0)/C = (1,0,0) and Ct= (0,0,1)/C =(0,1,0) 52 FIGURE 4.9) Line response for different couplings 53 FIGURE 5.1) a) Transmission line layout example, b) Coupling alternative 1. c) Alternative 2 59 FIGURE 5.2) a) Coupling 1 contour, b) Coupling 2 contour, c) Frequency responses 61 FIGURE 5.3) a) Example of a two color plot, b) Example of a three color plot. 63 FIGURE 5.4) a) Example of a feasible regions map 65 FIGURE 5.5) a) Effect of the earth resistivity on the modal plots 67 FIGURE 5.5) b) Effect of the conductors medium height on the modal plots. .. 67 FIGURE 5.6) Recommended phase to ground coupling for untransposed lines. ... 68 FIGURE 5.7) a) Recommended phase to phase coupling on untransposed lines, b) Second best coupling 69 FIGURE 5.8) Comparison of the couplings depicted in figures 5.7a and 5.7b. a) Two color plot, b) Three color plot 70 FIGURE 5.9) Recommended phase to ground coupling for single transposed lines. 71 FIGURE 5.10) Phase to phase recommended coupling for single transposed lines. 72 FIGURE 5.11) Couplings that should be avoided 72 FIGURE 5.12) Recommended phase to ground coupling in a double transposed line 74 FIGURE 5.13) Second best coupling 74 FIGURE 5.14) Two color plot comparing couplings depicted in figures 5.12 and 5.13 74 FIGURE 5.15) Complementary phase to phase coupling on a double transposed line 76 FIGURE 5.16) Recommended phase to phase coupling for a double transposed line 76 s  r  r  r  v  FIGURE 5.17)  Second best phase to phase coupling for a double transposed line. 76 FIGURE 5.18) Phase to ground recommended coupling for a three transposed line 78 FIGURE 5.19) Phase to phase recommended couplings for a three transposed line, a) Best coupling, b) Second best coupling 78 FIGURE 5.20) Contour map of mode 1 coupling on a double transposed line. . 80 FIGURE 5.20) Contour map of mode 1 coupling on a three transposed line. ... 80  vi  LIST OF TABLES TABLE TABLE TABLE TABLE TABLE TABLE  4.1) Polynomials of untransposed lines 4.2) Polynomials of single transposed lines 4.3) Polynomials of double transposed lines 4.4) Polynomials of three transposed lines 5.1) Polynomials of two transposed lines.nonconventional couplings 5.2) Polynomials of three transposed lines.nonconventional couplings  vii  55 56 57 58 82 83  ACKNOWLEDGEMENTS  To my parents Antonio Naredo and Caritina Villagran for their  encouragement  and their support that far exceed the mere parental duty.  To Professor Wedepohl (whose name is not Professor but Martin) for his endless patience  at sharing his knowledge  and his teachings  Engineering.  To  that trascend the field of  '  Professor Dommel for his readiness  to help as well as to share his vast  experience.  To  my former colleagues:  Leonardo Guardado, Felipe Gutierrez, Pablo Moreno,  Ricardo Romero and Jose Luis Silva; from whom I have received the benefits of our common professional interests.  To the National Research Council of Mexico (CONACYT) for the provision of a research studentship.  To the Instituto de Investigaciones Electricas de Mexico (HE) for the granting of a  leave  of  absence  along  with  the  provision of  through the Bank of Mexico.  viii  supplementary  financial aid  C H A P T E R 1. INTRODUCTION.  The  use of power lines as a medium for conveying communication signals goes  as far back as the early 1920s. This technique known as "power line carrier" (PLC),  when  properly designed,  is  highly  reliable; probably the  most  reliable  communication system on a single link basis, because of the ruggedness  of the  power lines.  The lack of understanding of the propagation phenomenon in multiconductor lines, on the other hand, previously precluded the better use of P L C systems. In the early days of power line communications, the conventional analytical tools of the power engineer were of no help in dealing with its analysis. classical  line  unidimensional  theory, lines.  used In  by  communication  engineers,  1963,  Wedepohl  the  set  for  basis  Neither was the it  only  for  considers  dealing  with  multiconductor lines [1]; this led him and his coauthors [2,3,4,5] to developing a new multiconductor line theory.  The  multiconductor or modal line theory is able to explain the phenomenon of  high  frequency  practical  electromagnetic  waves propagating in power  application to P L C design  is  still  not  very  lines;  however,  generalized.  its  One reason  might be that the theory is fairly new and it is still spread over several dozens of  papers.  Another reason  might  be  that  the  theory  makes  use  of matrix  calculus that may be intimidating for the uninitiated. As a consequence, misconceptions, day;  several  prior to the development of the theory, have persisted until this  some of them, when pertinent, will be refuted in this thesis. Another source  1  2 of misunderstandings is the oversimplification of the modal theory; this has also led to fallacies.  In this thesis, a series of results using multiconductor line theory are presented, with the intention of improving the understanding of the propagation phenomena in power lines and, at the same time, to providing a simpler way of analyzing power  line  carrier  systems.  Additionally,  a  new  technique  for assessing P L C  couplings is proposed and -by means of it- new results are obtained.  In chapter 2 recent advances in the calculation of electrical parameters of aerial lines  [6,7,8,9,10] are presented.  Then, in chapter 3, after  a summary of the  modal theory, some practical aspects concerning the calculation of the propagation modes of the line and their propagation constants are presented. Special emphasis is given to a set of formulae for calculating modal parameters of delta lines; they  were first published in Wedepohl's  1963  paper [1] and, since then,  they  have been largely overlooked. The simplicity of these formulae, together with the procedures of chapter  2 for calculating electrical line parameters,  suggests the  possibility of performing line analysis by means of programmable calculators.  It should be mentioned at this point, that the developements  -either in modal  theory or in electrical parameters calculation- are relevant to several other areas of power system analysis, such as: Transient behaviour of transmission systems [9,11]. Corona noise performance of power lines. Fault detection and location in transmission lines.  3 After  chapter  3,  the  attention  is  focused  on  three-phase  lines  in  delta  configuration. There are two reasons for this. The first one is that this type of lines is, if not the most common, among the most commonly found in practice. The second reason is that the delta lines and specially the horizontal one, which is  a  particular case  of  the  delta,  are  the  ones  that  present  the  worst  propagation problem -the so called modal cancellation effect [13].  A  simplified, but powerful method, is  the  one  proposed by  Walter H. Senn  [12,13,14]. The method presented in chapter 4 may be considered Senn's method, since  there  is  a  coincidence  between  the  results  derived from  it  and those  published by Senn. Whereas the line analysis based on the general modal theory requires lengthy computer programs hosted on a mainframe, Senn's technique is graphical; therefore, it is more suitable for field engineering work. It will become apparent, further on in the thesis, that both methods are necesary.  Senn's chapter  method 5,  is  used  as  the  basis  for  a  technique,  for comparing P L C coupling alternatives.  which is  proposed in  The technique is applied  there to make coupling recommendations for common line transposition schemes, as well as to analyze the performance of non-conventional couplings.  Finally, the conclusion of the work is given in chapter 6. Suggestions concerning future work to be done in this field are also included there.  C H A P T E R 2. E L E C T R I C A L CHARACTERISTICS OF O V E R H E A D TRANSMISSION  The  propagation  of  electromagnetic  waves  LINES.  on  overhead  transmission  lines  is  accurately described by means of the following differential equations:  dV  • ,  I I  a* ' and  -  (2.2)  d Y  known as the Telegrapher's Equations.  The  propagation phenomenon in power lines involves usually several conductors as  well  as  the  relationships and  earth [1,2].  plane;  thus,  equations  Z and Y are respectively  2 . 1 and the  2 . 2 are multidimensional  matrices of series impedance  of shunt admittance; both are given in per unit of length. V and I are  vectors t  composed of the voltages with the ground plane as reference  and of  the currents on each conductor of the line respectively.  Before  attempting  to  solve  equations  2 . 1 and  2 . 2 , the  electrical  parameter  matrices of the line, Z and Y, must be obtained.  t NOTE: The terms vector and column matrix are used here indistinctively. 4  5 2.1. E L E C T R I C A L  P A R A M E T E R S OF TRANSMISSION LINES.  The series impedance per unit of length of a transmission line can be considered as composed by four terms as follows [8,9]:  H = Zfr + l f + Z c Zg  depends  only  on  the  line  +  2  e  (2.3)  W  geometry;  it  is  therefore  called  impedance and it is related to the Maxwell's potential coefficients  geometrical  matrix "P" in  the following way [2]:  Z Z  e  0  - - J £ J £ P  ;  (2.4)  is the additional impedance due to the finite conductivity of the earth; Z  c  is  the impedance due to the conductors; and Z ^ is a term due to the presence of grounded ground wires.  The  shunt admittance per unit of length of an aerial line depends practically  only on the capacitance between the conductors and the ground plane. It is also related to the Maxwell potential coefficients matrix in the following way [2]:  Y = j v l i r e P ' *  2.5)  (  or, if the line has grounded ground wires:  Y  = jw  2rre  (p  +JP  GW  )  -l  .(2.6)  6 The  calculation  of the  general as indicated recently;  two  of  electrical  parameters  them  will  be  presented  here.  other one  wires impedance explicitly, as i n equation 2.3  The  lines is carried  in reference 2; however, some changes have been  impedance calculation [6,7,16]; the  2.2.  of aerial  One  consists  deals  with  out in suggested  the  earth  in expressing the ground  [8,9,10].  E A R T H IMPEDANCE C A L C U L A T I O N .  self  and  conducting  mutual  ground  (see  impedance figure  terms  2.1)  are  of  two  conductors  given  respectively  above by  a  perfectly  the  following  expressions:  (2.7)  (2.8) These terms are readily obtained by  means of the well known method of the  images, which is illustrated by figure 2.1; they correspond to the elements of the  7  geometrical impedance matrix.  2  «m i ~ t h conductor  r  j-th conductor  Ground plane  hi x i  3  Conductors images. FIGURE 2.1)  If the ground  is not a  Conductors above a perfect conducting ground. perfect conductor, the above expressions  have  to be  modified. Carson demonstrated [15] that the impedance could be expressed as the geometrical terms  2.1  and  2.2 plus correction terms; furthermore, he also found  that -after several simplifying assumptions- the correction terms would be given, for the self impedance by the following integral:  6&.  TT  and, for the mutual  •(2.9) impedance:  8  ^  1  Jo  «Wof *^^  (  2  where " CP" is the earth conductivity in Siemens/m and ' V is an integrating parameter.  Carson  integrals  cannot  be  solved  analytically.  Tables  as  well  as  series  expansions have been used in the past to handle them; more recently, they have been  dealt  with  by  means  of  numerical  integration.  A  remarkably  good  approximation to the Carson integral, which is also very simple to evaluate, was proposed by Dubanton in 1969 [6]. As the term  \j j w p . cr  that  appears  very  often  at  skin effect  studies,  plays  the  role  of  a  mean  penetration depth for fields and for currents, it seemed logical to place the plane of symmetry  of the images at  a complex  distance  "p" underneath the earth  plane in order to account for the effects of the finite ground conductivity; this is illustrated in figure 2.2.  For the self impedance, the following expression is obtained:  tit  = i^ii V  9  .(2.11)  dij 2r v  Ground plane Plane of symmetry  i' Conductor images.  j'  FIGURE 2 . 2 ) Complex depth of penetration, Since the first term corresponds to the geometrical impedance  (expression  2.7),  the second term must account for the earth correction factor. Dubanton used this method only for the self impedance calculation. In 1976  C. Gary extended it to  the calculation of the mutual impedances [7]:  1 w  2-TT Gary  made  also  1  J  5~  \dij J  fcir  comparisons  between  |_ the  T)^ above  correction formulae  .(2.12) and  the  10 numerical  calculation of the  Carson integrals.  The results  from  both  methods  were so similar that he suggested that the complex depth formulae could be the exact solution of the Carson integrals [6].  In 1981 Deri and Semlyen derived the above formulae as approximated solutions of the Carson integrals [16]. that the errors are negligible  They evaluated also the figures of error, finding at most frequencies,  except for a narrow band  where they become more noticeable. Inside that band, however, in most practical cases the error will be below  The error studies as, knowledge  of the  Dubanton-Gary formulae can be dismissed  in general, of the  3%, and in the worst case it will not exceed 9%.  it is  smaller than the  physical parameters  of the  one  in most practical  introduced by the limited  transmission lines  -specially  the  earth resistivity along a line.  2.3. I M P E D A N C E CORRECTION D U E TO GROUNDED  GROUND WIRES.  For a transmission line with " m " phase conductors and " n " ground wires, the matrix equations 2.1 and 2.2 can be partitioned as follows:  1  \rt\rn  (  2m « 1— L rr>  I  and  JU  I*  .(2.13)  11  dYIrr, m  .(2.14)  div,  Usually, the ground wires are not directly involved in the transmission communication  signals or of electric power; thus,  voltages or currents may  of either  the explicit knowledge of their  not be required; however, their influence must be taken  into account.  The be  ground  wires are usually grounded  assumed  dV /dx  =  n  that their voltage 0). Only  when  at each tower. In most studies, it may  profile all along  the separation  the line is zero  between  towers  (i. e., V  is close  n  and  to an even  multiple of half a wavelength, the assumption of zero voltage is not valid and a more complex solution due to Wedepohl and Wasley the  former  approach the ground  has to be used  wire terms are easily eliminated  [17]. With  from  equation  2.14 by erasing the last n columns, as they are multiplied by zero, as well as the last n rows, since they are not required:  Ah.  The  =  reduction  that dV /dx n  Y  of equation =  M  m  V  .  (2.i5)  2.13 requires some extra work. Under the assumption  0, the following expressions  are obtained  from 2.13:  12  m  .(2.16).  .(2.17) Expression 2.17 can be used to eliminate I  n  from equation  2.16  die  .(2.18)  The term "Zgw"  of expression 2.3 is thus:  m  It can be  shown in a similar way  .(2.19)  that the term  "Pgw  of expression 2.6 is  given by:  m  Another method for reducing the Z  matrix consists in inverting it; then as the  following relationship holds:  1, the last n  0 columns and rows can be ehminated in the same way  .(2.20) as it was  done  13 for the admittance matrix; the inversion of the reduced matrix will yield the impedance matrix with the ground wires term implicitly incorporated. This method is the traditional one [2]. In the next chapter it will become apparent that the explicit method has more advantages.  2.4.  E X A M P L E OF E L E C T R I C A L PARAMETERS  As  an example the electrical parameters  CALCULATION.  corresponding to the line depicted in  figure 2.3 are provided next.  Phase conductors. Outer conductor medium height. Central conductor medium height. Horizontal distance between conductors. Conductors. Bundle diameter.  15.24 m 23.62 m 6.248 m 2xll52-ACSR 0.45 m  Ground wires. Medium height. Horizontal distance Radius. Material  Ground resistivity.  36.17 m 7.874 m 0.489 cm Alumoweld 100.0 ohm-m  Frequency.  500.0 kHz.  FIGURE 2.3) Example of a 500 kV delta line.  Geometrical Impedance. (Ohm/km) 3571.89  833.05  609.02  561.47  509.96  833.05  3847.20  833.05  952.75  952.75  609.02  833.05  3571.89  509.96  561.47  561.47  952.75  509.96  6033.09  1397.19  509.96 .  952.75  561.47  1397.19  6033.09  Earth Return Impedance. (Ohm/km) 117.556  94.818  106.420  75.922  73.854  94.818  81.807  94.818  66.365  66.365  106.420  94.818  117.556  73.854  75.922  75.922  66.365  73.854  56.134  55.595  73.854  66.365  75.922  55.595  56.134  "142.927  110.457  124.047  85.918  82.969  110.457  93.522  110.457  73.896  76.896  124.047  110.457  142.927  82.969  85.918  85.918  73.896  82.969  61.476  60.774  82.969  76.896  85.918  60.774  61.476  Conductors Impedance. (Ohm/km) 1.330  0.0  0.0  0.0  0.0  0.0  1.330  0.0  0.0  0.0  0.0  0.0  1.330  0.0  0.0  0.0  0.0  0.0  12.756  0.0  0.0  0.0  0.0  0.0  12.756  1.330  0.0  0.0  0.0  0.0  0.0  1.330  0.0  0.0  0.0  0.0  0.0  1.330  0.0  0.0  0.0  0.0  0.0  12.756  0.0  0.0  0.0  0.0  0.0  12.756  Ground Wires Impedance. (Ohm/km) 22.924  28.450  22.877  28.450  31.450  28.450  22.877  28.450  22.924  100.886  167.471  100.249  167.471  277.995  167.471  100.249  167.471  100.886  Reduced Admittance,  (milli-mhos/km)  33.101  -5.635  -3.918  -5.635  32.613  -5.635  -3.918  -5.635  33.101  C H A P T E R 3. M O D A L ANALYSIS  OF MULTICONDUCTOR  TRANSMISSION  LINES.  The propagation equations 2.1 and 2.2 can be transformed as follows:  dx  1  2YV  (3.1)  YZI Each  expression represents  (3.2) a  system  of n  differential  equations, where each  equation involves all the n variables of voltage or of current.  The  modal approach for solving either 3.1  or 3.2  consists in transforming the  systen of n coupled equations to an n unidimensional or uncoupled solution is straightforward. This approach, mathematical  standpoint,  provides  a  system whose  apart of being convenient  valuable  physical  interpretation  from  the  of  the  propagation phenomenon.  In the first section of this chapter the results of the modal theory, that are relevant to the thesis, will be summarized; for detailed explanations, as well as for the proofs of these results, references 1, 2, 3, 4, 5, 18 and consulted. In section  3.2,  19 should be  practical aspects concerning the calculation of modal  parameters are presented. Section 3.3  is devoted  to the special case of modal  parameters of delta lines, which is central to the thesis.  17  18 3.1.  M O D A L SOLUTION OF T H E PROPAGATION  EQUATIONS.  3.1.1. E q u a t i o n for voltage.  Expression 3.1 is transformed into a decoupled system of equations by means of the matrix M , which diagonalizes the Z Y matrix product as follows:  .(3.3) where  ^  is a diagonal matrix, whose elements are the eigenvalues of Z Y and  the columns of M are its eigenvectors - also known as modal vectors.  Any  vector V of phase  voltages  may be regarded as  an assemblage  of the  eigenvectors. Let M j the i-th column of M , then:  where  is the contribution of the i-th mode to V . It follows then that the  transformation :  converts any vector V from the phase domain to the modal domain.  By applying 3.3 and 3.4 to 3.1:  19  dx As  A  1  .(3.5)  is diagonal, it is clear that expression 3.5 is already the desired system  of n decoupled equations. Its solution may be written as follows:  .(3.6) where exp(+/-X* x) is shorthand for: 3  O  t*,X  and  Vjjj  and V  m  g are integration constant  knowing the value of V ( x ) m  Yj  at two different points.  plays the role of the propagation  " OC j "  represents  vectors, which can be determined by  constant  of the i-th mode; its real part  the attenuation, and its imaginary  change of phase, which  part j3 j represents its  is related to the mode velocity. The presence of the  positive exponential term i n solution 3.6 is physically interpreted as a reflected wave traveling backwards.  Expression  3.6 may be transformed  to the phase domain by applying the inverse  20 of relation 3.4 to it  (t) = M tit (-Fx) M ^ V  +• jH e* (uV)N~ V l  F  f  B  (3.7)  If the following definition is introduced:  (3.8) it may be shown that expression 3.7 becomes:  (3.9) which is the  solution to equation 3.1.  matrix functions  Expressions 3.6  and of matrix calculus concepts;  and 3.9  details  make use of  about them may be  found in references 18 and 19.  In the  same  represents traveling  way  a wave  as in expression  3.6,  the negative  traveling forwards, and the  backwards.  Here  also,  the  exponential term of 3.9  positive  integration  one  constant  a reflected vectors  wave  may  be  determined from the knowledge of V(x) at two different points along the line. At the beginning of the line (x = 0) 3.9 becomes:  At x  = I (3.10)  solving for V g  Now, for a semi-infinite line, taking the limit as  :  replacing this result in 3.10:  V  F  =YCO)  Thus for the semi-infinite line, expression 3.9 becomes:  V(x)  (3.11)  This result is consistent with the physics of the propagation phenomenon, in the sense that an infinite fine does not produce reflected waves.  3.1.2. Equation for the current.  The  equation of currents 3.2  may  be solved either directly, as  the  voltage  equation, or from the voltage solution. Both approaches are complementary.  The first approach helps to establish the relationship between voltage modes and current modes. Let N be the matrix that diagonalizes YZ as follows:  22 In the same way as with the voltage equation, N is the matrix of modes of current, and 7\ ' is the diagonal matrix of eigenvalues.  It may be proved [19]  that:  ova E ' =1M  A' = A  The  second  approach  leads  1  to  the  concept  of  T  multidimensional  characteristic  admittance. Rewriting 2.1 as follows:  and from the value of V(x) obtained in 3.9:  1<*) = T ' f ' [ « p (- fx) V F - « p ( ^*)V ' a  Here, the term Z"^ {j/ plays the same role as the characteristic admittance in the unidimensional case; therefore it is refered to as "characteristic admittance" and it is denoted by Y . Its inverse, the characteristic impedance, by Z . c  c  3.1.3. Nonhomogeneous transmission systems representation.  The solutions of the voltage propagation equation 3.1 and of the current equation 3.2  lead  to  the  two  port representation  example, the chain matrix form:  of  a homogeneous  line  section;  for  23  (3.12)  or, the nodal form:  Vo'  'lo'  .(3.13)  J Non-homogeneous transmission and  inhomogeneities  part  is represented  (such as  systems may  be  broken into homogeneous sections  as, transpositions, faults, lumped a  two  port  network,  and  the  elements, etc.), each  two  port  sections  are  combined according to the system layout.  It seems, at  first  glance,  that  the  of transmission  mostly  of cascaded  matrix  representation; however, because of its poor numerical  form 3.13  sections is more  modeling  conveniently  done by  systems  means  composed  of the  chain  stability, the  nodal  is preferred [18].  3 . 2 . N U M E R I C A L  C O M P U T A T I O N  O F  P O W E R  L I N E  E I G E N V A L U E S  A N D  E I G E N V E C T O R S .  For  a transmission  line without  from relation 2.3, as follows [2]:  ground wires, the ZY  product  can  be  expressed,  24  zy »z»y +(z+Zc)Y ,  (3.14)  £  but from 2.4 and 2.5:  7 Y --fflVeU" where U denotes the unit matrix. If instead of ZY, only the second term of expression 3.14 is diagonalized:  (z z )y= E+  c  M"A7M")"\  it may be shown that the eigenvectors do not change [2]:  IH" = Itt and  that the new eigenvalues  7K-" are related to the Z Y matrix  eigenvalues  as follows:  A - A" - »>* There  are some  numerical  U  advantages on dealing  with  expression  than with ZY. A s the elements of Z g are much larger than those  3.15 rather of Z  e  and  Z - i t is not recommendable to form the Z matrix, for the information conveyed c  by Z  e  and Z  c  may be lost by numerical  all numerically close to  truncation. The eigenvalues  of Z Y are  -CO jo.6. The removal of Z g thus solves the numerical  truncation problem as well as accelerating the convergence.  25 A lossless line has repeated eigenvalues  and propagation constants:  where  c  is  the  speed  of  light.  It  follows  then  that,  in  addition  to  the  attenuation, the losses reduce the wave velocity slightly.  Where  ground wires  are concerned, the expression  analogous  to  3.14  takes a  somewhat more complicated form [8,9]:  ....(3.16)  3.3.  MODAL  PARAMETERS  OF  DELTA  TRANSMISSION  LINE  CONFIGURATION.  One  of the most common line configurations is the delta (see  figure 2.3). The  ZY product of this type of line is of the form:  r  1Y =  a d  b  c  e  d  b  a  .(3.17)  This form is valid also when the line has one ground wire at the center, or two of them located symmetrically with respect to the vertical axis of symmetry  26 of the line. The horizontal configuration is a particular case where all conductors have the same height above ground.  The  delta configuration  reference  is a special case of "odd" symmetry  [1]; where the derivation of analytical formulae for the eigenvalues and  eigenvectors  of 3.17 are also given. It may be shown that:  l  i  p  o  ^  i and  as described in  .(3.18)  corresponding:  A =  "fc,  0  0  0  U  0  0  0  a  3  with  e - o - c  - \/(a-rC-e)^-f  8bji  P =  - a - c  X = o4 c+ b  + \ / ( a t c-e?-+  8bd  .(3.19)  .(3.20)  .(3.21)  27  (3.22)  a-c  A  In  the  event  3  - a + c -+  that  the  line  (3.23)  is  completely  symmetrical,  with  e  =  a and  b = c = d: A i  =  X 2 ~ a — c, p = -2 and q = 1,  M becomes then a true Clarke matrix.  3.3.1. Example of eigenvalue/eigenvector calculation i n a delta line.  Consider the line depicted  in figure  2.3. From the impedance  and admittance  matrices derived in section 2.4, the corresponding modal parameters are derived next by means of formulae 3.18 to 3.23.  Matrix of eigenvectors of voltage. 1.0 + 0.0  l.O + O.Oj  1.0 + O.Oj  -3.559-0.062j  0.0 +O.Oj  0.812-0.006J  1.0 + 0.0  -l.O + O.Oj  1.0 + O.Oj  28  Matrix of eigenvectors of current. 0.0928 + 0.0005J  0.5 +O.Oj  0.4069+ 0.0123J  -0.2287-0.0029J  0.0 + O.Oj  -0.2287-0.0029J  0.0928 + 0.0005J  -0.5 + O.Oj  0.4069 + 0.0123J  Modal propagation constants. attenuation  velocity  (dB/km)  (km/s)  Mode 1  0.1954  299,172.  Mode 2  0.1904  298,984.  Mode 3  2.1877  291,192.  3.4. R E M A R K S .  Whereas section further  on in  3.1 provides the concepts the  thesis,  section  3.2  of modal analysis that are required  focuses  on  practical aspects  of modal  parameters calculation.  Delta lines occur very frequently in practice; from a practical point of view, the formulae for calculating modal parameters of delta lines given in section 3.3 are very valuable. In power line communications, the delta lines -and, specially the horizontal ones- are the most likely to present propagation problems; therefore, they  require special  analysis,  the  consideration. In the field of frequency  modal parameters  of  lines  have  to  be  domain transient  evaluated  for  different  frequencies, typically from 128 to 1024 times; thus, when formulae 3.19 to 3.23  29 are applicable, substantial savings of computation time are possible [11].  C H A P T E R 4. G R A P H I C A L METHOD FOR PREDICTING F R E Q U E N C Y R E S P O N S E OF D E L T A LINES.  It was shown in the previous chapter that any vector of voltages or currents in a transmission line could be regarded as a linear combination of the line natural modes.  Since  the  modal  velocities  generally  differ,  the  relative  phase  angles  between modes change as they propagate. These phase shifts cause fluctuations of the signal amplitude along the line.  Sometimes  the components of two modes, which were in phase on the coupled  conductors at the sending end, arrive at the receiving end with phase reversal. In the worst case, the phase reversal occurs when the modulii of the components are equal and the  signal is  "modal cancellation", and the  lost entirely. This  phenomenon is  frequencies  the  where  signal is  refered to as totally  lost are  known as "cancellation poles".  When a coupling arrangement is selected, it is desirable to ensure that there are no poles in the PLC frequency band; this, however, may not be easy to achieve. Pole location is very sensitive to physical changes of the line, as for example the conductor sag variations due to temperature shifts; measurements thus fail in finding them. Computer programs, on the other hand, because of the number of parameters  involved  in  the  propagation  phenomenon,  are  not  a  practical  alternative for locating poles.  In  this  chapter  a  graphical  method  for  30  predicting  delta  line  responses  is  31 presented. This method -first proposed by Senn et. al. [12,13,14], in addition to frequency  response  predictions,  provides  accurate  information  concerning  the  location of cancellation poles. Although Senn's method is restricted to delta lines, its importance is justified as these lines are perhaps the most commonly used, as well as by the fact that they are the ones that present the most severe modal cancellation effects. Since the method does not require a digital computer for its application, it is recommended for engineering work.  4.1. REFLECTIONLESS W A V E  The  PROPAGATION.  input/output relationship of a transmission line involves the voltage  vectors  as well as the current vectors. As has been pointed out in chapter 3, either the currents or the voltages may be eliminated by considering the line terminations. Senn's  method  assumes  reflection-free  propagation,  which  is  equivalent  to  assuming that the line is terminated at both ends in its characteristic impedance. For  most practical cases it may be considered that the  line terminations are  fairly close to this perfect matching condition; thus, for an homogeneous line of length 1, expression 3.20 becomes:  v  €  (4.D  - MLoyr'v,  were L stands for exp(- T 1).  Except  for  perfectly  symmetric  lines,  the  transpositions  always  produce wave  reflections; however, these may be neglected at carrier frequencies [13]. Therefore for  a line  divided by a transposition  in two homogeneous sections of lengths  32 \l and I 2 , the following expression may be applied:  V< = ML,«TMLiM"V. ,r  where  Lj  =  exp  L£ =  (-IP l j )  exp  (- f l£.  transpositions, expression 4.2 may be generalized  ..•(4.2)  For a  bigger  " m " of  number  as follows:  V< - WL.MTT ' T (TMlLiM"|.V„ ,  .(4.3)  i  When  dealing  with  PLC  systems, one is interested  i n the voltage  v  r  at the  receiver's input as a function of the voltage v^ at the output of the transmitter. These  scalar  voltages -v  r  and v - may t  be related  to V L = V ( 1 )  and V Q = V ( 0 )  respectively in the following form:  tit,  or  .(4.4)  and  0  or  -  07  £  33  "V  where  C  r  =  0  and  receiving and  C v t  C  t  .(4.5)  t  are vectors  that  describe  the coupling  transmitting ends respectively. From  4.4  and  connections 4.5  at the  expression  4.3  yields: m  (=1  The line insertion loss may  .(4.6)  J  1  be already obtained  from 4.6; however, before  doing  this, it is convenient to factorize the mode 1 loss term. This is acomplished by expressing each Lj in the following form:  1  0  0  1 t eip  O  K O L '  .(4.7)  Recalling that "1" is the total length of the line:  expression 4.6 becomes thus:  17. Now,  =*rp('r*t)  C f ^T|lll.'iW: }-C Ar T  T  ,  ,  i  £  t  .(4.8)  from the consideration that the receiving end coupling impedance is equal  34 to the one at the transmitting end, the line insertion loss is:  .(4.9)  t=l  The first term of 4.9 is indeed  the attenuation of a pure mode 1 signal; it is  therefore refered to as theoretical minimum attenuation or as mode 1 attenuation; it will be represented, henceforth, by " A j " .  The second term of expression 4.9, known as supplementary loss:  As -  7jog^  i0  |(CllllL.K ]T {TiKLiM-" ]c | H  .  ,  i  i  (4  10)  accounts for the coupling losses as well as for the modal interaction effects. As this is far more complicated  than the mode 1 term, most of the attention is  devoted to it.  4.2. SUPPLEMENTARY LOSSES IN DELTA TRANSMISSION LINES.  It  was  shown  in section  3.4  that  for delta  transformation matrices are the following:  transmission  lines,  the modal  35  i  P where  0  -2,  9  2,  "p" and "q" are given  by formulae  3.19  -P  and 3.20.  Within  the PLC  frequency range (from 30 to 500 kHz) and for horizontal lines: p = -2.0 and q = 1.0 If these approximations are applied to M and M"* in 4.11,  they become  the  Clarke transformation matrices.  Whereas  it  is  possible  to  approximate  the  modal  constants have to be as accurate as possible,  since  vectors, it is the  their  propagation  slight  difference  between them that determines the cancellation poles location.  Since mode 3 attenuation is very high, it practically vanishes  within the first  few kilometers of line. Mode 3 may therefore be disregarded and M and M"l become:  i  JH =  - IT,  5  -i  i  0-3  .(4.12)  36 4.2.1. Homogeneous lines.  By  applying the modal transformation matrices 4.12  to an  homogeneous line, the  supplementary loss term becomes:  As  Two  =  1 i -% o 1 - i  -10  o " "1 -z r o -3 ^  ~i  things should be noted here; the first is that as the third row  of L'  are  not  needed  anymore, they  have been  eliminated; the  and  column  second one  is  that the following definition has been used:  (4.13) By  performing the matrix products in 4.12,  As As  (p,  0  +P,K)/K  .(4.14)  an example, consider the transmission system of figure 4.1. Here:  As =  The  =-10(^,  a first degree polynomial is obtained:  10  lo  matrix products yield:  1" 0 0  T  "1 1 " -2. 0  1 Ol [i -2 l i 0 *] [> -*J 0  1' 0 0  37 It is clear that different couplings will  produce different polynomials; table 4.1  provides their coefficients for the most common couplings -phase to ground and phase to phase in differential mode (push-pull).  4.2.2. Transposed lines.  Consider first the case of a transposed  line where the two homogeneous sections  have equal length; the following expression may be deduced from 4.10:  A s = 2.06o  | <C H Vo IM'"F JN to M T  3lo  H  C  f  (4.15)  if the following definition is used:  with 1 the length of each homogeneous section. Expression 4.15 yields a second 0  degree polynomial in X:  (4.16)  38  As an example, consider the system depicted in figure 4.2; here:  1  '  \ ~  -  O  4-  o  o  "i  r  0  -z  0—  _1 '1  O  0  '1  _o yj [3  1"  l  LOO  4.  o] T i -z l ~ |  1  o  o  X  £  o  -3J  o o  -2  3  O  1  '1  "E- (x> = C- 1 -6X  /i2  It is possible to show that for a line divided in "m" homogeneous "m-l"  transpositions,  the  supplementary  loss  term  involves  an  sections by m-th  degree  polynomial:  .(4.17)  Very often in practice, transpositions are spaced at unequal distances; in these cases,a polynomial expression as 4.17 can also be obtained. This is shown next for  a line consisting of two  unequal sections;  the  generalization for a bigger  number of sections is straightforward.  Suppose that a line of length "l^t" is divided into two line sections of lengths \l and I2. Let the ratio I1/I2 be approximated by means of two integers "nj" and "n2" in the following manner:  FIGURE 4.1) Example of c a r r i e r c o u p l i n g on nontransposed l i n e .  FIGURE 4.2)  Example of c o u p l i n g on a s i n g l transposed l i n e .  40  .(4.18) where Now,  the common factors between  n j and  n£  have been already  eliminated.  a section of length  I -  TOT  .(4.19)  n, +  is chosen as basis. In a similar way  as in 4.13, "X" may  be defined as:  X 4 exp[-£.(*»-*,\l For the first line section:  1  0  0  x  ™1  and, for the second one:  i ,"2.  o  the supplementary loss term thus becomes:  \  0  o  x  "  n i  1  o  0  x  41 It is clear that this expression n  l  +  n  a polynomial whose degree  is  at most  2•  From expression 4.18, the  yields  I1/I2  however,  ratio is  is  it is obvious that -in general- a better approximation to  achieved  by  choosing  bigger  values  for nj  and T12; this,  inconvenient from the numerical standpoint, for the degree  of the  polynomial will increase accordingly. It is clear then, that in these cases there has to be a trade off.  Perhaps the most common case of lines with unequal section lengths are those where the transpositions  are located  length; this transposition scheme show,  respectively,  at  1/6-th 3/6-ths and 5/6-ths of the line  is depicted in figure 4.3d. Figures 4.3a,  4.3b  and  4.3c  the most common layouts of lines with zero, one  and  two transpositions. The polynomials associated to all these line schemes, for  the most conventional coupling arrangements, are provided in tables 4.1, 4.2, and  4.3  4.4  4.2.3. Insertion loss calculation.  Once the polynomial coefficients  for a specific line layout are available, the line  response to a frequency excitation can be obtained through the following steps: 1.  Obtain the electrical parameters Z and Y of the line by means  of the  methods given in reference 2 as well as in chapter 2. 2.  From the electrical parameters derive the modal propagation constants using the  formulae  given  in  section  3.4.  Only  42  i/2  V 3  ^  l/t —  FIGURE  4.3)  -  t  f  Transmission line transposition layouts most commonly f o u n d i n p r a c t i c e , a) U n t r a n s p o s e d l i n e . b) One t r a n s p o s i t i o n , c ) Two t r a n s p o s i t i o n s , d) T h r e e t r a n s p o s i t i o n s unequal s p a c i n g .  43 )f  I =  0< i  + jjf$i and Y 2  =  +  J^ 2  8 1 1 - 6  required.  3.  Given the basic homogeneous section length "lo," obtain X:  4.  Calculate the supplementary loss term as:  A = -20«og |(Z: s  |o  1=0 5.  Calculate the mode 1 attenuation:  where "1" is the total line length. 6.  Obtain the total insertion loss:  A  = A ,  +  A  (4-20)  s  4.2.4. Modal cancellation poles.  Since oc j as well as "1" (the total length of the line) are always non-negative and  finite, the mode 1 term:  A, can  The  -  ^oao^  i 0  o") •  e  only take non-negative finite values.  supplementary loss term -on the other hand- may become  infinite.  It is  apparent from expression 4.17 that this can only happen at the root values of  44  £py  .  1=0 By  recalling the definition of X:  x £  and  t-Mc»-0]  the fact that:  it becomes clear that not  all the  roots  of P ( X ) m  have  physical meaning. Only  those roots such that:  o < |xl < 1 are related to poles; these will be henceforth  Tables 4.1, Their values  4.2,  4.3  and  4.4  are specified by  that are related to "X"  %  ^  €^p  include the  referred to as poles.  poles corresponding to each  means of two  new  variables: " Ac*. " and  polynomial. "AO  ",  as indicated as follows:  (-  A a  4J  (  4  2  1  )  therefore:  (4.22)  45  Ap  A©  *.C  - ^  (4  . 3) 2  may be interpreted as the phase change suffered by mode 2 with respect  to mode 1 as they travel through a basic line length "1 ".  Acx thus represents  0  the attenuation difference (in Nepers) between mode 2 and mode 1 for the basic line length.  4.3. SENN'S METHOD  FOR E V A L U A T I N G INSERTION LOSSES.  The steps required for evaluating a line response to a carrier signal are given in sub-section  4.2.3; alternatively, Senn [12,13,14] proposed a graphical method  that is to be presented next.  From the definitions 4.22  and 4.23  be regarded as a function of A CK.  it follows that the supplementary loss may and of A ©  ; it may thus be plotted for a  range of values of these two variables as it is shown in figure 4.4a.  Since three-dimensional plots -as the one in figure 4.4a- are not very practical, Senn proposes the use of contour map representations the  contour  map  in  figure  4.4b  which  dimensional graph.  Relation 4.17 can be expressed as follows:  instead. See for example  corresponds  to  figure  4.4a  three  46  r FIGURE 4.4)  Plot of the function A = 201ogin|(X2-6X-3)/8| a) Three dimensional graph, b) Contour representation. s  47  .(4.24) In  order  to generate  the contours, a constant  asigned to A ,  then  there  its direct  g  preclude  value  (i. e., a level  value) is  "X" has to be solved from 4.24; the absolute value solution  as a polynomial  equation. The method  bars here  proposed makes use of the following definition:  where z* is the complex conjugate of z. Equation 4.24 thus yields:  By  recalling 4.21,and after some algebraic manipulations:  Assuming  that  becomes clear generated  /A.©  is given  K  Z  |0  C A s / i 0 > )  (4.25)  and, i f " exp(- A a )" is replaced by "Y", it  that 4.25 is a 2m  degree polynomial  by assigning successive values to A ©  from  A  contour  can thus be  0° to 360°,and solving  this 2m degree polynomial each time.  In a similar way  as with the poles, not all the roots of 4.25 have physical  meaning; only those that are real and satisfy the inequality: 0  <  V  <  ±  48 belong to a contour segment.  Since the contour maps are obtained for general values  of  A ex.  and A©  -  ,  they do not depend on the specific dimensions or the electrical properties of the lines;  therefore  they  are  generic  to  delta  lines  with  the  same  transposition  scheme and coupling arrangements.  Once the contours are available, specific values for determined. In figure 4.5a figure  4.5b;  the  A&  frequency  is  AcX  and  A©  must be  is plotted against Aa for the line described in varied  continuously  and  the  earth  resistivity  assumes 5 different values. The supplementary loss term may be obtained easily by  superposing  figure  4.5a  curve  -also  known  as  modal  curve-  to  the  corresponding contour map.  Although the modal curve has to be elaborated usually for each particular case, the same curve may be used several times for evaluating different couplings by just superposing it with the  contour maps corresponding to each coupling. An  example is provided next.  4.3.1. Example.  For  the  line  scheme  of figure 4.6,  couplings providing metallic continuity:  C j : C =(l,0,0)/C =(0,0,l) t  r  it  is  desired  to  compare  the  following  49  Conductor radius 1.598 cm. Gnd. wire radius 0.489 cm. Gnd. wire material Alumoweld Earth r e s i s t i v i t y 300 Ohm-m Line section length .... 30 km.  fe)  FIGURE 4.5) a) Graph of A a - A e for d i f f e r e n t . .. frequencies and d i f f e r e n t earth r e s i s t i v i t i e s , b) l i n e data for the graph 4.5a.  50 C : 2  C = ( 0 , l , 0 ) / C = (l,0,0) t  "  r  "  0. C : 3  C =(0,0,1)/C = (0,1,0), t  r  |<  against the discontinuous one: C : 4  The  50krri5  C =(0,l,0)/C =(0,l,0). t  FIGURE  r  line  dimensions  and conductors  data  >|<  50fcms — ^ |  4.6) Example.  are those  of figure  4.5b; the earth  resistivity is 300 Ohm-m.  The  theoretical minimum  attenuation, as well  obtained from the line data; they  as the modal curve  are readily  are plotted, respectively, in figures 4.7a and  4.7b.  The  contour  maps corresponding  to couplings  4.8a  and 4.8b, with the modal curve  and C  already superposed.  2  are given in figures The contour  map of  coupling C 3 is equal to that of C . For C 4 a contour map is not required -the 2  supplementary loss is a constant equal to 9.5424 dB.  The  total  insertion  loss  for each  coupling  is obtained  by  adding the  supplementary loss terms to the theoretical minimum attenuation; the results are plotted i n figure 4.9. Note that -contrary to the common sense, the discontinuous metallic path is the best; it is, i n fact, the only one free of poles.  360o  /  *  270O  51  / 350 khz  250 k Hz AO  180o  /  ]  50 k H z  90o /  50 kHz  Oo 0.  © I 0.  1  10.  1 100.  — I  Frequency  20.'  A  1 ..200-  1  30.  1 300.  1  '  1 400.  kHz.  FIGURE 4.7) Example, a) Modal curve. b) theoretical minimum attenuation.  • 500.  52  \: A  ©  1  \  \ \  /  tool  MOO  Yl •  AIB.  \ Aa  SO  A ©  AaFIGURE  4.8)  Contour maps with modal curve superposed, a) Coupling C t = ( 1 , 0 , 0 ) / C = ( 0 , 0 , 1 ) . a) Couplings Ct=(0,1,0)/C =(1,0,0) and Ct=(0,0,1)/C =(0,1,0). r  r  r  53  o 0.  100.  200. FREQUENCY  300.  400.  (kHz)  FIGURE 4 . 9 ) Line response for d i f f e r e n t couplings.  500.  54 4.4. R E M A R K S .  A  simplified  method  for PLC response  prediction has  been  presented  in this  chapter.  By assuming reflectionless  propagation, it was  possible to express the insertion  loss term as the sum of the theoretical minimum attenuation of the line and of a  supplementary  transformations  led  loss to  term.  The  expressing  further the  assumption  supplementary  of  loss  constant in  terms  modal of  a  polynomial whose roots correspond to the modal cancellation poles of the line.  With the  supplementary loss term in analytical form,  a set  of contour maps  may be derived. The contours, together with another type of curves - the modal curves, facilitate the line response calculation.  The computer is required only to generate the contour maps. Afterwards it is not needed any longer; hence, the method is suitable for field applications. Two additional advantages of the graphical method are: 1.  A single modal curve may be used to evaluate different couplings.  2.  The proximity of poles,  which may cause subsequent  detected from the contour maps.  problems, is readily  T A B L E 4 . 1 - P O L Y N O M I A L S OF U N T R A N S P O S E D LINES. COUPLING Trnsm./rceiv. (1,0,0)/(1,0,0) (0,0,1)/(0,0,1)  POLYNOMIAL  POLES  (3X+D/6  9.5424 <180o  (1,0,0)/(0,0,1) (0,0,1)/(1,0,0)  (3X-D/6  9.5424 <0o,<360o  (1,0,0)/(0,1,0) (0,1,0)/(1,0,0) (0,1,0)/(0,0,1) (0,0,1)/(0,1,0)  (-2)/6  none  (0,1,0)/(0,1,0)  (4)/6  none  (1,-1,0)/(1,-1,0) (0,1,-1)/(0,1,-1)  (X + 3)/4  none  (1,-1,0)/(1,0,-1) (1,0,-1)/(1,-1,0) (0,1,-1)/(1,0,-1) (1,0,-1)/(0,1,-1)  (X)/2  none  (1,-1,0)/(0,1,-1) (0,1,-D/(1,-1,0)  (X-3)/4  none  (1,0,-1)/(1,0,-1)  X  none  T A B L E 4.2- POLYNOMIALS O F SINGLE TRANSPOSED LINES  COUPLING Trnsm./rceiv. (1,0,0)/(1,0,0) (0,0,1)/(0,0,1)  POLYNOMIAL  POLES  (3X2+ 1)/12  4.77 000,2700  (1,0,0)/(0,1,0) (0,1,0)/(0,0,1)  (3X - l)/6  9.54 <0o,360o  (1,0,0)/(0,0,1)  (3X2 + 6X-D/12  16.21 <0o,360o  (0,1,0)/(1,0,0) (0,0,1)/(0,1,0)  (3X+D/6  9.5424 <180o  (0,1,0)/(0,1,0)  (-D/3  none  (0,0,1)/(1,0,0)  (3X2-6X-D/12  16.2102 <180o  (1,-1,0)/(1,-1,0) (0,1,-1)/(0,1,-1)  (X2 + 3)/8  none  (1,-1,0)/(1,0,-1) (1,0,-1)/(0,1,-1)  (X2 + 3X)/4  none  (1,-1,0)7(0,1,-1)  (X2 + 6X-3)/8  6.67<0o,360o  (1,0,-1)/(1,-1,0) (0,1,-1)/(1,0,-1)  (X2-3X)/4  none  (1,0,-1)/(1,0,-1)  (X2)/2  none  (0,1,-1)/(1,-1,0)  (X2-6X-3)/8  6.6677<180o  57  T A B L E 4 . 3 - P O L Y N O M I A L S OF D O U B L E T R A N S P O S E D LINES.  COUPLING Trnsm./rceiv. (1,0,0)/(1,0,0) (0,0,1)/(0,0,1)  POLYNOMIAL  POLES  (3X3-9X2-3X+l)/24  6.29 <180o 13.55<0o,360o  (1,0,0)/(0,1,0) (0,1,0)/(0,0,1)  (3X2 + 6X-D/12  (1,0,0)/(0,0,1)  (3X3-3X2 + 9X-D/24  18.79 <0o,360o  (0,1,0)/(1,0,0) (0,0,1)/(0,1,0)  (3X2 + 1)/12  4.77 <90o,270o  (0,1,0)/(0,1,0)  (3X-D/6  9.54<0o,360o  (0,0,1)/(1,0,0)  (3X3-15X2-3X-D/24  11.94 < 1800+/-66.18o  (1,-1,0)/(1,-1,0) (0,1,-1)/(0,1,-1)  (X3-3X2-9X + 3V16  10.30<0o,360o  (1,-1,0)/(1,0,-1) (1,0,-1)/(0,1,-1)  (X3 + 3X)/8  none  (1,-1,0)/(0,1,-1)  (X3 + 3X2 + 15X-3V16  14.33<0o,360o  (1,0,-1)/(1,-1,0) (0,1,-1)/(1,0,-1)  (X3-6X2-3X)/8  6.67<180o  (1,0,-1)/(1,0,-1)  (X3-3X2)/4  none  (0,1,-1)/(1,-1,0)  (X3-9X2 + 3X-3)/16  4.62<180+/-105.07o  1  16.21 <0o,360o  TABLE 4.4-POLYNOMIALS OF THREE TRANSPOSED LINES. COUPLING Trnsm./rceiv.  POLYNOMIAL  POLES  (1,0,0)/(1,0,0)  (3X6-21X4-15X2 +1)/48  1.49 < 900,2700 12.115<0o,180o,360o  (3X5 + 3X4-6X3 + 6X2 + 3X-l)/24  5.7 <180o 12.36<0o,360o  (1,0,0)/(0,0,1)  (3X6 + 6X5-15X4-12X3-3X2 + 6X-1)/48  1.40 < 135.50,224.50 8.857<Oo,360o 13.4<Oo,360o  (0,1,0)/(1,0,0) (0,0,1)/(0,1,0)  (3X5-3X4-6X3-6X2 + 3X +1)/24  5.7 <0o,360o 12.36<1800  (0,1,0)/(0,1,0)  (3X4 + 6X2-1)/12  8.106<0o,180o,360o  (0,0,1)/(1,0,0)  (3X6-6X5-15X4+ 12X3-3X2-6X-D/48  1.4046 <44.509o,315.49o 8.857<180o 13.4<180o  (1,-1,0)/(1,-1,0) (0,1,-D/(0,1,-1)  (X6-154-21X2+ 3)/32  8.835<0o,180o,360o  (1,-1,0)/(1,0,-1) (1,0,-1)/(0,1,-1)  (X6 + 3X5-6X4-6X3-3X2 + 3X>)/16  0.8189< 132.560,227.440 6.882<0o,360o  (1,-1,0)/(0,1,-1)  (X6 + 6X5 + 3X4- 12X3 + 5X2 + 6X-3)/32  1.2<35.40,324.60 2.5475<180o 6.46<0o,360o  (1,0,-1)/(1,-1,0)  (X6-3X5-6X4 + 6X3-3X2-3X)/16  0.8189<47.44o,312.56o 6.88<180o  (1,0,-1)/(1,0,-1)  (X6-6X4-3X2)/8  3.333<90o,270o  (0,1,-1)/(1,-1,0)  (X6-6X5 + 3X4+ 12X3-15X2-6X-3)/32  5.375<0o,360o 10.15<180o  (0,0,1)/(0,0,1) (1,0,0)/(0,1,0) (0,1,0)/(0,0,1)  (0,1,-1)/(1,0,-1)  C H A P T E R 5. COUPLING RECOMMENDATIONS.  Although the  modal theory or - in the  case of delta lines  Senn's graphical  method may be used for comparing coupling alternatives, it is desirable to have a  set  of  general  recommendations  or  guidelines  for  selecting  adequate  line  couplings.  Prior to the  development  of the modal theory, coupling recommendations were  based on simplistic line concepts.  It was found, afterwards, that most of them  were incorrect; however, the recommendations are still widespread. One of these rules, for instance,  advises  the use of the couplings that provide a continuous  metallic path between the transmitter and the receiver; a counter-example to this rule was presented in section 4.3.1.  Conductor 1_  c:ond. 2  FIGURE 5.1) a) Transmission l i n e layout example. b) Coupling alternative 1. c) Alternative 2. 59  60 Another rule recommends that, for phase to phase couplings, the two conductors which stay  closest to each other for most of the distance  along the line be  chosen; for example, according to this rule, in figure 5.1a conductors 2 and 3 must be selected. This coupling, represented in figure 5.2b, hereafter referred to as coupling 1, is to be compared against coupling 2 on figure 5.2c.  By assuming the fine data provided in Figure 4.5b, a modal plot is obtained. In figures 5.2a and 5.2b, this plot is superposed to the contour maps corresponding to coupling 1 and coupling 2. The resulting line responses are plotted in figure 5.2c; it is clear there that coupling 2 is much better than coupling 1. It may be concluded, thus, that the abovementioned rule is unfounded.  Some  more recent  studies [20,21] introduce the  concept  of optimum coupling;  however, in a strict sense, optimum couplings very seldom exist. In reference 21, coupling 2 of the previous example is presented as the optimum phase to phase arrangement  for  single  transposed  lines.  This  coupling  is  indeed  the  most  recommendable in that case for, among other things it is free of poles; however, it cannot be said that it is optimum. For instance, in figure 5.2c, coupling 1 performance  is  much better  at frequencies  below  120  kHz and slightly  better  above 320 kHz.  The purpose of the first section couplings  along  technique  is  with  based  a on  is  technique Senn's  to introduce the for  generating  graphical  method.  concept  the In  of recommended  recommendations; section  5.2  this  coupling  recommendations are produced for the transposition schemes that are supposed to  61  FIGURE 5.2) a) Coupling 1 contour, b) Coupling 2 contour. c) Frequency responses.  62 be the most commonly found in practice. Finally, in section 5.3 of  the  performance  non-conventional couplings is analyzed.  5.1. P R O P O S E D METHOD F O R ASSESSING COUPLING A L T E R N A T I V E S .  The  type  of recommended  couplings  of concern here is the one that,  in addition  to providing low losses, minimizes the risk of modal cancellation.  It  was  shown  in  chapter  4 that  the  line  insertion  loss  is  composed  by  two  terms: A Whereas the  the  mode  their  1  supplementary  loss term  term  equal;  remains  supplementary  compare  =  two  loss  terms  couplings,  A  A  changes  thus,  solely.  their  +  x  s  from  coupling Here,  coupling  comparisons  it  corresponding  one  is  proposed  to  can  be  that,  supplementary  the  based  in  loss  other,  order terms  on to be  subtracted: -^(coupling  The  difference  colors;  one  better  than  example,  color  two  between  two  1,  that  and  s  in the the  the  color plot of figure  plot  provides  couplings;  a  however,  quick it  pi g m  Ao--A0  difference other  C 0 U  is  2)  plane by means of two positive,  color  indicating  5.3a,  the  as coupling 1 and that on figure 5.1c  color two  indicating  coupling  for the  been chosen  The  may be represented  1) " A (  the  e.,  coupling  opposite.  coupling of figure  As 5.1b  2  is an has  as coupling 2.  overview  of  may  misleading.  be  i.  the  comparative In  performance  figure  5.3,  for  10.0  20.0  30.0  ATTENUATION DIFFERENCE Aa  I 0.0  1 1  I 10.0  •  I 20.0  >•  !  1 30.0  ATTENUATION DIFFERENCE Aa  FIGURE 5 . 3 ) a) Example of a two color p l o t . b) Example of a three color  plot.  64 instance,the  black region covers  more than half of the A a - A ©  plane; it may  seem then, that coupling 1 is better than coupling 2. A look at their frequency responses in figure 5.2c shows that this appreciation is incorrect. The first point to notice  is that  some of the  differences  between the  two  couplings  may be  irrelevant; it can be seen at figure 5.2c that above 300 kHz both responses are very similar. Another point to consider is that not all regions of the A a - A © plane are equally important; in the current example, the central region containing the pole of coupling 1 seems to have a high probability of occurrence in the practice.  As in PLC communications a difference of 3 dB is considered as unimportant, a three color plot -as the one in figure 5.3b, is introduced in order to remove the irrelevant differences. There, the third region in gray color may be considered as a neutral zone where the response  differences  are comprised in the  +/-  3 dB  range. The black and white regions designate thus only meaningful differences.  In order to establish the regions of the A o - A ©  plane that are relevant for the  coupling comparisons, a second type of plot, which henceforth will be referred to as "feasible regions map", is introduced. An example is provided in figure 5.4.  The  shaded  region of figure  5.4  was  produced by varying three  of the  line  physical parameters -frequency, earth resistivity and conductors medium height.  It is important in practice to consider the variations of the earth resistivity and of  the  conductor  height  for  two  reasons.  Firstly,  these  two  parameters  are  0.0  10.0  20.0  30.0  ATTENUATION DIFFERENCE Aa  L I N E DATA: -Medium heights of phase conductors 12-18m -Horizontal distance -between conductors 9.0 m -Conductor radius 4x1.05 cm. -Gnd. wires None -Earth resistivities ... 30-3000 Ohm-m -Frequencies 50-450 k H z -Line section length .... 30 km.  FIGURE 5 . 4 ) a) Example of a feasible regions map.  66  affected by the climate -the conductor sag depends on the temperature and the earth resistivity on the humidity.  Secondly, the earth resistivity parameter, due  to its nature, involves a considerable uncertainty. Figure 5.5a shows the influence of the earth resistivity on the modal curves of the line specified by table 5.1, and  The  figure 5.5b shows the effects of varying the conductors medium height.  feasible regions  maps, i n addition to establishing the  meaningful  regions for the coupling comparisons, help to detect the possibility of falling into a cancellation pole; this this is very important  since, as it will become apparent  in the next section, it is not always possible to find pole-free couplings.  5.2.  COUPLING CONSIDERATIONS FOR COMMON LINE CASES.  The  method described i n the previous  couplings  section is applied here to the selection of  for the most common transposition schemes; i . e., those depicted in  figure 4.3.  It is found here that only i n the most simple  case -untransposed line phase to  ground coupling, the recommended arrangement is optimum. It is also found that, for lines with two or more transpositions it may not be possible to find pole-free couplings; for these cases, obviously, the recommendations cannot be considered of general validity, and the use of feasible region maps is strongly recommended.  The  contour maps for all the couplings mentioned along this section are provided  in the appendix.  FIGURE 5 . 5 ) a) Effect of the earth r e s i s t i v i t y on the modal p l o t s . LINE:HEGHTS  f lZ.Or. •j3 Z  / / /  a•  /'•/  J ^> Z  <  J it.C  'lfc.0m  /' * ' i t  73 tl  'iso  kHt  ///As •>kHi  Ear4h  rcs/jrtt/.  iooonm  -  '50  0.0  10.0 20 0 ATTENUATION DIFFERENCE Aa  FIGURE 5.5)b) E f f e c t of the conductors medium height on the modal plots.  68 5.2.1. Untransposed lines. Phase to ground coupling.  As it was mentioned before, this is the most simple case found i n practice. The following coupling arrangement: ( 0 , 1 , 0 ) / ( 0 , 1 , 0 ) depicted i n figure supplementary  5.6, is the recommended  losses -according  to Senn's  one. This coupling i s optimum; its simplified method-  are constant and  equal to: A s = 3.5218. These losses are due only to mode conversion at both end of the line.  FIGURE  5.6)  Recommended phase to ground coupling for untransposed l i n e s .  69 Phase to phase coupling.  For phase to phase differential mode coupling (push-pull), the arrangement: ( 1 , -1 , 0 ) / ( 1 , -1 , 0 ), represented considered  i n figure  5.7a, is the recommended  optimum. It may be seen  from  one. This  coupling cannot be  the two and three color plots i n  figures 5.8a and 5.8b, that the coupling: ( 1 , -1 , 0 ) { ( 0 , 1 , -1 ), represented i n figure 5.7b, has a very similar performance and in some regions, small though, it is better.  FIGURE 5.7) a) Recommended phase to phase coupling on untransposed l i n e s , b) Second best coupling.  FIGURE 5 . 8 ) Comparison of the couplings depicted in figures 5 . 7 a and 5 . 7 b . a) Two color plot, b) Three color p l o t .  71  5.2.2. Single transposed lines. Phase to ground coupling.  The  coupling  of figure  5.9 is the best  recommended, mainly  because  it  is  pole-free. The other alternatives may have better performance i n small regions of the A a - A 6  plane which correspond  to low frequencies  and/or very  high  earth  resistivities; additionally alll them present cancellation poles.  0  0 FIGURE  5.9) Recommended phase to ground coupling for single transposed l i n e s .  Note that section 4.3 provides a specific example of this coupling.  72 Phase to phase cooupling. The  arrangement depicted in Figure 5.10  is the recommended differential mode  coupling for single transposed lines. There are some other alternatives that are also  free  of  poles;  however,  the  two  color plots  coupling has a better performance in most of the small  regions  where  the  other  alternatives  are  indicate  that  Figure  5.10  A a - A 0 plane; moreover, the better,  correspond  to  low  frequencies and/or to very high earth resistivities. The two couplings depicted in figure  5.11  should be avoided. The coupling on Figure 5.11a  has been already  analyzed in the example at the beginning of the chapter.  FIGURE 5.10) Phase to phase recommended coupling for single transposed l i n e s .  FIGURE 5.11) Couplings that should be avoided.  73 5.2.3. Lines with two transpositions. Phase to ground coupling.  As it may be seen from table 4.3, in this case none of the couplings is free of poles; for this reason, the  coupling recommended here (figure  considered as generally valid, t convenient  to  check,  5.12)  Before choosing a carrier frequency,  by means  of a feasible  regions  map, that  cannot be it is  there  thus is no  danger of modal cancellation.  As an example, the coupling of figure 5.13 is the one whose performance is the closest to that of the recommended one.The two color plot comparing these two couplings is  shown in figure 5.14.  Apart from the  fact that the  white area  where the recommended coupling performs better is bigger, its poles are slightly closer to the rigth hand side of the Act-A©  plane; this means that they are  less likely to happen.  5.2.4. Phase to phase coupling.  Here the selection of a coupling is much more difficoult than in the previous cases. There are actually two pole-free couplings: ( 1 , -1 , 0 ) / ( 1 , 0 , -1 ) which is equivalent to: ( 1 , 0 , -1 ) / ( 0 , 1 , -1 ),  tNote: Reference [21] refers to this coupling as optimum.  FIGURE 5.12) Recommended phase to ground coupling in a double transposed l i n e .  0.0  10.0  20.0  30.0  ATTENUATION DIFFERENCE Aa  FIGURE 5.14) Two color plot comparing couplings depicted in figures 5.12 and 5.13.  75 and ( 1 , 0 , -1 ) / ( 1 , 0 , -1 ); Unfortunatelly, their losses grow very quickly as  increases. The first coupling,  depicted in figure 5.15, is better than the second one.  The  coupling of figure  5.16  has  plane; however, it is not pole-free, t performance to that of 5.16.  losses for most of the A a - A ©  the smallest  The coupling of figure 5.17  is similar in  With these two couplings the possibility of modal  cancellation exists for high earth resistivities  and for long lines with medium  earth  this  resistivities.  For figure  5.17  coupling,  danger  also  exists  at  low  frequencies. As the supplementary losses of figure 5.15 coupling are small at the region on the  left  hand side  of the Aa-A©  plane -which is  precisely  where  couplings 5.16 and 5.17 have their poles, it is possible to consider the former as the  complement of the  every  application be  latter  ones; however,  supported by means  it is  of its  strongly  recommended that  corresponding feasible  regions  map.  5.2.5. Lines with three transpositions.  It  may  be  seen from  table  4.5  that  all the  three-transposed  line  couplings  present cancellation poles; thus, the selection of frequencies should be aided by a feasible regions map.  For  phase  to ground coupling, the recommended arrangement is represented in  tNOTE: At reference 21 this coupling is refered to as optimum coupling without warning about the possibility of modal cancellation.  FIGURE 5 . 1 5 )  FIGURE 5 . 1 6 )  FIGURE 5 . 1 7 )  Complementary phase to phase coupling on a double transposed l i n e .  Recommended phase to phase coupling for a double transposed l i n e .  Second best phase to phase coupling for a double transposed l i n e .  77 figure 5.18; the recommended one for phase to phase coupling is the one given in figure 5.19a; although, the coupling of figure 5.19b is as nearly as good as that in figure 5.19a.  The  three suggested couplings have the common characteristic that their poles lie  close to the left hand side of the ^ a - A 0  plane, which means that the possibility  of modal cancellation exists for lines with high earth resistivity or for very long lines with medium earth resistivity.  5.3. NONCONVENTIONAL  The  coupling  '•  COUPLINGS.  arrangements  analyzed  in  the  previous  section  are  the  ones  traditionally used in the power sector. However, other arrangements are possible -they are referred to here as nonconventional couplings.  ln  order to find better couplings (i. e.,  pole-free)  in lines with two and three  transpositions, two nonconventional phase to phase alternatives were analyzed: 1.  Common  mode  (push-push)  transmission  with  differential  mode  (push-pull)  reception 2.  Differential mode transmission with common mode reception.  Tables 5.1  and 5.2  with  cancellation  their  provide the polynomials of the analyzed couplings poles.  None  of  the  analyzed  alternatives  attractive as to justify the departure from the conventional practices.  together  resulted  as  78  FIGURE 5.18) Phase to ground recommended coupling for a three transposed l i n e .  FIGURE 5.19) Phase to phase recommended couplings for a three transposed l i n e , a) Best coupling, b) Second best coupling.  79 Another nonconventional  coupling arrangement that was analyzed is the so called  mode 1 coupling. This is an alternative several carrier equipment manufacturers advocate for. In mode 1 coupling the carrier signal is injected to the three phase conductors of the line with the following current (or voltage) distribution: ( 1 , - 2 , 1 ) .  The  intention with this coupling is to put as much energy as possible into mode  1 form, which is usually the mode with the least losses.  Figure 5.20 shows the contour map of mode 1 coupling on a  double-transposed  line. Note that it has cancellation poles. This map can be compared with the one  of the recommended phase to ground coupling given in the appendix; the  difference between them  is a constant  value  of 3.52 dB. The insertion loss  difference between mode 1 coupling and the recommended phase to phase coupling -although  it is not constant-  may be approximated  which is very accurate for most of the A a-A©  to the figure of 2.5 dB,  plane.  It may be concluded from the above results that, since mode 1 coupling does not eliminate the possibility of modal cancellation, an improvement of 2.5 dB would hardly justify the additional expense of the three-phase coupling.  Figure  5.21  shows  three-transposed recommended  line;  the contour note  that  map this  for the mode  coupling  presents  1  more  coupling poles  on  a  than the  phase to ground coupling of figure 5.18, or than the phase to  phase coupling of figure 5.19. As A a  increases the supplementary loss difference  between mode 1 coupling and the recommended phase to ground one tends to  80  aio  FIGURE 5.20)  FIGURE 5.21  Contour map of mode 1 coupling on a double transposed l i n e .  Contour map of mode 1 coupling on a three transposed l i n e .  81 3.5; whereas its difference with respect to figure 5.19a coupling tends to 2.5 dB. By  the same token  as with double-transposed  seem that atractive for three-transposed  lines.  lines, mode 1 coupling does not  82  T A B L E 5.1-POLYNOMIALS OF DOUBLE TRANSPOSED LINES. NONCONVENTIONAL COUPLINGS. COUPLING Trnsm./rceiv. (1,1,0)/(1,1,0) (0,1,1)/(0,1,1)  POLYNOMIAL  POLES  (3X3-9X2-3X+ l)/48  6.29 <180o 13.55<0o,360o  (1,1,0)/(1,0,1) (1,0,1)/(0,1,1)  (3X2+D/24  4.77 <90o,270o  (1,1,0)/(0,1,1)  (3X3-15X2-3X-D/48  11.93 < 113.820,246.180  (1,0,1)/(1,1,0) (0,1,1)/(1,0,1)  (3X2 + 6X-D/24  16.21 <0o,360o  (1,0,1)/(1,0,1)  (3X-D/12  9.54<0o,360o  (0,1,D/(1,1,0)  (3X3-3X2 + 9X-D/48  18.78 <0o,360o  (1,-1,0)/(1,1,0)  (X3 + X2 + 7X-D/16  8.46< 100.90,259.10  (0,1,-1)/(0,1,1)  (X3-7X2-X-U/16  17.1< 00,3600  (1,-1,0)/(1,0,1)  (X2 + 4X-D/8  12.54<180o  (1,0,-1)/(0,1,1)  (X3-4X2-X)/8  12.54<0o,360o  (1,-1,0)/(0,1,1)  (X3-X2-X+D/16  0<0o,180o,360o  (1,0,-1)/(1,1,0)  (X3-2X2 + X)/8  0< 0o,360o  (0,1,-1)/(1,0,1)  (X2-2X+U/8  0< 0o,360o  (1,0,-1)/(1,0,1)  (X2+X)/4  0<180o  (0,1,-1)/(1,1,0)  (X3-5X2-5X-1)/16  0<180o 15.3<0o360o :  83 T A B L E 5.2-POLYNOMIALS OF T H R E E TRANSPOSED LINES. N O N C O N V E N T I O N A L COUPLINGS. COUPLING Trnsm./rceiv. (1,1,0)/(1,1,0) (1,0,1)/(0,1,1)  POLYNOMIAL  POLES  (3X6-21X4-15X2+l)/96  1.49 <90o,270o 12.116<0o,180o,360o  (1,1,0)/(1,0,1) (1,0,1)/(0,1,D  (3X5-3X4-6X3-6X2 + 3X +1)/48  5.7 <0o,360o 12.36<180o  (1,1,0)/(0,1,1)  (3X6-6X5-15X4+ 12X3-3X2-6X-l)/96  1.40 <44.48o,315.52o 8.857<180o 13.4<180o  (1,0,1)/(1,1,0) (0,1,D/(1,0,1)  (3X5 + 3X4-6X3 + 6X2 + 3X-D/48  5.7 <180o 12.36< 00,3600  (1,0,1)/(1,0,D  (3X4 + 6X2-l)/24  8.106<0o,180o,360o  (0,1,1)/(1,1,0)  (3X6 + 6X5-15X4-12X3-3X2 + 6X-D/96  1.4046 < 135.50,224.50 8.857<0o,360o 13.4<0o,360o  (1,-1,0)/(1,1,0)  (X6-4X5-3X4 + 8X3 + 3X2-4X-1)/32  0< Oo, 1800,3600 12.54<180o  (0,1,-1)/(0,1,1)  (X6 + 4X5-3X4-8X3 + 3X2 + 4X-l)/32  0<0o,180o,360o 12.54<0o,360o  (1,-1,0)/(1,0,1)  (X6-X5-6X4 + 2X3-3X2-X> )/16  2.33<67.73o,292.27o 11.8<180o  (1,0,-1)/(0,1,1)  (X5 +3X4-2X3 +6X2+ X-1»/16  7.K180O  9.37<0o,360o  (1,-1,0)/(0,1,1)  (X6 + 2X5-9X4-4X3-9X2 + 2X +1)/32  0< 108.10,251.20 8.21<0o,360o 12.24<180o  (1,0,-1)/(1,1,0)  (X5-3X4-2X3-6X2 + X+D/16  5.7<180o 12.36< 0o,360o  (o,i,-i)/(i,o,i)  (X6 + X5-6X4-2X3-3X2 + X)/16  2.33< 112.270,247.73 11.8<0o,360o  (1,0,-1)/(1,1,0)  (X5-2X3 + X)/8  0< 0o,180o,360o  (0,1,-1)/(1,-1,0)  (X6-2X5-9X4 + 4X3-9X2-2X+ l)/32  12.25<0o,360o 8.2K180O  C H A P T E R 6. CONCLUSIONS  A  new  method  for  comparing  coupling  alternatives  in  power  line  carrier  communication systems has been proposed.  The  method has  been applied to practical  recommendations has been generated.  cases,  and a new  In general, the  coincide with those proposed by the IEC in reference  set  of coupling  coupling recommendations [21]; however,  guide does not mention that some of these couplings are not  the TEC  100% safe, nor  does it provide alternatives; furthermore it is suggested there that these couplings are optimum.  To  the  best  of  the  author's  knowledge,  coupling  recommendations  for  three  transposed lines have not been produced before.  The method has been applied also to the study of non-conventional couplings. As it  is  mentioned  in chapter  5,  none  of  the  considered  couplings  resulted  as  attractive as to justify its adoption. This conclusion applies also to the so called mode 1 coupling, which is in its way of becoming a standard practice in North America [22]. Mode 1 is considerably more expensive than the conventional phase to ground or phase to phase couplings.  The method for comparing coupling alternatives is based on a graphical technique for  evaluating power line frequency  responses  proposed by  Senn  [12].  Senn's  technique has been wholy developedin chapter 4. Several gaps that were left out  84  85 in the related publications [12,13,14,20] are presented in detail in chapter 4. The results  obtained  with  the  method  therein described coincide  with those published  by Senn.  In  chapters  parameters that  it  2  and  are  seems  programmable  a  series  of  presented.  These  procedures  that  3,  it  is  possible  calculators.  It  procedures  now  should  be  to  for  simplify  calculating phase the  perform line  mentioned  that  and modal  computations analysis this  by  type  so  much  means of  of  analysis  usually requires a mainframe.  6.1. F U T U R E R E S E A R C H  Along  the  work  became  constant  research  modal  reported  apparent.  in  Three  transformation  this of  thesis,  them  matrices,  are the  several  topics  mentioned other  two  that  next; with  require one  further  deals  with  improvements  to  Senn's method. Constant modal transformation matrices  The  possibility  always  of  attractive,  using  frequency  specially  for transient  drawn some attention recently  From  the  studies  done  for  invariant  transformation  studies  matrices  of transmission  lines.  has  been  This  has  whereas  the  [23].  this  thesis,  it  became  evident  that,  86 transformation matrices of aerial lines are fairly  independent  of frequency,  they  depend heavily on the line geometry, t The methods proposed i n references 2 and 3, and i n section 3.2 of the thesis, may be applied to substantiate  the use of  constant transformations as well as to generate them. Coupling vectors in Senn's method  The way i n which the coupling vectors C to  now- i n Senn's  method,  assumes  r  and C have been implemented -up  implicitly  t  that  the unused  phases are  grounded (at carrier frequencies); accurate modeling of line terminating impedances at  high frequencies  -on the other  hand- is out of question  in most practical  cases. A s an alternative, it is suggested here that the coupling vectors and their associated polynomials be obtained under the assumptions that the unused phases are,  first, terminated  circuit;  the contour  in the line characteristic impedance and, second, in open maps  can then  be elaborated  from  the three  resulting  polynomials on a worst case basis. Pole trajectories in the &a-A0 plane.  In delta lines, the bigger the height of the central conductor is with respect to the external conductors, the less accurately the line modes resemble the Clarke components.  In the example  given in sections  2.3 and 3.4, for instance, p  becomes closer to -3.5 instead of -2.0 and q becomes 0.8 instead of 1.0. Since the value of p affects the line polynomials, it is suggested here that the range  t N O T E : Recalling that the studies here deal with carrier frequencies  87 of feasible 6 Q-A0  values  plane be  for p plotted.  be  determined  and,  from  i t , pole trajectories in the  PAGE IN  88  PAGE  PAGE  88  OMITTED NUMBERING  OMISE  DANS L A P A G I N A T I O N  REFERENCES.  [  1]  Wedepohl.L.  M.,"Application of  Travelling-wave  Phenomena  Matrix  in Polyphase  Methods  to  the  Solution  Systems", Proc. IEE, vol.  of 110,  Dec. 1963, pp. 2200 2212. [  2]  Galloway  R.H., Shorrocks  W.B., Wedepohl  L.M.,"Calculation  of Electrical  Parameters for Short and Long Polyphase Transmission Lines". Proc. IEE, vol. I l l Dec 1964 pp. 2051-2059. t 3] Wedepohl L.M., "Electrical Characteristics of Polyphase Transmission Systems with  Special  Reference  to  Boundary  Value  Calculations  at  Carrier Frequencies." Proc. IEE, Vol. 112, No. 11, November [  4]  Wedepohl  L . M . "Wave  Propagation  in  Nonhomogeneous  Power-line  1965. Multiconductor  Systems Using the Concept of Natural Modes." Proc. TEE, Vol. 113, No. 4, April [  5]  1966  Wedepohl  L . M . , Wasley  R.G.,  "Propagation  of  Carrier  Signals  in  Homogeneous , Nonhomogeneous and Mixed Multiconductor Systems." Proc. IEE., Vol. 115, No. 1 January 1968 [ 6] Dubanton C , "Calcul Approche des parametres Primaires et Secondaries d' une Ligne de Transport. Valeurs Homopolaires.", Bulletin de la Direct, des Et. et Rech. E. D. F., No. 1, 1969, pp. 53-62. [  7]  Gary  C.  Frequence  "Approche Complete  de  par  Matrices  Utilisation  des  la  Propagation Complexes"  Multifilaire E.D.F.  en  Bull,  Haute de  la  Direction des Etudes et Recherches serie B, No. 3/4 1976 pp 5-20. [ 8] L . M . Wedepohl, Personal communication.,  1981  t 9] Frausto J . , Naredo J . L . , De La Rosa R., "Introduction to the Modern  89  Techniques  of  Power  System  Transient  Calculation."  26-th  Midwest  Symposium on Circuits and Systems, Aug. 1983. [10]  Naredo  J.  L.,  Silva  J.  L.,  Romero  R.,Moreno  P.,  "Application  of  Approximated Modal Methods for PLC System Design". IEEE Transactions on Power Delivery, Vol. PWRD-2 No.1, Jan. 1987, pp. 57-63. [11]  Moreno P., De La Rosa R., Naredo J . L . , " Computation of Transmission Line Transients  in the  Frequency Domain and its  Comparison with  Method of the Characteristics.", (Proceedings of the LASTED,  the  International  Symposium High Technology in the Power Industry., Aug 20-22 1986, pp. 234-237. [12] Senn W. H . , "A New Approach to Determine the Carrier Signal Attenuation on Horizontal HV Lines Both under International  Conference  Normal and Abnormal Conditions."  on Large High Voltage Electric Systems,  CIGRE  35-03, 1976 Session, Aug.-25/Sept.-2, [13]  Eggimann  F.,Senn  W.,Morf  K.,"The  Transmission  Characteristics  of  High-Voltage Lines at Carrier Frequencies" Brown Boveri Rev. 8 1977 [14]  Senn  W.  H., "Power  Line  Carrier  Signal  Propagation  Under Abnormal  Conditions." IEE Conference on Developements in Power System Protection., Conference Publication No. 125, March 1975, pp. 168-174. [15]  Carson J . R., "Wave Propagation in Overhead Wires with Earth Return." Bell Syst. Tech. J . 1926, pp. 539-554.  [16]  A. Deri,  G. Tevan, A. Semlyen,  A. Castanheira,  "The Complex Ground  Return Plane, a Simplified Model for Homogeneous and Multi-layer Earth Return.",  IEEE  pp.3686-3693.  Transactions  on  PAS,  Vol.  PAS-100,  Aug.  1981,  91 [17]  Wedepohl  L . M . , Wasley  R.  G., "Wave  Propagation  in  Multiconductor  Overhead Lines - Calculation of Series Impedance for Multilayer Earth.", Proc. IEE, Vol. 113, No. 4, Apr. 1966, pp. 627 - 632. [18] Wedepohl L. M . , "Theory of Natural Modes in Multiconductor Transmission Lines.",  Notes  of  the  Course  ELEC  552,  The  University  of British  Columbia. [19] Pipes L . A.,"Matrix Methods for Engineering.", Prentice-Hall, 1963. [20]  Senn W. H . , Morf K. P.,"Optimum Power Line Carrier Coupling Coupling Arrangement  on  Transposed  Single  Circuit  Power  Lines.",  International  Conference on Large High Voltage Electric Systems.(35-02), 21-29  August  1974. [21]  IEC,"Planning  of  (Single-sideband)  Power  Line  Carrier  Systems.",  International Electrotechnical Commission IEC Report. Publication 663/1980. [22]  "Notes  of  Northwest  Power  Pool  Relay  and  Communication  Meeting.", Vancouver, B. C , Canada, September 23, [23]  Marti  J.  R.,  Dommel  H.  W.,  Marti  Engineers  1986.  L . , Brandwajn  V.,  "Approximate  Transformation Matrices for Unbalanced Transmission Lines.", Submitted to the 9-th Power Systems Computation Conference-PSCC, Aug. 30 - Sept. 4 1987, Lisbon, Portugal.  APPENDIX  Contour maps of the couplings considered in chapters 4 and  92  5  6  IO  15  EO  85  30  104  3BO  o  B  to  is  eo  es  30  106  107  O  B  10  IS  BO  E5  30  


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items