SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES OF AN UNDERGROUND CABLE SYSTEM by Navaratnam Srivallipuranandan B.E.(Hons.), University of Madras, India, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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, 1986 C Navaratnam Srivallipuranandan, 1986 November 1986 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 The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 6 n/8'i} SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES OF AN UNDERGROUND CABLE ABSTRACT This thesis describes numerical methods for the: evaluation of the series impedance matrix and shunt admittance matrix of underground cable systems. In the series impedance matrix, the terms most difficult to compute are the internal impedances of tubular conductors and the earth return impedance. The various form u hit- for the interim!' impedance of tubular conductors and for th.: earth return impedance are, therefore, investigated in detail. Also, a more accurate way of evaluating the elements of the admittance matrix with frequency dependence of the complex permittivity is proposed. Various formulae have been developed for the earth return impedance of buried cables. Using the Polhiczek's formulae as the standard for comparison, the formula of Ametani and approximations proposed by other authors are studied. Mutual impedance between an underground cable and an overhead conductor is studied as well. The internal impedance of a laminated tubular conductor is different from that of a homogeneous tubular conductor. Equations have been derived to evaluate the internal impedances of such laminated tubular conductors. (ii) Table of Contents Abstract — - - - li Table of Contents - iij List of Table - •• - V List of Figures - -:— VI List of Symbols - Viii Acknowledgement , - •- ix 1. INTRODUCTION .—.:. 1 2. SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES 2.1 Basic Assumptions 6 2.2 Series Impedance matrix [Z] for N Cables in Parallel 7 2.2.1 Submatrix [Z„] 9 2.2.2 Skin Effect 13 2.2.3 Internal Impedance of Solid and Tubular Conductors 14 2.2.4 Submatrix [Z,-yJ - •• 15 2.2.4.1 Proximity Effect 16 2.2.4.2 Proximity Effect of a Single-Phase Circuit of Two Identical Conductors 17 2.2.4.3 Shielding Effect of the Sheath 17 2.2.4.4 Elements of Submatrix [Z„] 19 2.3 Shunt Admittance Matrix [K]; for N Cables in Parallel 22 2.3.1 Leakage Conductance and Capacitive Suceptance 22 2.3.2 Frequency Dependence of the Complex Permittivity 23 2.3.3 Submatrix 27 2.4 Conclusion 29 3. COMPARISON OF INTERNAL IMPEDANCE FORMULAE 3.1 Exact Formulae for Tubular Conductors -•- 30 3.2 Internal Impedance of a Solid Conductor ..— 31 3.3 Internal impedance of a Tubular Conductor ; 36 3.4 Conclusion — -•- 47 (iii) 4. EARTH RETURN IMPEDANCE 4.1 Earth Return Impedance of Insulated Conductor 50 4.2 Earth Return Impedance in a Homogeneous Infinite Earth 50 4.3 Earth Return Impedance in a Homogeneous Semi-Infinite Earth 52 4.4 Formulae used by Ametani, Wedepohl and Semlyen 57 4.5 Effect of Displacement Current Term and Numerical Results 61 4.6 Cable Buried at Depth Greater than Depth of Penetration 64 4.7 Mutual Impedance between a Cable Buried in the Earth and an Over• head Line or vice versa 67 4.8 Conclusion - 69 5. LAMINATED TUBULAR CONDUCTORS 5.1 Internal Impedance of a Laminated Tubular Conductor 70 5.1:1 Internal Impedance with External Return 70 5.1.2 Internal Impedance with Internal Return 72 5.2 Application to Gas-Insulated Substations 73 5.2.1 Case i: Core and Sheath not Coated 74 5.2.2 Case ii: Only Sheath Coated 74 5.2.3 Case iii: Only Core Coated 76 5.2.4 Case iv: Both Core and Sheath Coated 77 5.2.5 Stainless Steel Coating 79 5.2.6 Supermalloy Coating 82 5.2.7 Comparison between Stainless Steel and Supermalloy Coatings 82 5.3 Conclusion 85 6. TEST CASES 6.1 Single-Core Cable 86 6.2 Three-Phase Cable 91 6.3 Shunt Admittance Matrix 93 7. CONCLUSION 94 APPENDIX A 96 APPENDIX B : 98 APPENDIX C 103 APPENDIX D 106 REFERENCES : 114 (Iv); List of Tables 3.1 Internal Impedance of a Solid Conductor 33 3.2 Internal Impedance Za of a Tubular Conductor 38 3.3 Mutual Impedance (Za(,) of a Tubular Conductor with Current Return• ing Inside 43 3.4 Internal Impedance Zy of a Tubular Conductor with Current Returning Outside -. , 44 4.1 Solution of PoIIaczek's Equation by Numerical Integration and Using Infinite Series 58 4.2 Earth Return Self Impedance with and without Displacement Current Term .' 1 61 4.3 Earth Return Self Impedance as a Function of Frequency 64 '• 5.1 Resistivity and Relative Permeability of Coating Materials 79 5.2 Skin Depth of Stainless Steel 79 5.3 Skin Depth of Supermalloy 82 6.1 Impedances of Single Core Underground Cable 87 6.2 Mutual Impedance between Two Cables with Burial Depth of 0.75m and Separation of 0.30m 91 (V) List of Figures 1.1 Potential Difference V, between Core and Sheath and F2 between Sheath and Earth 3 2.1 Basic Single Core Cable Construction 7 2.2 Loop Currents in a Single Core Cable 9 2.3 Potential Difference between Two Concentric Conductors 10 2.4 Three Conductor Representation of a Single Core Cable : 10 2.5 Sheath with Loop Currents Ix and I2 ... •- 15 2.6 Two Cable System , -. 16 2.~! Circuit Arrangement of Primary, Secondary and Shielding Conductors, with Shielding Conductor Grounded at Both Ends".. 18 2.8 Transmission System Consisting of a Single Conductor and a Cable ; 21 2.9 Cross-Section of a Coaxial Cable 23 2.10 (a),(b) - Measurements of e'(<o) and ("(<*) of an OiMmpregnated Test Cable at 20°cC 24 2.11 Values of e'(o>) and «"((•)) Obtained from the Empirical Formula 26 2.12 Polarization-Time Curve of a Dielectric Material 27 3.0 Loop Currents in a Tubular Conductor 30 3.1 (a),(b) - Impedance of a Solid Conductor as a Function of Frequency 3.2 (a),(b) - Errors in Wedepohl's and Semlyen's Formulae for a Solid Conductor 35 3.3 Cross-Section of a Tubular Conductor 30 3.4 (a),(b) - Impedance Za of a Tubular Conductor (with Internal Return): as a Function of Freqency 39 3.5 Errors in Wedepohl's, Schelkunoff's and Bianchi's Formulae for Za 40 3.6 Errors in Wedepohl's and Schelkunoff's Formulae for Zab 42 3.7 (a),(b) - Impedance Zb of Tubular Conductor (with External Return) as a Function of Frequency ..' 45 3.8 Errors in Wedepohl's, Schelkunoff's and Bianchi's Formulae for Zb :.. „ , 46 4.1 Electric Field Strength at Point P 51 4.2 Error in Replacing a Conductor of Finite Radius by a Filament Con• ductor —-- 53 4.3 Solution of Real and Imaginary Part of Equation (4.9), for a Freqency of 1MHz. > - - «... 56 4.4 Relative Error in the Evaluation of Carson's Formulae with an Asymptotic Expansion 59 (vi) 4.5 Error in the Earth Return Self-Impedance if the Displacement Current is Ignored 62 4.6 (a),(b) - Earth Return Self Impedance as a Function of Freqency , 65 4.7 Errors in Earth Return Self Impedance 60 4.8 Differences in Resistance Values of Semi-Infinite and Infinite Earth Return Formulae : 68 5.1 (a),(b) - Numbering of Conductor Layers to Find the Internal Impedances of a Laminated Tubular Conductor 71 5.2 Representation of the nt th Layer 71 5.3 Core and Sheath not coated 74 5.4 Inner Surface of the Sheath only, Coated 75 5.5 Core Alone Coated 76 5.6 Core as well as Inner Surface of the Sheath Coated 77 5.7 Dimensions of the Bus Duct in a Gas-Insulated Substation <• 78 5.8 (a),(b) - Variation of Resistance; and Inductance with Frequency for the Four Cases; Stainless Steel Coating, Thickness, 0.1mm 80 5.9 (a),(b) - Variation of Resistance and Inductance with Freqency for the Four Cases; Stainless Steel Coating, Thickness 0.5mm 81 5.10 (a),(b) - Variation of Resistance and Inductance with Freqency for the Four Cases; Supermalloy Coating, Thickness 0.01mm 83 5.11 (a),(b) - Variation of Resistance and Inductance with Frequency for the Four Cases; Supermalloy Coating, Thickness 0.05mm 84 6.1 Errors in Ametani's and Wedepohl's Approximations in Zcc 88 6.2 Errors in Ametani's and Wedepohl's Approximations in Zgc 89 6.3 Errors in Ametani's and Wedepohl's Approximations in Zef 90 6.4 Errors in Ametani's and Wedepohl's Approximations in the Mutual Impedance between Two Cables 92 A.l Three-Phase Cable Set-up for the Study 96 A. 2 Basic Construction of Each Single Core Cable 96 B. l The Relative Directions of the Field Components in a Coaxial Transmission Line 98 B. 2 Loop Currents in a Tubular Conductor 101 C.