Analysis of approximate formulas of internal impedance for the study of high-frequency transients in high-voltage underground cables

José Miguel García Guzman, Jacinto Torres Jiménez, Francisco Javier Ortega Herrera, Alfonso Lozano Luna


This study presents an analysis of approximate formulas of internal impedance for the study of high-frequency transients in high voltage underground cables. The analysis is conducted in the frequency domain, and converted to the time domain performed using a Fast Fourier Transform. The cable is modeled as a multiconductor transmission line, using modal analysis and two-port network theory. The approximate internal impedance formulas analyzed are those proposed by Gary and Wedepohl-Wilcox, used in modeling overhead and underground transmission lines, respectively. To conduct the analysis, formulas are compared with the exact model of internal impedance and the relative error of each formula is determined for real and imaginary values. The results show that the approximate formula proposed by Gary for modeling overhead transmission lines can be used to analyze high-frequency electromagnetic transient phenomena in underground cables with acceptable results. In order to carry out a comparative study of the approximate formulas of internal impedance, a series of case studies are presented, in which the magnitude of the transient voltage, the transit time and wave attenuation are determined in different lengths of single-phase underground line.


underground cables; relative error; internal impedance; approximate formulas; high-frequency transients;


J. Torres, “Análisis transitorio de redes de tierra en el dominio de la frecuencia”, Tesis de maestría, Instituto Tecnológico de Morelia, Morelia, Mich., 2004.

F. Pollaczek, “Uber das feld einer unendlich langen wechsel stromdurchflossenen Einfachleitung”. Electrishe Nachrichten Technik, Vol. 3, No. 9, pp. 339‑360,

L. V. Bewley, “Critique of Ground Wire Theory”.AIEE Transactions, Vol. 50, No. 1, pp. 1‑22, Mar.,1931.

P. L. Bellaschi, “Impulse and 60-Cycle Characteristics of Driven Grounds”. aiee Transactions, Vol. 60, pp.123‑128, Mar., 1941.

R. Rudenberg, “Fundamental Considerations on Ground Currents”. Electrical Engineering, Vol. 64, pp.1‑13, Ene., 1945.

E. D. Sunde, “Earth Conduction Effects in Transmission Systems”. Dover Publications Inc., Nueva York, 1968.

L. M. Wedepohl y D. J. Wilcox, “Transient and Harmonic Induction in Underground Cables Systems”, Proc. iee, Vol. 120, pp. 253‑260, 1973.

L. Marti, “Simulation of Transients in Underground Cables with Frequency Dependant Modal Transformation Matrices”, IEEE Transactions on Power Delivery, Vol. 3,

No. 3, pp. 1099‑1110, 1988.

N. S. Nahman, y D. R. Holt, “Transient analysis of coaxial cables using the skin effect approximation A + B s", IEEE Transactions on Circuit Theory, Vol. 19, No. 5, pp. 443-451, 1972.

A. Semlyen y A. Deri, “Time domain modeling of frequency dependent three phase transmission line impedance”, IEEE Transactions on Power Apparatus and Systems, Vol. 104, No.6, pp. 1549‑1555, 1985.

L. M. Wedepohl y D. J. Wilcox, “Transient analysis of underground power transmission systems: System-model and wave propagation characteristics”, IEE Proceedings

on Generation, Transmission and Distribution, Vol. 20,No. 2, pp. 253‑260, 1973.

S. Vujević, V. Boras y P. Sarajčev, “A novel algorithm for internal impedance computation of solid and tubular cylindrical conductors”, International Review of Electrical Engineering (IREE), Vol. 4, No. 6, Part B, pp.1418‑1425,2009.

W. Mingli y F. Yu, “Numerical calculations of internal impedance od solid and tubular cylindrical conductors under large parameters”, IEE Proceedings on Generation, Transmission and Distribution, Vol. 151, No. 1, pp. 67-72, 2004.

D. W. Knight, “Practical continuous functions and formulae for the internal impedance of cylindrical conductors”. Version 2.04, Mar., 2010. [En linea]. Disponible en:

J. Torres, A. I. Jardines y J. L. Guardado, “Análisis Transitorio de Cables Subterráneos”. Memorias de la XI Reunión de Otoño de Potencia, Electrónica y Computación (ROPEC,2009), Morelia, Mich., México, 2009, pp. 106‑110.

W. Johnson y M. Gram, High-Speed signal propagation: advanced black magic. Saddle River, NJ, Prentice‑Hall, 2003, pp. 67‑73.

C. Gary, “Approche complete de la propagation multifilaire en haute frequence par utilization des matrices complexes”, edf Bulletin de la Direction des Etudes et Recherches, No. 3‑4, pp. 5‑20, 1976.

F. A. Uribe, “Evaluacion Algoritmica de la Integral de Pollaczek y sus aplicaciones en el Analisis de Transitorios Electromagnéticos en Sistemas de Transmision Subterránea”, Tesis de doctorado, Centro de Investigación y de Estudios Avanzados del IPN, Guadalajara, Jal., México,

Análisis de formulaciones aproximadas de impedancia interna 23

O. Saad, G. Gaba y M. Giroux, “A Closed-Form Approximation for Ground Return Impedance of Underground Cables”, IEEE Transactions on Power Delivery, Vol. 11, No. 3, pp. 1536‑1545, Jul., 1996.

Unión Fenosa Internacional, “Conductores Desnudos de Aluminio con Alma de Acero (ACSR)”. Anexo 2, 2004.

F. A. Uribe, J. L. Naredo, P. Moreno y L. Guardado, “Electromagnetic Transients in Underground Transmission Systems Through the Numerical Laplace Transform”.

International Journal of Electrical Power and Energy Systems, Vol. 24, No. 3, pp. 215-221, Mar., 2002.

comments powered by Disqus

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Revista indexada en:



Línea gratuita nacional

01 8000 420101


Facultad de Ingeniería
Avenida Caracas no. 37-15 
Bogotá, D.C.


(57) (1) 3323565

(57) 3004956353

Revista en OJS implementada por Biteca Ltda.