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Version 02.04.2007

BROAD RANGE ALGORITHM FOR THE EVALUATION OF CARSON’S INTEGRAL

USER’S MANUAL

By:

F. Alejandro Uribe Abner Ramirez

e-mails: fauribe@ieee.org, abner.ramirez@cts-design.com CONTENTS:

1. INTRODUCTION………………………………………………… 1 2. PROGRAM FILES…………………………………………………2 3. DESCRIPTION OF THE ROUTINES ………………………….... 3 4. EXAMPLES………………………………………………………..4

I. INTRODUCTION The algorithms described in this manual solve efficiently and accurately Carson’s integral for any practical configuration of overhead transmission lines. The algorithms can be used by practical engineers having available the geometry of the line and by power systems researchers who may like work with parametric solutions. This manual describes the algorithms, the program files, and their way of use. The user has two choices; both of them yield as output the ground impedance. The first one has as input data the geometry of the line. The second one performs the calculation based on parametric data. The program is of public domain and can be freely used by anyone and downloaded from either http://www.gdl.cinvestav.mx/jcinv/minisitios/abner/ or http://www.cucei.udg.mx/~furibe/ We kindly ask the user to make reference to: [1] A. Ramirez and F. Uribe, “A broad range algorithm for the evaluation of Carson’s integral”, IEEE Trans. on Power Delivery, vol. 22, no. 2, pp. 1188-1193, April 2007.

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II. PROGRAM FILES The routines for the algorithms below are in Matlab v7 code. Choice 1. (based on geometry) Main program: carson_algorithm.m Auxiliary programs: one_osc.m oscillations.m no_oscillations.m Choice 2. (parametric solution) Main program: carson_broad_range.m Auxiliary programs: one_osc.m oscillations.m no_oscillations.m

III. DESCRIPTION OF THE ROUTINES 3.1. Ground return earth impedance through Carson’s integral (based on geometry) % function [Zcar, Qcar] = carson_algorithm(w,hi,hj,x,rhog) % % Routine in Matlab-V7 for calculating the self or mutual ground return % impedances of an overhead transmission line by solving Carson´s integral % with the algorithm described in the paper: % % A. Ramirez and F. A. Uribe, "A broad range algorithm for the evaluation of % Carson´s Integral", IEEE Trans. on Power Delivery, Vol.22, No.2, pp. % 1188-1193,2007. % % Running the program, % % Zcar = carson_algorithm(w,h1,h2,x,rhot,miurel), % % where: % Zcar - self or mutual ground return impedance p.u. length (ohm*m), % w - angular frequency (equal to 2*pi*f) (rad/seg), % p - Skin Effect thickness, % hi - conductor "i" height, % hj - conductor "j" height, % x - horizontal distance between conductors % for the self impedance set it equal to zero,

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% rhog - is the ground resistivity (ohms*m), % miu0 - Is the ground permeability (H/m), % % [Version 1.0-revised 02 Apr/2007], 3.2. Broad range ground return earth impedance through Carson’s integral (parametric solution) % carson_broad_range.m % % Evaluates Carson's integral for a broad range of line configurations. % % The used concepts are described in the paper: % % A. Ramirez and F. A. Uribe, "A broad range algorithm for the evaluation of % Carson´s Integral", IEEE Trans. on Power Delivery, Vol.22, No.2, pp. 1188-1193,2007. % % To run the program type at the Matlab prompt: carson_broad_range.m, % % % delta - Skin Effect thickness, % h - conductor average height, % Qcar - Carson's integral % p - Normalized parameter p = 2h/abs(delta) % eta - Normalized parameter eta = x/2h % % [Version 1.0-revised 02 Apr/2007],

IV. EXAMPLES 4.1. Choice 1. (based on geometry) Consider the following parameters (see Fig. 1 from the paper):

m 15=ih , m 20=jh , m 3=x , m 202 =h , m-S 100/1=σ , rad/s 102 3×= πω We type: >>[Zcar, Qcar] = carson_algorithm(w,hi,hj,x,1/σ) Solution: Zcar = 0.0008 + 0.0024i Qcar = 0.9591 - 0.3351i 4.2. Choice 2. (parametric solution) The user just has to type: >>carson_broad_range Then, the user should be able to obtain the results as in Fig. 1 below (as shown also in Fig. 4 from the paper referenced above)

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Fig. 1. a) Integrand, real part b) Integrand, imaginary part

In case the user wants to calculate Carson’s integrand for a different range, lines 24 and 25 of the program have to be modified accordingly.

V. ADDITIONAL ROUTINES For comparison purposes, we have added an additional routine. This routine compares the proposed algorithm with Carson’s series (version EMTP) and adaptive quadrature Gauss/Lobatto numerical integration for the parametric case (broad range). The routine is named: comparison.m In addition to Fig. 1, the user should be able to obtain the results shown in Fig. 2.

Fig. 2. a) Impedance, real part b) Impedance, imaginary part