Fiber Optic Cable Transmission Losses with Perturbation Effects

Transcription

1 Fiber Opti Cable Transmission Losses with Perturbation Effets Kampanat Namngam 1*, Preeha Yupapin 2 and Pakkinee Chitsakul 1 1 Department of Mathematis and Computer Siene, Faulty of Siene, King Mongkut s Institute of Tehnology Ladkrabang, Bangkok 10520, Thailand 2 Light-wave Tehnology Researh Center (LTRC), Department of Applied Physis, Faulty of Siene, King Mongkut s Institute of Tehnology Ladkrabang, Bangkok 10520, Thailand *Corresponding author. knkampan@kmitl.a.th ABSTRACT The transmission losses in a fiber opti able hanging on an eletrial power tower (EPT) in terms of able radius of urvature is investigated. The onstruted mathematial modeling is formed, relative to the able parameters, suh as sag and hanging height whih an be used to design and predit the industrial able for EPT span, based on the transmission losses with the minimum tension and ost. The stability of the analytial solution with small perturbation in terms of able lengthening and inlined EPT is studied. The results obtained show that the transmission losses are still valid to use in transmission line within the Southeast Asia region. Key words: Mathematial modeling, Fiber opti able design INTRODUCTION Generally, types of fiber opti ables are divided into indoor and outside-plant ables. There are several types of outside-plant ables suh as aerial, underground, submarine and onduit ables (Koehler, 1989; Lee and Hong, 2001), where eah of them is suitably employed in different environmental onditions, i.e., different part of the world. The able parameters suh as able length, weight, EPT span, transmission losses and the surrounding environment are ommonly involved in the able design and assembly. The mathematial method known as alulus of variation is one of the subjets that an be used to design and optimize the fiber opti able parameters (Allard, 1990; Mathis and Vontobel, 2000), while the least ation priniple is also popularly used in several subjets that the problem requires the minimum spae and time variations. In this work, the aerial fiber opti able is required to use under Thailand environment where the optimum in optial power, transmission loss and installation ost are the basi requirements. The analytial solution and the small perturbation in terms of able lengthening are studied. MATHEMATICAL MODELING When a fiber opti able is hung on the EPT span, i.e., the able is drawn by gravitational fore, then the able shape is formed into a atenary s urve. Mathematial

2 modeling of this urve is desribed by using a geometrial form as shown in Figure 1 when x is an EPT half-span, y is a height of hanging able, l is a able length for the EPT half-span (dotted line), h is a sag and is a distane between a minimum point of the hanging able and ground, respetively. h l y ground level span x Figure 1. Shemati presentation of able on the EPT span. The height of hanging able for eah EPT span, whih the atenary s equation is y = osh x (1) by Taylor s theorem y = n = 0 1 x 2 n (2n)! 2 n-1 (2) when x is a onstant and is a positive number, let = 1 x be a ritial point where a is a real a number, thus - (2n -1) a 2 n = 0 n = 0 (2n)! (3) the solution is a 1.2. Obviously, y'' > 0 for all > 0 and x 0. Therefore = 1 x is a minimum point, this value ours at a minimum tension. 1.2 The able marobend loss by hanging an be expressed as Allard, 1990: bend = 10log log R 2 b (4) where b is a ore radius, is an index elevation, R is a bend radius and is the index profile parameter. In this problem, the transmission losses of hanging able are onsidered in terms of radius of urvature, where the radius of urvature at original equal is represented by radius R. The bend loss is onsidered from radius of urvature, thus b,, are remiss. Let

3 transmission loss be formed, the radius of urvature as. loss = 10 log. (5) PERTURBATION EFFECTS The stability of solution with small perturbation in terms of able lengthening and the EPT are studied, when this two ases our with installation. Perturbation with Cable Length The able length for eah EPT half-span forms a atenary urve is l = sinh x (6) when the length of a fiber opti able, l is prepared for installation for the EPT half-span x, but the EPT half-span is perturbed by, thus the effet for parameter, P is hanged as equal to. With this parameter, P is affeted by other values with respet to analytial method (dotted line) as shown in Figure 2. l h P y P P Figure 2. Shemati presentation of the able perturbation effet. Sine the length of a able from analyti and perturbation is equal, by using Taylor s theorem expand as x 2 n -1 = (x + ) 2 n-1 (7) (2n-1)! 2 n-2 n = 1 2n-2 n=1(2n-1)! P n = 1 x + where P = + and = a 1 x. Varying the parameters x and, the solution is for eah EPT half-span, whih affet other values as shown in Table 1. The EPT span is inreased but the able length does not hange. Thus the tension of able, T is alulated in terms of perentage of hange as

4 T - T P x 100 = wy - wy P x 100 = y - y P x 100. T wy y (8) Table 1. Results of small perturbation of the able length. = 0.1 span sag distane height loop loss loss/length %tension = = Perturbation with EPT When the EPT are inlined by angle, 1 with the vertial, the shape of the EPT is studied as two ases based on analytial. Where the small perturbation with the EPT ours by symmetry shape, the first ase is shown in Figure 3.

5 x 1 1 l 1 1 y P1 Figure 3. Shemati presentation of the able perturbation effets (first ase). For the dotted line, the length of a able is alulated by analytial method, and the value in this paper does not hange for EPT. Thus this parameter, 1 affets other values as y P 1 = y sin (90-1 ), (9) and P 1 = + 1, (10) where 1 = (y sin (90-1 )) 2 - l 2 -. (11) Table 2. Results of small perturbation effet with EPT. = 1 span sag distane height loss loss/length %tension

6 = = The seond ase, the EPT are inlined by ontrary angle with the first ase, 2 with the vertial, shown in Figure 4. x 2 2 l 2 2 y P2 Figure 4. Shemati presentation of able perturbation effets (seond ase). The results are alulated for some small perturbation with inlined by angle, 2 and the EPT span, shown in Table 3.

7 Table 3. Results of small able perturbation with ontrary EPT. = 1 span sag distane height loss loss/length %tension = = DISCUSSION Cable losses are onsidered in respet to only the able radius of urvature, whih means the able bend loss may be inreased if inluding the other parameters suh as b,,. The able tension is alulated by using the minimum onditions, therefore, in general the other parameters suh as fiber opti able strength, small values of wind and temperature effets, i.e., perturbation may be involved. The result of able perturbation is shown in Table 1. The small perturbation with able length is inreased for eah EPT span, then distane between a minimum point of the hanging able, sag, height of hanging able, transmission losses, losses per unit length and perentage of tension are inreased, while the loop storage is dereased. The EPT span is inreased for eah small perturbation of transmission losses, losses per unit length and tension whih approah to the analytial results. Results of perturbation with EPT are shown in Tables 2 and 3, where both inlined EPT for the angles

8 are the same for eah EPT span, height of hanging able and tension. The small perturbation from angle is inreased for eah span. when the distane between a minimum point of the hanging able, transmission loss and losses per unit length are inreased, while the able sag and height of hanging are dereased. The inlined EPT by ontrary angle with the previous ase, the small perturbation from angle is inreased for eah EPT span, when the distane between a minimum point of the hanging able, transmission losses and losses per unit length are dereased, but sag and height of hanging are inreased. The EPT spans are inreased for some small perturbation, transmission losses and losses per unit length that approah to the results from analytial method. In this study, length of fiber opti able in terms of sag, span, distane between a minimum point of the hanging able, loop storage and height ould be presented in terms of kilometer, meter, i.e., arbitrary unit (a.u.) and transmission losses ould be presented in terms of deibel (db). CONCLUSION In onlusion, results of atenary problem of the hanging able is studied in terms of able radius of urvature. The transmission losses are investigated for some EPT span, the results of analytial method with the perturbation effets for some EPT span are shown. The optimum parameters obtained have shown that the transmission losses are still valid for transmission link, with power transmission losses of 10 db remain in the range of fiber opti able length of 20 km. The ontinuing work is foussed on the onstrution of the fiber opti able that will result in the minimum installation ost and power transmission loss of able on the EPT span within the Southeast Asia region (Thailand). The other types of fiber ables will be installed and tested. The fiber loss with lightning effets will also be investigated. ACKNOWLEDGEMENTS The authors wish to thank the Faulty of Siene, King Mongkut s Institute of Tehnology Ladkrabang for providing failities to arry out this researh work. REFERENCES Allard, F. C Fiber opti handbook for engineers and sientists. MGraw-Hill, New York. Koehler, C. W Analysis of basi position loop ontrol systems: Conferene of IEEE on Industry Appliations Soiety Annual Meeting. Vol.2: Lee, Y. S., and S. I. Hong A Study of maintaining ontrol of tension for a atenary s able. Proeeding of ISIE International Symposium on Industrial Eletronis. Vol.1: Mathis, H., and P. O. Vontobel Shape optimization for a retangularly onstrained small loop antenna. Proeeding of International Zurih Seminar on Broadband Communiations. p