J. Electrical Systems x-x (xxx): x-xx. Regular paper. Reflected Signal on a Nonuniform Overhead Transmission Line at High Frequency

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1 A. Boudjeaa S. Tahi B. Bennaane T. B. Berbar B. Lehouidj J. Electrical Systes x-x (xxx): x-xx Regular paper Reflected Signal on a Nonunifor Overhead Transission Line at High Frequency The effect of the sag of an overhead transission line is nuerically characterized in the case of two conductors above lossy earth at high frequency. The technique uses the chain ABCD atrix for a transission line of two conductors above ground and we present a generalized recurrence forulation of the reported input atrix ipedance for a finite length section of ulticonductor transission line. Then the nonunifor transission line can be subdised into sections of unifor transission line. This technique is used to calculate input ipedance and reflection coefficient of an incurved overhead two-conductor transission line over the 1-30MHz range. Keywords: power line counications, ulticonductor transission line, propagation constant, characteristic ipedance, quasi-tem approxiation, overhead lines, sag. 1. Introduction The power line is originally built for power delivery at low frequencies (50 or 60Hz), however it can be used siultaneously as a counication ediu at high frequencies (1-30MHz). Then an iportant research effort is ade in any countries to enhance the perforance of BPL (Broad band Power Line). One of the iportant liitations of this counication channel is the ultiple reflections that reduce the effective available bandwidth. This work is essentially focused on the reflection coefficient of the incurved overhead two wires transission line. It is interesting to know the electrical characteristics of a Multiconductor Transission Line (MTL) in several configurations. The odal ethod is introduced in 1926 by J.R. Carson [1] for the industrial frequencies and extended by M. D'Aore & al [2] and R.G. Olsen & al [3] to the high frequency range (1-30MHz). This ethod is used for the study of coupling between parallel cylindrical conductors (Figure 1) in the frequency doain. Diaeter, conductity and agnetic pereability of each conductor wire are explicitly taken into account, and the influence of electrical and agnetic ground characteristics can be evaluated. Using the extended ethod, P. Airshahi & al [4] show that the transission capacity of the MTL is very high in ideal and theoretical situation: binary capacity of 1Gbps over 1K is possible. In the case of overhead parallel wires, the structure is not strictly TEM because ground and wires are not perfect conductors. However, under soe conditions, their behaor is quasi-tem ode and the radiation is supposed negligible. C.R. Paul [5] clarifies the widely used decoupling technique for the deterination of propagation constants and characteristic ipedances. Corresponding author : A. Boudjeaa Faculté d'electronique et d'inforatique,université des Sciences et de la Technologie Houari Bouédiène, Alger-Algérie Boudjeaa.Abdelkri@hotail.co, Copyright JES 2010 on-line : journal.esrgroups.org/jes

2 In real situation, the overhead transission line is incurved and the distance between the lowest level of the incurved wire and the ideal straight line is called the sag (fig. 2.a). It is the consequence of dilatation and weight of the conductors[6]. In the situation of a nonunifor MTL, we use ethod usual in icrostrip patches and antennas analysis: ipedance atching [7], icrostrip patch antenna [8] [9], two-conductor icrostrip antenna [9] [11]. The nonunifor MTL is subdised into eleentary unifor MTL then the properties of the chain ABCD atrix can be used for the deterination of ipedance, voltage, current and power of the considered structure. Fig. 1. Parallel Two-conductor transission line with loss 2. Incurved Two-Conductor Transission Line 2.1. Description Voltages and currents waves are propagating along the two conductor transission lines. Figure 2.b represents the voltages and currents of a set of two coupled parallel conductors, the ground is also a conductor. Where: ˆ ( z ) = [ V V ] t Vector of phasor line voltage (with V, 1 2 I ˆ = 1, vector 2 V = and I = I ˆ(0 ) the respect to the reference ground) at the distance z fro the origin, and ( z) [ I I ] t of phasor line current at the distance z fro the origin. ˆ0 Vˆ(0 ) voltage and current phasors at the origin. ˆ0 2

3 (a) Gˆ Y ˆ v ˆ Gˆ i (b) Fig. 2. Overhead two-conductor line with sag. (a) Description. (b) Electrical odel y h(z 0 )=h ax h (z 1 )=h(z 5 ) h (z 2)=h(z 4) h in l l l l l l z 0 z 1 z 2 z 3 L z 4 z 5 z 6 z Fig. 3. Subdision of overhead two-conductor line with sag ( N=6) Fig. 4. Electrical odel of subdised two-conductor transission line (N = 4) 3

4 We consider ˆ in ( z) the input ipedance atrix for any position z along the twoconductor line [5]: Vˆ ( z) = ˆ in ( z) Iˆ( z) (1.a) 0 At the origin (z = 0) we have the following relationship: Vˆ = ˆ Iˆ (1.b) in 0 We can identify the curve of the overhead line (Fig. 2.a) with the following function: ( acosh( h / h ) (2z / 1) ) h( z) = h cosh L in ax in (2) The incurved line (Fig. 3) is subdised into N sections of length l that ust be a sall fraction of the wavelength of the signal. Then the nuber N and the length l verify the following conditions: fax N 16L (3.a) c c l (3.b) 16 f ax To have pertinent frequency response, the nuber N f of frequency values ust verify this condition on the range [f ax,f in ]: ( fax fin ) N f 32L (4) c 2.2. ABCD-Matrix for a nonunifor transission line The electrical odel (Fig. 4) can be forulated as follows: 1 Vˆ( L) Gˆ ˆ ˆ v V0 = (5.a) Iˆ( L) ˆ ˆ Y G ˆ i I0 1 ˆ ˆ 1 Gv ˆ Y M 1 Gˆ i = Aˆ = Yˆ C ( h = N M ) Bˆ 1 Bˆ ˆ C ( h ) Yˆ ( h ) Aˆ ˆ ( h C C ) (5.b) (5.c) Aˆ Bˆ 4 ( l ˆ( γ )) = cosh (5.d) h ( l ˆ( γ )) = sinh (5.e) h

5 γˆ : Propagation atrix Ẑ C : Characteristic ipedance atrix of the MTL ˆ ˆ 1 Y C = C : Characteristic adittance atrix of the MTL The eleents of the propagation and the characteristic ipedance atrices are calculated using the forulas presented by [2] and used by [4] under the usual quasi-tem restrictions: d << λ, hi << λw, a i << hi, a i << d (Fig. 1). Where λ and λ w denote wavelengths in vacuu and in conductors respectively. We notice that hyperbolic sin and cos of atrices appears in (5.d) and (5.e), the author of [12] presented powerful and efficient nuerical algorith to calculate the Recurrence ipedance relationship They are evaluated with the following recurrence expressions. Starting with the right extreity (Fig. 4), a atched load is connected at interface N + 1 : ˆ ˆ ˆ + = ( ) (6) N 1 L = C h ax Then, we copute successively the input ipedance at each interface as follow (index is decreented fro N + 1 to 1): ˆ ˆ ˆ ˆ (7.a) 1 1 = X Y C, 1 X ˆ ˆ ˆ ˆ ˆ (7.b) = A 1 + B 1 C, 1 Y ˆ ˆ ˆ ˆ ˆ (7.c) = B 1 + A 1 C, 1 This step is repeated until we reach the feed location where ( = 1) ˆ ˆ in = Input ipedances and reflection coefficients The attenuation of the two-conductor transission line can be deduced by coparing the total active power P in entering into the syste with the power P L flowing into the load. The input ipedance atrix derived fro recurrent relation (7) can be written: ˆ in11 in 12 in = (8) in21 in22 Fro Figure 5.a, the coon ode input ipedance is defined as: V inc = (9) I Where V = V 01 = V02 and I = I 01 + I02 5

6 Gˆ Y ˆ v ˆ Gˆ i (a) Gˆ Y ˆ v ˆ Gˆ i (b) Fig. 5. Excitation ode of the MTL. (a) Coon ode. (b) Differential ode. Moreover, fro Figure 5.b, the differential ode input ipedance is defined as: V = Dif ind I (10) Where V Dif = V 01 V02 and I = I 01 = I02 Finally, the expression of the coon and differential ode input ipedances are respectively: + in11 in22 in12 in21 inc = (11.a) in 11 + in22 in 12 in21 ind = + (11.b) in11 in22 in12 in21 The reference ipedance atrix is: ˆ N11 N12 N = (12) N 21 N 22 Coon and differential ode reference ipedances are respectively: + N11 N 22 N12 N 21 Nc = (13.a) N11 + N 22 N12 N 21 Nd = + (13.b) N11 N 22 N12 N 21 The reference ipedance atrix can be chosen arbitrary but the better choice is the characteristic ipedance atrix of the first segent of the MTL. 6

7 R(Ω) h ax =10 sag=1 sag=2 sag=3 sag= (a) h ax =10 sag=1 sag=2 sag=3 sag=4 8 X(Ω) (b) Fig. 6. Coon ode input ipedance of the two-conductor line above ordinary soil. (a) Resistance. (b) Reactance. 7

8 R(Ω) h ax =10 sag=1 sag=2 sag=3 sag= (a) X(Ω) h ax = sag=1 sag=2-2.6 sag=3 sag= (b) Fig. 7. Differential ode input ipedance of the two-conductor line above ordinary soil. (a) Resistance. (b) Reactance. 8

9 sag=1 sag=2 sag=3 sag=4-50 Γ (db) (a) sag=1 sag=2 sag=3 sag=4-80 Γ (db) (b) Fig. 8. Reflection coefficient of the two-conductor line above ordinary soil. (a) Coon ode. (b) Differential ode. 9

10 sag=1 sag=2 sag=3 sag=4-50 Γ (db) (a) sag=1 sag=2 sag=3 sag=4-80 Γ (db) (b) Fig. 9. Reflection coefficient of the two-conductor line above wet soil. (a) Coon ode. (b) Differential ode. 10

11 Then, the coon and differential reflection coefficients are respectively: inc Nc Γ c = (14.a) inc + Nc ind Nd Γ d = (14.b) ind + Nd These expressions are used to study lines that have a curvature, it becoes then easy to quantify the effect of the sag of an overhead line on the frequency response. 3. Nuerical application A nuerical application is ade to quantify the effect of the sag at high frequency. During siulation, it is assued in all cases, spacing between wires is d = 1, the axial height of the wires is equal to h ax = 10, and the transission line has a span of L = 100. We consider average (σ g = 0.005S/, ε g = 13) and wet (σ g = 0.01S/, ε g = 30) soils. The results for coon and differential odes are presented. In the case of ordinary soil, we copare (Fig. 6 and 7 ) the input ipedance of the unifor straight line (h ax = 10) with that of the incurved line for any values of the sag (1, 2, 3 and 4 ). Oscillations appear for the input ipedance in the curved line case. This effect is ore pronounced for a large sag. But it is less iportant for the differential ode (Fig. 7). The figures 8 and 9 show respectively the reflection coefficient for an ordinary and a wet soils. We notice that the soil characteristic has a negligible effect on the reflection coefficient. However, the latter is less than -20dB for the coon ode and -50dB for the differential ode. It is concluded that the reflection coefficient increases with the sag. This work clearly shows that the sag of an overhead line has a significant influence on the reflection coefficient and therefore on the useful bandwidth at high frequencies (1 to 30 MHz) of a signal transitted over this kind of line. References [1] J.R. Carson, Wave propagation in overhead wires with ground return, Bell Syst. Techn. Journal, Vol.5, 1926, pp [2] M. D'Aore, M.S. Sarto, A New Forulation of Lossy Ground Return Paraeters for Transient Analysis of Multi-Conductor Dissipative Lines, IEEE-T-Power Del., Vol.12, n 1, January 1997, pp [3] R.G. Olsen, M.D. Wu, High Frequency Propagation Losses on an Open Wire Transission Line Above Dissipative Earth, IEEE-T-Broadcast., Vol.34, n 2, June 1988, pp [4] P. Airshahi, M. Kavehrad, High-Frequency Characteristics of Overhead Multiconductor Power Lines for Broadband Counications, IEEE Journal on Selected Areas in Counications, Vol.24, n 7, July 2006, pp [5] C.R. Paul, Decoupling the Multiconductor Transission Line Equations, IEEE-T-MTT, Vol.44, n 8, August 1996, pp [6] M. Muhr, S. Pack, R. Schwarz, S. Jaufer, Calculation of Overhead Line Sags, 51st Internationales Wissenschaftliches Kolloquiu, Technische Universität Ilenau, Septeber 2006 [7] S.J. Orfanidis, Electroagnetic Waves and Antennas, Rutger University, USA, 2008 [8] G. Dubost, Linear Transission Line Model Analysis of Arbitrary Shape Patch Antenna, Electronics 11

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