A transmission line conductor structure can be formed either as single layer or multiple layers of strands. The conductor is produced in the form of a stranded structure, produced by the assembly of substructure known as the strand. The strand is the basic structure for the formation of the conductor and the arrangements which comprises of the central core and the helically curved strands in various layer(s). Therefore, the stranded conductor can be treated as composite structure of the assemblies of strands, arranged in layers and each layer is constituted by a helically wound
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profile of a number of strands with a lay angle and the helical strands lay arrangement is alternated in successive layers with opposite lay angle direction. This means that, the stranding is done in the right and left lay direction in the arrangement of alternating layers.
Figure 2.5: The four-layered conductors
For a four-layered conductor as shown in figure (2.5), comprise of a centre core strand, a first layer of 6-strand with left hand lay direction and the second layer of 12-strand in the right-hand layer direction. The normal arrangement indicated that the next successive layers has an addition of 6 strands with an opposite lay angle and so on till the fourth or the outer layer is reached.
2.4.2 Lay lengths and Lay angles
The geometric arrangement of the helical strands of conductors is done in a particular layer with specific lay values. The arrangement of strands in a given layer is a function of its lay length and the lay angle, either in right or left lay direction. The right hand lay arrangement is taken as positive and the left hand lay negative. Figure (2.6), illustrates the helical length (pitch length)and the layer angle and these parameters are very vital in defining the path the strands takes in the conductor geometric formulation.
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Figure 2.6: The pitch length and lay angle
The length of lay or pitch length is the path of the helix formed by a strand in a layer, which is the distance along the conductor for one complete turn of the strand along the axis of the helical path defined with respect to the conductor neutral axis.
For given layer, i, the pitch length PL (i), and the lay angle αi, can be calculated by using the following equations:
i i i
L
P R
tan
2 ……… (2.1)
i
L i i i
i P
d
D
,1
tan
……… (2.2)
Usually the manufacturers may provide these parameters and where provided, the geometric formulation of the conductor will be found using these equations. In the absence of some of these parameters, in this study, these values were obtained based on theoretical evaluation as documented in [56]. The evaluation of these parameters was done using the ratio of the length of lay of a given layer to the diameter of that layer enveloping the strands. This ratio is known as the
“theoretical layer ratio” of the layer. When the concentric lay rule is observed, there is a certain value of lay ratio that results in perfect packing of the strands of the layer such that there are neither gaps nor interference between strands. The value for the “theoretical lay ratio” can be obtained as indicated by C. Rawlins [56]:
1 tan
9 1 1 3
1
2 1 2
,
i i i
i i
i L
n n n
D P
.……….……. (2.3)
In case the number of strands in the first layer; just above the core strand, if it is a 6-strands layer, equation (2.3) will fail to give a real value for the lay ratio. This is because with 6 strands at first layer the solution to this equation is undefined. For this layer, the lay ratio is determined using
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figure (2.7). The graph is obtained by plotting the fill ratio and lay ratio against the number of strands in a given layer. The fill ratio is obtained as the ratio of diameter strand in a given layer to the diameter of strand in the next layer. The interpolation of the graph using the number of strand and the fill ratio gives a layer ratio for a particular strand distribution. In most cases, the size of the central strand and that on the first layer is assumed to be equal, thus the fill ratio for that layer equal to one, the trace along the graph using these two variables is used to determine the lay ratio.
Figure 2.7: Lay fill ratio as function of lay ratio and number of strands [56]
2.4.3 Geometric Description of a Conductor Cross-Section
The conductor, the centreline of generic strands in a given layer can be described as a circular helix path, with radius R, and lay angle α. A typical cross-section of the conductor is made of strands, arranged as represented in figure (2.8). In the top corner of figure (2.8), define the coordinate system used as a reference for the geometric description of the conductor. This Cartesian coordinate system is the right-handed system and positive x-direction determine path of formation of the conductor, the y- and z-axis determine the transverse and normal directions respectively.
This global coordinate system was used throughout this study.
Cross-sections of the strands are usually round shaped. Consider figure (2.8), which is a cross- section of a conductor, made up of strands of circular cross-section. The stranded bare conductor cross-section with a total cross-section area AT, consisting of i-layers (i = 0, 1, 2, …, N), of strands of radius ri (di = 2ri) with lay angle αi, wrapped over the centrally located core strand, of radius
r
0(d0 = 2r0).
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Each particular layer consists of n number of strands. Using the compact arrangement, the number of strands in each layer will be defined as ni = 1, 6, 12, 24, … This may not necessary be the case, as other number of strands can be formed, in the case of Tern conductor, ni = 1, 6, 9,15, and 24.
The position of each strand in a given layer is defined as P (i, j), where (j = 0, 1, 2, …, ni).
Therefore, P (0, 0) defines the position of the core strand and others positions of strands in each layer are defined on yz-plane in figure (2.8) in the anti-clockwise direction along the x-axis. The positions of each strands as located at its cross-section can be determined by:
i j Ri iP , cos ……….……. (2.4)
where o
i o
i 360n :360
:
0
Figure. 2.8: The conductor cross-section
The outer radius and the conductor cross-section can be calculated respectively by 2
2
0 1
1
d d r d
N
i i N
N
.……….……. (2.5)
N 2T r
A ……… (2.6)
The radius between layers and the radius of the circular path that passes through the centre of strands located at i-layer (excluding single layer conductor) from the origin, are calculated by
1 0 0 1 ,
N
i i i
i r d
R and 1
1
0 0
,
N ii i
i r d r
R ……… (2.7)
In the case of a single layer conductor, these values are obtained as:
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0 1 ,
0 r
R and R1r0r1 ..…..……… (2.8)
For the strands arrangement in various layers, parallel to the x-axis, where the distance for the strand centre are defined from the centre of the core. The position of the centre line of each strand in a given layer and the distance along these centres along the curvilinear axis of helix from the conductor neutral axis which is located in the same direction to the x-axis are given as:
ii i i
i r
R,1 , 1sin and
i
i L i
i P
r x x
R 2
sin )
( ……… (2.9)
Where
i incremental helix angle (wrap angle), which is the angular position of strand from the z-axis in the anti-clockwise direction.This form of descriptions given above is applicable to any conductor with strands of circular cross- section.