Another approximation is given by the following formula due to Wedepohl and Wilcox [1]:
where γ is Euler’s constant.
For mutual ground-return impedances, one should apply D1, x, and Y as in Equation 3.12, while for self-ground-return impedances one should instead use Equation 3.13.
An additional approximation for calculating ground-return impedances, known as the Infi nite Earth Model [6], is obtained by considering the ground as a hollow cylinder of infi nite external radius. On taking a = r in Equation 3.5 along with the limit of b that tends to infi nity, the following expression is obtained for self Zg:
( ) ( )
The expression corresponding to mutual Zg is [6]:
( )
where D1 is given by Equation 3.12.
3.4.1.4 Accuracy of Ground-Return Impedance Evaluation Methods
Pollaczek integral in Equation 3.11 does not possess a closed-form solution, and its numeri-cal integration cannot always be obtained reliably by general purpose integration routines.
An integration method that guarantees its accurate solution is described in Uribe et al. [25].
This method, however, requires a considerable amount of computations. In consequence, it is not recommended here for its incorporation into CC routines. Instead, this method is adopted as reference to evaluate other methods as well as approximate formulas. It has been used, for instance, to assess solutions to Pollaczek Integral by the Gauss-Lobato quadrature routine [26], as well as by the Monte-Carlo quadrature method [27].
Pollaczek integral, in principle, is a function of fi ve physical variables: three electrical ones, μ, ω, and σ, and two geometrical ones, Y and x. The latter variables are defi ned by Equations 3.12 and 3.13. Uribe et al. [25] have shown, however, that this dependency can be reduced to the following two dimensionless variables only:
ξ = Y p (3.29)
η =x D2 (3.30)
where p = 1/m is the complex thickness of the Skin Effect layer in the ground.
To assess approximate methods for calculating ground-return impedances, fi rst, the following expression is adopted:
∈ =% g-REF− g-APPROX ×
g-REF
Z Z 100
Z (3.31)
where
∈% is the percent relative error
Zg-REF is the ground-return impedance obtained from the reference method
Zg-APPROX is the ground-return impedance obtained with the method being assessed Then, the two dimensionless variables ξ and η, defi ned by Equations 3.29 and 3.30, are introduced. By doing this, the error term ∈% defi ned by Equation 3.31 becomes a function of these two dimensionless variables only. Next, the following ranges are established for ξ and η to encompass most practical cases:
−4≤ ξ ≤ 2
10 10 (3.32)
−3≤ η ≤ 2
10 10 (3.33)
The ranges have been obtained by considering the variations shown in Table 3.4 for physi-cal variables D2, x, ω, and σ. Magnetic permeability μ is considered here as constant and is equal to that of vacuum (μ0 = 4π × 10−7 H/m). Finally, the ranges defi ned by Equations 3.32 and 3.33, along with error formula (Equation 3.31), are used to produce error contour-maps, whose usefulness is illustrated in the following example.
Example 3.1
Consider a system of two cables with an external radius of r = 0.04 m, buried at a depth of Y = 2 m, with a horizontal distance of x = 2 m between them, the ground conductivity being σ = 0.1 S/m. Suppose that, for a particular study, the range of relevant frequencies is between 60 Hz and 200 kHz. For the self-impedance case, at a frequency of 60 Hz, Equations 3.13 and 3.29 result in ξ = 0.0275, while Equations 3.12 and 3.30 yield η = 0.01. These two values correspond to point P1 in Figures 3.9 and 3.10. As the frequency is varied continuously from 60 Hz to 200 kHz, trajectory P1−P2 is drawn in these two fi gures. For P2, at 200 kHz, ξ = 1.589 and η remains con-stant. This trajectory shows the global errors in the evaluation of Zg as a particular approximation is employed. A comparison of errors between Carson Integral approximation and SGG formula for the self-impedance evaluation is provided in Figure 3.11a. One can observe here that for the particular case in consideration, the Carson approximation can produce errors above 10%, while those for the SGG formula are below 1%. Trajectory P3−P4 in Figures 3.9 and 3.10 corresponds to the values of ξ and η for the mutual impedance evaluation within the range of frequencies from 60 Hz to 200 kHz. Values of ξ for P3 and P4 are equal to those of P1 and P2, respectively, while η takes a constant value of 0.05. Figure 3.11b shows the error plots for the two methods being considered here. Note that the approximation by Carson produces an error greater than 10% at frequencies above 10 kHz, whereas the one for the SGG formula is less than 5% for the entire TABLE 3.4
Variation Ranges for Physical Variables
0.2 ≤ Y ≤ 2 × 102 (m) 2π ≤ ω ≤ 2π × 106 (rad/s)
10−4 ≤ σ ≤ 1 (S/m)
10−2 ≤ x ≤ 103 (m)
ξ
Example 3.1: Error map for evaluating Zg by the Carson Integral Method.
ξ
Example 3.1: Error map for the SGG formula.
Frequency (Hz)
Example 3.1: (a) Plot of error vs. frequency for self-ground-return impedance. (b) Plot of error vs. frequency for mutual ground-return impedance.
range of frequencies. Figure 3.12a and b provides the plots for the self-ground-return resistance and the self-ground-return inductance as obtained by the adopted integration method (labeled as Pollaczek in these fi gures), by the Carson approximation, and by the SGG formula. Figure 3.13a and b provides the corresponding plots for the mutual ground-return resistance and the mutual ground-return inductance.
3.4.2 Pipe-Type Cables
3.4.2.1 Series Impedance Matrix
Figure 3.14 shows the cross section of a pipe-type cable with a cradle confi guration. Since the penetration depth into the pipe at 50/60 Hz is usually smaller than the pipe thickness, it is reasonable to assume that the pipe is the only return path and the ground-return current can be ignored. In this case, an infi nite pipe thickness can be assumed. A technique to take into account the ground-return current is detailed in [2,6].
For each coaxial cable in the pipe, the impedance matrix for circulating currents given in Equation 3.14 can be used.
The matrix elements are calculated using Equation 3.15, except for Z33, which is replaced by
= + − +
33 bb(armor) i(armor pipe) aa(pipe)
Z Z Z Z (3.34)
where Zbb (armor) is obtained from Equation 3.6.
0
Example 3.1: (a) Plot of self-ground-return resistance vs. frequency. (b) Plot of self-ground-return inductance vs. frequency.
Example 3.1: (a) Plot of mutual ground-return resistance vs. frequency. (b) Plot of mutual ground-return induc-tance vs. frequency.
Pipe Coaxial
cables
FIGURE 3.14
Cross section of a pipe-type cable.
Since the conductor geometry of a pipe-type cable is not concentric with respect to the pipe center, the formula for Zi(armor–pipe) is somewhat complicated compared with Equation 3.10:
μ is the permeability of the insulation between the armor and the pipe R is the radius of the pipe
r is the radius of the armor of interest
d is the offset of the coaxial cable of interest from the pipe center On the other hand, Zaa(pipe) is calculated from the following expression:
∞
m is given in Equation 3.7
μ = μ0μr is the permeability of the pipe Kn′ ( · ) is the derivative of Kn( · )
To take into account the mutual impedance among the coaxial cables in a pipe (the three-phase case is shown here), the 9 × 9 impedance matrix for circulating currents given in Equation 3.18 has to be built. Since an infi nite pipe thickness is assumed, Zg,ab, Zg,bc, and Zg,ca are replaced by Zp,ab, Zp,bc, and Zp,ca (the subscript “p” designates a pipe) and they are deduced by substituting the phase indexes a, b, and c into i and j in the formula below.
di is the offset of the i-phase coaxial cable from the pipe center dj is the offset of the j-phase coaxial cable from the pipe center
θij is the angle that the i-phase and the j-phase cables make with respect to the pipe center
Equations 3.35, 3.36, and 3.37 are from Brown and Rocamora [28]. A method to take into account the saturation effect of a pipe wall was presented in Brown and Rocamora [29].
3.4.2.2 Shunt Admittance Matrix
The inverse of Y3×3 in Equation 3.21 multiplied by jω gives the p.u.l. potential coeffi cient matrix of each coaxial cable in the pipe. If potential coeffi cients of phases a, b, and c are denoted as Pa, Pb, and Pc, the potential coeffi cient matrix of the whole cable system, includ-ing the pipe, is written in the form
×
where the submatrices Pab, Pbc, and Pca consist of nine identical elements which can be calculated by substituting the phase indexes a, b, and c into i and j in the following formulas [28]:
where ε is the permittivity of the insulation between the armors and the pipe.
Finally, the p.u.l. shunt admittance matrix is calculated as follows:
× = ω −×1
9 9 9 9
Y j P (3.41)