4 1 CANTO TERCERO
48 ARTE POÉTICA
d
x
x x
V Z I (2.3a)
−d ( )ω = ω ω ( ) ( ) d
x
x x
I Y V (2.3b)
where Z(ω) and Y(ω) are respectively the series impedance and the shunt admittance matrices per unit length.
The series impedance matrix of an overhead line can be decomposed as follows:
ω = ω + ω ω ( ) ( ) j ( )
Z R L (2.4)
where Z is a complex and symmetric matrix, whose elements are frequency dependent.
Most EMT programs are capable of calculating R and L taking into account the skin effect in conductors and on the ground. This is achieved by using either Carson’s ground impedance [11] or Schelkunoff’s surface impedance formulas for cylindrical conductors [12]. Other approaches base the calculations on closed form approximations [13,14]. Refs.
[15,16] provide a description of the procedures.
The shunt admittance can be expressed as follows:
ω = + ω ( ) j
Y G C (2.5)
where the elements of G may be associated with currents leaking into the ground through insulator strings, which can mainly occur with polluted insulators. Their values can usu-ally be neglected for most studies; however, under a corona effect, conductance values can be signifi cant. That is, under noncorona conditions, with clean insulators and dry weather, conductances can be neglected. As for C elements, they are not frequency dependent within the frequency range that is of concern for overhead line design.
The main formulas used by commercial programs for calculating line parameters (impedance and admittance) in per unit length and suitable for computer implementation are presented in this section.
2.2.2 Calculation of Line Parameters 2.2.2.1 Shunt Capacitance Matrix
The capacitance matrix is only a function of the physical geometry of the conductors.
Consider a confi guration of n arbitrary wires in the air over a perfectly conducting ground.
Assuming the ground as a perfect conductor allows the applica-tion of the image method, as shown in Figure 2.4. The potential vector of the conductors with respect to the ground due to the charges on all of them is
=
v Pq (2.6)
where
v is the vector of voltages applied to the conductors
q is the vector of electrical charges needed to produce these voltages
P is the matrix of potential coeffi cients whose elements are given by
ε0 is the permittivity of free space
ri is the radius of the ith conductor and (see Figure 2.4)
( ) ( )
When calculating electrical parameters of transmission lines with bundled conductors ri
may be substituted by the geometric mean radius of the bundle:
( )
−n is the number of conductors rb is the radius of the bundle
Finally, the capacitance matrix is calculated by inverting the matrix of potential coeffi cients
= −1
C P (2.10)
2.2.2.2 Series Impedance Matrix
The series or longitudinal impedance matrix is computed from the geometric and electric characteristics of the transmission line. In general, it can be decomposed into two terms:
= ext+ int
Application of the method of images.
where Zext and Zint are respectively the external and the internal series impedance matrix.
The external impedance accounts for the magnetic fi eld exterior to the conductor and comprises the contributions of the magnetic fi eld in the air (Zg) and the fi eld penetrating the earth (Ze).
2.2.2.2.1 External Series Impedance Matrix
The contribution of the Earth return path is a very important component of the series impedance matrix. Carson reported the earliest solution of the problem of a thin wire above the earth [11] in the form of an integral, which can be expressed as a series [11,17–23].
The calculation of the electrical parameters of multiconductor lines is presently per-formed by using the complex image method [24], which consists in replacing the lossy ground by a perfect conductive line at a complex depth. Deri et al. extended this idea to the case of multilayer ground return [14], showing that the results from this method are valid from very low frequencies up to several MHz.
All the solutions provided in those works are valid when the conductors can be con-sidered as thin wires. For practical purposes it can be said that the so-called thin wire approximation is valid when (r/2h)ln(2h/r) << 1, being r the conductor radius and h the con-ductor height [25].
Consider again a confi guration of n arbitrary wires in the air over a lossy ground. Using the complex image method, (see Figure 2.5), the external impedance matrix can be written as follows
and the complex depth p is given by
= ωμ σ + ωεe e e the traditional defi nition of the complex depth is
= ωμ σe e
Geometry of the complex images.
Defi ning p as in Equation 2.14 makes the Earth impedance of a single conductor calcu-lated with the complex image method equal to that calcucalcu-lated with a simple and accurate expression given by Sunde [26].
Multiplying each element of Equation 2.12 by Dij/Dij, the external impedance can be cast in terms of the geometrical impedance, Zg, and the Earth return impedance, Ze:
= +
2.2.2.2.2 Internal Series Impedance
When the wires are not perfect conductors, the total tangential electric fi eld in the wires is not zero; that is, there is a penetration of the electric fi eld into the conductor. This phenom-enon is taken into account by adding the internal impedance. The internal impedance of a round wire is found from the total current in the wire and the electric fi eld intensity at the surface (surface impedance) [27]:
Zcw is the wave impedance in the conductor given by
= ωμ
γc is the propagation constant in the conducting material
γ = ωμ σ + ωεc j c( c j c) (2.21)
The conductivity, permittivity, permeability, and the radius of the conductor are denoted by σc, εc, μc, and rc, respectively.
For low frequencies |γcrc|<< 1; using series expansions of the Bessel functions the inter-nal impedance is [27]
⎡ ⎛ ⎞ ⎤ ωμ
≈ ⎢⎢⎣ + ⎜ ⎟⎝ ⎠δ ⎥⎥⎦+ π
2 c
LF dc
1 1
48 8
Z R r j (2.22)
where
δ = ωμ σc c
2 (2.23)
is the skin depth in the conductor. The second term in the brackets is a correction useful for rc/δ as large as unity, while the imaginary term corresponds to a low-frequency internal inductance and Rdc is the direct current resistance given by
=π σ
dc 2
c c
R 1
r (2.24)
At very high frequencies |γcrc| >> 1, and using asymptotic expressions of the Bessel func-tions, the internal impedance becomes
≈ π σ
HF
c c c
1 Z 2
r p (2.25)
where pc is the complex penetration depth for the conductor and is given by
= ωμ σ
c
c c
p 1
j (2.26)
Note that Equation 2.25 can be interpreted as the complex resistance of an annulus defi ned by the conductor perimeter and the complex penetration depth. Using different approxi-mations for different frequency ranges produces discontinuities in the calculated imped-ance. To avoid such discontinuities the following expression can be used for the whole frequency range:
= 2 + 2
int dc HF
Z R Z (2.27)
For the case of bundled conductors Zint must be divided by the number of conductors in the bundle. Finally, the internal impedance matrix for a multiconductor line with n phases is defi ned as follows:
( )
= …
int diag Zint,1,Zint,2, ,Zint,n
Z (2.28)
Galloway et al. [17] and Gary [24] provided formulas for the internal impedance, which take into account the stranding of real power conductors. Results from these formulas differ from those obtained using Equation 2.25 only by a multiplicative constant. Being the internal impedance a small part of the total impedance, using Equation 2.27 provides results within measurement errors.