CHARGE-POTENTIAL RELATIONS FOR MULTI-CONDUCTOR LINES

Section 3.5 in the last Chapter 3, equations (3.38) to (3.40) describe the charge-potential relations of a transm ission line w ith n conductors on a tower. The effect of a ground plane considered as an equipotential surface gave rise to Maxwell's Potential coefficients and the general equations are

The equivalent radius or geometric mean radius of a bundled conductor has already been discussed and is

r_eq = R(N. r/R)^1/N ...(4.31)

where R = bundle radius = B/2 sin ( Nπ/ )^,

B = bundle spacing (spacing between adjacent conductors) r = radius of each sub-conductor,

and N = number of conductors in the bundle.

The elements of Maxwell's potential coefficients are all known since they depend only on the given dimensions of the line-conductor configuration on the tower. In all problems of interest in e.h.v. transmission, it is required to find the charge matrix from the voltage since this is also known. The charge-coefficient matrix is evaluated as

] 2 /

[Q πe₀ = [P]^–1 [V] = [M] [V] ...(4.32)

or, if the charges themselves are necessary,

[Q] =2πe₀[M][V] ...(4.33)

In normal transmission work, the quantity Q/2πe₀ occurs most of the time and hence equation (4.32) is more useful than (4.33). The quantity Q/2πe₀ has units of volts and the elements of both [P] and [M] = [P]^–1 are dimensionless numbers.

On a transmission tower, there are p phase conductors or poles and one or two ground wires which are usually at or near ground potential. Therefore, inversion of [P] becomes easier and more meaningful if the suitable rows and columns belonging to the ground wires are eliminated. An example in the Appendix illustrates the procedure to observe the effect of ground wires on the line-conductor charges, voltage gradients, etc. On a 3-phase ac line, the phase voltages are varying in time so that the charges are also varying at 50 Hz or power frequency.

This will be necessary in order to evaluate the electrostatic field in the line vicinity. But for Radio Noise and Audible Noise calculations, high-frequency effects must be considered under suitable types of excitation of the multiconductors. Similarly, lightning and switching-surge studies also require unbalanced excitation of the phase and ground conductors.

4.4.1 Maximum Charge Condition on a 3-Phase Line

E.H.V. transmission lines are mostly single-circuit lines on a tower with one or two ground wires. For preliminary consolidation of ideas, we will restrict our attention here to 3 conductors excited by a balanced set of positive-sequence voltages under steady state. This can be extended to other line configurations and other types of excitation later on. The equation for the charges is, For a 3-phase ac line, we have

V₁ = 2Vsin(wt+φ), V₂ = 2Vsin(wt+φ−120°) and V₃ = 2Vsin(wt+φ+120°)with

V = r.m.s. value of line-to-ground voltage and w = 2πf,

f = power frequency in Hz. The angle φ denotes the instant on V₁ where Differentiating with respect to θ and equating to zero gives

) Substituting this value of θ in equation (4.35) yields the maximum value of Q₁.

An alternative procedure using phasor algebra can be devised. Expand equation (4.35) as

0 It is left as an exercise to the reader to prove that substituting θ_m from equation (4.36) in equation (4.35) gives the amplitude of (Q₁/2πe₀)_max in equation (4.38).

Similarly for Q₂ we take the elements of 2nd row of [M].

max 0 2/2 )

(Q πe = 2V[M₂₂² +M₂₁² +M₂₃² −(M₂₁M₂₂ +M₂₂M₂₃ +M₂₃M₂₁)]^1/2 ...(4.39)

For Q₃ we take the elements of the 3rd row of [M].

max 0 3/2 )

(Q πe = 2V[M₃₃² +M₃₁² +M₃₂² −(M₃₁M₃₂ +M₃₁M₃₃+M₃₃M₃₁)]^1/2 ...(4.40) The general expression for any conductor is, for i = 1, 2, 3,

max 0) 2 /

(Q_i πe = 2V[M_i²₁+M_i²₂+M_i²₃−(M_i1M_i2+M_i2M_i3+M_i3M_i1)]^{1/2 ...(4.41)}

4.4.2 Numerical Values of Potential Coefficients and Charge of Lines

In this section, we discuss results of numerical computation of potential coefficients and charges present on conductors of typical dimensions from 400 kV to 1200 kV whose configurations are given in Chapter 3, Figure 3.5. For one line, the effect of considering or neglecting the presence of ground wires on the charge coefficient will be discussed, but in a digital-computer programme the ground wires can be easily accommodated without difficulty. In making all calculations we must remember that the height H_i of conductor i is to be taken as the average height. It will be quite adequate to use the relation, as proved later,

Hav = Hmin + Sag/3 ...(4.42)

Average Line Height for Inductance Calculation*

The shape assumed by a freely hanging cable of length L_c over a horizontal span S between supports is a catenary. We will approximate the shape to a parabola for deriving the average height which holds for small sags. Figure 4.10 shows the dimensions required. In this figure,

H = minimum height of conductor at midspan S = horizontal span,

and d = sag at midspan.

Fig. 4.10 Calculation of average height over a span S with sag d.

The equation to the parabolic shape assumed is

y = H + (4d/S²) x² ...(4.43)

The inductance per unit length at distance x from the point of minimum height is L = 0.2 ln (2y/r) = 0.2 [ln 2y – ln r] ...(4.44) Since the height of conductor is varying, the inductance also varies with it.

The average inductance over the span is L_av =

∫

−

−

2 /

S/2

) ln 2 (ln 2 . 1 ^S 0

dx r S y

*The author is indebted to Ms. S. Ganga for help in making this analysis.

S

d

H

x y

y = H + (4d/S )x²

=

∫

^{/2 −} If the right-hand side can be expressed as ln (2 Hav), then this gives the average height for inductance calculation. We now use some numerical values to show that Hav is approximately equal to

Fig. 4.11 Inductances and capacitances of 400 kV horizontal line.

(a) Consider H = 10, d = 10. Using these in equation (4.47) give

ln 20 + ln (1 + 1) – 2 + 2 tan^–1 1 = 3 + 0.6931 – 2 + π/2 = 3.259 = ln 26 = ln(2 Hav) H_av = 26/2 = 13 = 10 + 3 = H + 0.3d

(b) H = 10, d = 8. ln 16 + ln 2.25 – 2 + 2 1.25tan⁻¹ 0.8 ^{= ln 24.9} H_av = 12.45 = H + 2.45 = H + 0.306d

(c) H = 14, d = 10. H_av = 17.1 = 14 + 3.1 = H + 0.3d

These examples appear to show that a reasonable value for average height is H_av = H_min + sag/3. A rigorous formulation of the problem is not attempted here.

Figure 4.11, 4.12 and 4.13 show inductances and capacitances of conductors for typical 400 kV, 750 kV and 1200 kV lines.

Ls

0.02 0.03 0.04 0.05

Outer-Inner

0.02 0.03 0.04 0.05

nF/km

Cond. Dia., metre Cond. Dia., metre

400-kV Horizontal H = 15 m, S = 12 m N = 2, B = 0.4572 m

In document Extra High Voltage Ac (Page 91-95)