Overhead Line Construction and Conductors

Chapters 1 to 4 treated the transmission line as a symbol. Chapter 3 drew it as a single stroke on the single-line diagram; Chapter 4 gave that stroke a per-unit impedance and connected it to the rest of the network. Nothing so far has said what the stroke is made of. Part 2 of this book computes the four parameters — resistance, inductance, capacitance and conductance — that fill in the impedance, and every one of those calculations begins from a geometry: the radius of the conductor, the spacing between phases, the height above earth. This chapter fixes that geometry, and explains why it is what it is.

An overhead line is a deceptively short list of parts. Bare metal conductors carry the current. Insulator strings hang them from steel cross-arms, and the strings are long enough that the conductor cannot flash over to the earthed tower. One or two earth wires run along the tower tops, above everything, to intercept lightning. Towers stand a few hundred metres apart; between them the conductor hangs in a curve, and the depth of that curve — the sag — is what keeps the mechanical tension within safe limits while still leaving statutory clearance to the ground below.

Each of those parts is sized by a different constraint, and the constraints pull against each other. Raise the tension to reduce the sag and the towers can be shorter, but the conductor fatigues and may snap in winter. Increase the phase spacing and the flashover risk falls but the line inductance rises, as Chapter 6 will show. Choose a fatter conductor and the loss falls, but so does the surface electric field, which is the point of the exercise above 220 kV — Chapter 15 returns to it under the name of corona.

Chapter 2 established the central economic fact of transmission: for a three-phase line the volume of conductor material varies as \(1/(V^2\cos^2\phi)\). That settled the choice of voltage. It did not settle the choice of system. Direct current, single phase, two phase and three phase all deliver power along wires, and each can be arranged with or without an earthed neutral. Ten arrangements are worth comparing, and the comparison is only honest if every one of them is asked to do exactly the same job under exactly the same restriction.

Four conditions are held identical across all ten systems.

The fourth condition deserves the most care, because it is the one students most often get wrong. In an overhead line the conductors hang in air, far apart, and the dielectric that is closest to breaking down is the air path from a conductor down to the earthed tower or to the ground. The limiting quantity is therefore the maximum potential difference between any conductor and earth. In an underground cable the conductors sit side by side inside a common sheath with a few millimetres of insulation between them, so the chief stress is between conductor and conductor, and the comparison must instead fix the maximum voltage between conductors. The two bases give different answers. Everything in this section is the overhead basis; Chapter 9 takes up cables, where the ranking changes.

With the basis fixed, the calculation for each system is mechanical and always the same four steps: find the load current from the power and the available voltage, write the loss as \(nI^2R\) over the \(n\) conductors, solve for the cross-sectional area that the loss budget permits, and multiply that area by the total conductor length to obtain the volume.

One system is chosen as the yardstick. Universal practice takes the simplest of all — the two-wire d.c. system with one conductor earthed — and calls its volume \(K\). Every other result is quoted as a multiple of \(K\), so the comparison reduces to a single column of numbers.

Two-wire d.c., one conductor earthed. One conductor sits at earth potential and the other at \(V_m\); the working voltage between them is therefore \(V_m\), and that is all the arrangement offers. The load current is \(I_1 = P/V_m\), and with two conductors each of resistance \(\rho l / a_1\),

Two-wire d.c., mid-point earthed. Now the supply is split about an earthed centre point, so one conductor sits at \(+V_m\) and the other at \(-V_m\). The insulation to earth is stressed exactly as before, but the working voltage between the conductors has doubled to \(2V_m\). This is the single most instructive step in the whole comparison, because doubling the useful voltage at no insulation cost halves the current and — the loss being fixed — quarters the metal.

Three-wire d.c. The two outers are held at \(+V_m\) and \(-V_m\) with an earthed neutral wire between them. If the two halves of the load are balanced, the neutral carries no current at all, and the outers behave exactly as in the mid-point-earthed system: \(a_3 = a_2\). The neutral is nevertheless installed, since the load will not stay balanced, and standard practice gives it half the cross-section of an outer. The extra half-conductor is pure overhead in the balanced case:

Every a.c. system carries the same two penalties relative to d.c. The insulation limit \(V_m\) is a peak value, so the useful r.m.s. voltage is smaller by \(\sqrt2\); and the load draws current at power factor \(\cos\phi\), so a given real power needs a current larger by \(1/\cos\phi\). Both penalties enter the volume squared. Every a.c. result below therefore carries the factor \(1/\cos^2\phi\), and the comparison between a.c. systems is a comparison of the numerical coefficients in front of it.

Single-phase, two-wire, one conductor earthed. The r.m.s. voltage available is \(V_m/\sqrt2\), so \(I_4 = \sqrt2 P/(V_m\cos\phi)\) and

Single-phase, two-wire, mid-point earthed. As in the d.c. case, earthing the mid-point doubles the working voltage to a peak of \(2V_m\), that is \(\sqrt2\,V_m\) r.m.s., and quarters the volume to \(K/(2\cos^2\phi)\).

Single-phase, three-wire. Add a neutral of half section to the previous arrangement. With a balanced load the neutral carries nothing, the outers are unchanged, and the volume becomes \(2.5\,a_5 l = 5K/(8\cos^2\phi)\) — a 25% penalty over the two-wire mid-point system, in exchange for the ability to supply two voltage levels.

Two-phase, four-wire. Two independent single-phase circuits in quadrature, each carrying \(P/2\), each with its own pair of conductors at \(\pm V_m\). Because each pair sees the full \(\sqrt2 V_m\) r.m.s. while carrying only half the power, the current per conductor falls to \(P/(2\sqrt2 V_m\cos\phi)\):

Two-phase, three-wire. The one arrangement in the list whose neutral is genuinely busy. Two outers, each at \(V_m/\sqrt2\) r.m.s. to the common return, carry currents \(I_8 = P/(\sqrt2 V_m\cos\phi)\) that are \(90^\circ\) apart; the return therefore carries their phasor sum, of magnitude \(\sqrt2 I_8\). Holding the current density constant makes the neutral \(\sqrt2\) times the section of an outer, so its resistance is \(R_8/\sqrt2\):

since \((2+\sqrt2)^2/2 = (6+4\sqrt2)/2 = 5.828\) and \(5.828/4 = 1.457\). The two-phase three-wire system is the second worst in the table, and for the clearest of reasons: it is the only one asked to carry current in a conductor that produces no useful power.

Three-phase, three-wire. Take the conductors star-connected for the calculation, so each carries a phase voltage of \(V_m/\sqrt2\) r.m.s. and a third of the power. Then \(I_9 = \sqrt2 P/(3V_m\cos\phi)\), and

Three-phase, four-wire. Add a neutral of half section. Balanced three-phase currents sum to zero — the result Chapter 3 derived from \(1+a+a^2=0\) — so the neutral is idle and the outers are unchanged, giving \(3.5\,a_{10}l = 7K/(12\cos^2\phi)\).

The comparison of Section 5-4 fixed how many conductors to string and at what voltage; it left \(\rho\) as a symbol. Choosing the metal is a separate optimisation, and the honest criterion is not conductivity but cost per unit of conductance, at the required mechanical strength, at the required weight.

Copper is the best available conductor after silver, takes a high tensile strength when hard-drawn, and resists corrosion. It also has a density of about 8890 kg/m³ and a price per tonne that has never been low. Aluminium conducts only 61% as well volume-for-volume, but weighs less than a third as much. Those two numbers combine into the fact that ended the argument.

An aluminium conductor of the same resistance is 28% fatter and weighs half as much. Half the weight means lighter towers, longer spans and cheaper foundations; the greater diameter is, if anything, an advantage, because it lowers the electric field at the conductor surface and postpones corona. The single defect is mechanical: hard-drawn aluminium has roughly half the tensile strength of hard-drawn copper and creeps under sustained load, so a long span of pure aluminium sags unacceptably.

The fix is to separate the two duties. ACSR — aluminium conductor, steel reinforced — puts a core of galvanised steel strands, which carry the mechanical tension, inside layers of aluminium strands, which carry the current. The steel contributes almost nothing electrically, and since it sits at the centre where the current density is lowest in any case, nothing is lost. A designation such as 54/7 means 54 aluminium strands over a 7-strand steel core.

Where the mechanical demand is milder, the steel can be dispensed with. AAAC (all-aluminium alloy conductor) uses an aluminium–magnesium–silicon alloy that is stronger than pure aluminium and slightly less conductive; it has no bimetallic corrosion problem and is favoured in coastal and industrial atmospheres. ACAR reinforces aluminium with alloy strands instead of steel. ACSS uses fully annealed aluminium over a high-strength core so that the steel takes essentially all the load, which permits very high operating temperatures and therefore higher current — a retrofit strategy when a right of way cannot be widened.

The catalogue of ACSR sizes is standardised, and Indian practice names them after animals. The four in the table below account for most of the country's transmission network.

A conductor of 500 mm² drawn as a single solid rod would be 25 mm across and effectively unbendable; it could not be coiled on a drum or dead-ended at a tower. Every practical conductor is therefore stranded: many thin wires laid up helically in concentric layers, each layer wound in the opposite sense to the one beneath so that the whole assembly does not unwind under tension.

Concentric stranding follows a fixed arithmetic. A central strand is surrounded by 6 strands, then 12, then 18, and so on, each successive layer taking 6 more than the last. With \(n\) layers counted including the central strand,

where \(d\) is the diameter of one strand. The overall-diameter rule follows by walking outward: each new layer adds one strand diameter on each side.

Stranding has two electrical consequences. The first is minor and easily stated: because every strand except the central one follows a helix, its true length exceeds the length of the conductor by one or two per cent, and the d.c. resistance is higher than \(\rho l/A\) by the same fraction. Manufacturers' tables already include this, which is why the catalogue \(R_{dc}\) never quite matches a hand calculation from the area.

The second is the one that matters for the rest of Part 2. Magnetically, a stranded conductor is not a solid cylinder. It is a bundle of parallel filaments, and the flux linking the bundle depends on the distances between the filaments as well as on their individual radii. Chapter 6 will derive the general result, but the definition can be stated now because it is a geometrical fact about the conductor alone, independent of what the line is doing.

Example 4 evaluates this for the seven-strand conductor and finds \(D_s = 2.177\,r_{\text{strand}}\), which is a little less than the physical outer radius \(3r_{\text{strand}}\) — as it must be, since the definition averages distances that include the small self-distances. For ACSR the steel core is normally excluded from the calculation, on the grounds that it carries negligible current, and manufacturers publish the resulting \(D_s\) directly; the values in the table of Section 5-5 are of that kind.

Of the four line parameters, resistance is the only one that dissipates energy, and it is the one most sensitive to conditions of operation. The starting point is elementary,

and two corrections turn it into something usable. The first is temperature. Over the range a conductor actually experiences, resistivity rises very nearly linearly with temperature, so \(R_t = R_0(1+\alpha_0 t)\). Plotting \(R\) against \(t\) gives a straight line that, extrapolated backwards, meets the temperature axis at \(t = -1/\alpha_0\). Calling that intercept \(-T\) turns the correction into a ratio of two lengths along the same straight line, which is far easier to use than the coefficient form:

An aluminium conductor at its summer operating temperature of 75 °C therefore has resistance \((228+75)/(228+20) = 1.222\) times the catalogue value — a 22% increase that must not be left out of a loss or voltage-drop calculation.

The second correction is the skin effect. Under alternating current, the flux inside the conductor links the inner filaments more completely than the outer ones, so the inner filaments have a larger internal inductance. Since all filaments are in parallel and share the same terminal voltage, current redistributes toward the surface where the impedance is lower. The effective area is reduced and \(R_{ac} > R_{dc}\). The natural scale of the effect is the skin depth,

A Zebra conductor has an outer radius of 14.3 mm, comparable with \(\delta\), so the effect is real but modest: \(R_{ac}/R_{dc}\) for ACSR at 50 Hz typically lies between 1.02 and 1.05. It rises with frequency, with conductor size and with permeability, which is one more reason not to make a solid steel conductor. The hollow-core and expanded-core constructions used at extra-high voltage exploit the same physics from the other side: metal placed at the centre of a large conductor is doing very little, so it may as well be replaced by a light former.

A related redistribution, the proximity effect, arises when the return conductor is close enough for its field to distort the current density in this one. On overhead lines the phase spacing is metres and the effect is negligible; in the closely packed cores of an underground cable it is not, and Chapter 9 returns to it.

An insulator has to do two things at once: hold the mechanical load of the conductor, and stand off the line-to-earth voltage against both a dry power-frequency stress and the far higher transient of a lightning or switching surge. It must do the second in rain, in fog and under a coating of industrial dust, which is why the shape of every insulator is a sequence of sheds — the underside of each shed stays dry, so the leakage path along the surface is several times the straight-line distance.

Four types cover practice. Pin insulators are mounted rigidly on a spindle on the cross-arm and are economic up to about 33 kV; above that they become clumsy and expensive. Suspension insulators are discs with metal caps and pins, connected in a flexible string, with the conductor hanging from the bottom; the number of discs is simply increased with voltage, and a damaged disc can be replaced individually. Strain insulators are suspension strings mounted horizontally where the line dead-ends or turns a sharp corner and the string must take the full pull. Shackle insulators are used on low-voltage distribution.

A rough working figure is one standard disc per 11 kV of line voltage, so an 11 kV line uses one, a 66 kV line four or five, a 132 kV line nine or ten, and a 400 kV line twenty-three or so. But a string of \(n\) identical discs does not divide the voltage into \(n\) equal parts, and the reason is a stray capacitance that cannot be designed away.

Each disc has a self-capacitance \(C\) between its cap and its pin. Each metal link between two discs also has a capacitance to the earthed tower, which is nearby; call it \(kC\), with \(k\) typically between 0.1 and 0.2. The self-capacitances form a series chain from the cross-arm to the conductor, but at every intermediate node some of the current is diverted to earth through the shunt capacitance, so the current in the chain is not the same all the way down. It grows toward the conductor, and since each disc has the same \(C\), the voltage across a disc is proportional to the current through it.

Number the discs 1 at the cross-arm and \(n\) at the conductor. Applying Kirchhoff's current law at the node between disc 1 and disc 2, the current entering disc 2 is the current leaving disc 1 plus the shunt current drawn from that node, which is set by the voltage of the node above earth — namely \(V_1\):

At the next node the shunt capacitance sees the whole voltage accumulated so far, \(V_1+V_2\):

The pattern continues; every step adds a larger increment, so \(V_1 < V_2 < V_3 < \cdots < V_n\) always. The unit nearest the line conductor takes the largest share, is the first to flash over, and is the one that fails in service. The quality of the distribution is measured by

Three remedies follow directly from the derivation, and each attacks a different term. Making the cross-arm longer moves the tower metal away from the string and reduces \(k\) — cheap, but limited by tower economics. Capacitance grading gives the lower discs a larger \(C\), so that the same increasing current produces the same voltage in every unit; it works, but requires several different disc types on one string and is rarely done. Static shielding — a large metal grading ring, bell-mouthed toward the conductor, fitted at the bottom of the string — introduces a capacitance from each link to the line conductor that injects current in the opposite sense to the earth capacitance, cancelling it. The grading ring is standard on every extra-high-voltage string, and it also protects the bottom discs from power-arc damage.

Supports carry the conductors at a height that guarantees clearance and hold the phases apart at a spacing that guarantees no flashover between them. Wooden and concrete poles serve distribution; steel lattice towers serve transmission, and are classified by the angle of deviation they are designed for — type A for a straight run up to 2°, B up to 15°, C up to 30°, D up to 60°, with special dead-end towers at line terminations and river crossings.

What hangs between two towers is not a straight line. A flexible conductor of uniform weight suspended between two points takes the shape of a catenary, and the vertical distance from the line joining the supports down to the lowest point is the sag. The sag is not an accident to be minimised; it is a design variable, because it fixes the tension, and the tension is what the conductor and the tower have to survive.

Take supports at the same level, a span \(L\), a conductor of weight \(w\) per unit length, and let \(T\) be the horizontal component of tension — constant along the conductor, since no horizontal force acts on it between supports. Put the origin at the lowest point and consider the portion from there out to a general point \((x,y)\). It is in equilibrium under three forces: the horizontal pull \(T\) at the origin, the tension along the curve at \((x,y)\), and the weight \(wx\) of the portion acting at its centroid, a distance \(x/2\) from the origin. Taking moments about \((x,y)\),

The curve is a parabola — the first term of the catenary, and accurate to better than half a per cent whenever the sag is under about 10% of the span, which covers all normal practice. Setting \(x = L/2\) gives the sag at the support.

Real conductors do not hang in still, warm air. Ice builds a concentric sleeve of thickness \(t\) around a conductor of diameter \(d\), whose weight per unit length is \(w_i = \rho_{\text{ice}}\,\pi t (d+t)\) and which acts vertically downward, adding to \(w\). Wind presses on the projected area of the iced conductor, \(w_w = p\,(d+2t)\) per unit length, and acts horizontally. The conductor hangs in the plane of the resultant.

Only the vertical component reduces ground clearance, so it is the vertical sag that the statutory clearance is measured against, while the slant sag is what determines the tension the tower must resist. When the two supports are at different levels — a hillside, a river bank — the lowest point of the curve is no longer at mid-span. Writing \(x_1\) and \(x_2\) for the horizontal distances from the low point to the lower and higher supports, with \(x_1+x_2 = L\) and a height difference \(h\),

If \(Th/(wL)\) exceeds \(L/2\), the formula returns a negative \(x_1\), which is the algebra's way of reporting that the lowest point lies outside the span altogether — the conductor rises the whole way from the lower support, and the erection procedure has to be checked for uplift on that tower.

Unless stated otherwise, compare systems at equal power, distance, loss and maximum voltage to earth, and take \(\rho_{Cu} = 1.72\times10^{-8}\;\Omega\)·m, \(\rho_{Al} = 2.83\times10^{-8}\;\Omega\)·m.