Electronics Guide

Overhead Lines

An overhead line hangs bare conductors from insulators on towers or poles and uses the air as insulation. Most conductors are stranded aluminum, which conducts about 61 percent as well as copper but is so much lighter that a conductor of equal resistance weighs about half as much. The workhorse design, aluminum conductor steel reinforced (ACSR), wraps aluminum strands around a steel core that carries much of the tension. All-aluminum and aluminum-alloy conductors serve many distribution lines, and high-temperature designs such as aluminum conductor steel supported (ACSS) and composite-core conductors carry more current with less sag. Shield wires above the phases intercept lightning, and many carry optical fibers.

Two physical limits shape the design. A conductor sags as it heats and must keep a safe clearance above the ground. And the electric field at its surface must stay below the level at which air ionizes, roughly 3 MV/m under standard conditions; near that gradient a conductor goes into corona, which wastes energy, produces audible noise and radio interference, and degrades hardware. Lines at 345 kV and above therefore usually carry bundles of two or more subconductors per phase, which spread the charge over a larger effective surface.

Underground Cables

A cable replaces the air with solid insulation a few centimeters thick. A high-voltage cable has a stranded copper or aluminum conductor; a semiconducting conductor screen that smooths the field at the conductor surface; the main insulation; a semiconducting insulation screen; a metallic sheath or wire screen that carries charging and fault current and holds the outer surface at ground potential; and a protective jacket. Extruded cross-linked polyethylene (XLPE) is the usual insulation for new cable, ethylene propylene rubber (EPR) is common at distribution voltages, and older networks still contain paper-insulated lead-covered cable and fluid-filled cable in steel pipes.

Cables cost considerably more than overhead lines of similar rating, but they are the practical choice beneath dense cities, across water, and where an overhead route cannot win approval. Electrically, a cable is the opposite of an overhead line: its conductor sits a few centimeters from a grounded screen, so its inductance is low and its capacitance high, which produces the charging-current limit discussed below.

The Inductance of a Round Conductor

Current in a conductor produces magnetic flux inside and outside it. For a long, straight, nonmagnetic round conductor of radius r whose current I is spread uniformly over its cross section, the internal flux linkage is μ₀I/(8π) per meter, and the external flux linkage out to a distance D is (μ₀I/2π) ln(D/r). With μ₀/2π = 2 × 10⁻⁷ H/m, the total per meter is

λ = 2 × 10⁻⁷ I (1/4 + ln(D/r)) = 2 × 10⁻⁷ I ln(D/r′) Wb/m, where r′ = e^−1/4 r ≈ 0.7788 r

The fictitious radius r′ is the conductor's geometric mean radius (GMR): a thin-walled tube of radius r′, which has no internal flux, would have the same inductance. The uniform current density assumed here is a reasonable approximation at power frequency.

Three-Phase Lines and Transposition

For a balanced three-phase line whose conductors sit at the corners of an equilateral triangle of side D, the flux from the other two phases combines so that each phase has an inductance of 2 × 10⁻⁷ ln(D/r′) H/m. Real lines rarely have equal spacing. The middle phase of a flat, horizontal line is closer to both outer phases than they are to each other, so each phase has a different inductance, and balanced currents produce unbalanced voltage drops.

Transposition removes the difference on average. The line is divided into three sections of equal length, and the phases exchange positions at transposition structures so that every phase occupies every position for one-third of the length. The per-phase inductance of the transposed line is then

L = 2 × 10⁻⁷ ln(GMD/GMR) H/m, where GMD = (D_ab D_bc D_ca)^1/3

and the geometric mean distance (GMD) is the geometric mean of the three phase spacings. At 60 Hz, the per-phase reactance is X = 0.0754 ln(GMD/GMR) Ω/km; at 50 Hz, the coefficient is 0.0628. Spacing enters through a logarithm, so doubling every distance adds only 0.0523 Ω/km at 60 Hz. Many lines are transposed only partially or not at all; detailed studies then use the full matrix of self and mutual phase impedances, and the remaining asymmetry appears as negative-sequence and zero-sequence voltage that grows with load.

Stranded Conductors and Bundles

A stranded conductor behaves as a group of parallel subconductors. Its GMR is the geometric mean of all distances between strands, with each strand's own r′ as its distance to itself. Seven identical strands give 0.726 times the overall radius, and ideal concentric layers approach the solid value of 0.7788: about 0.758 for 19 strands and 0.768 for 37. Multilayer ACSR can exceed the solid value, because its steel core carries little current, and current confined to an outer ring has a larger GMR. In practice, engineers take GMR from the manufacturer's table.

A bundle is treated the same way. For subconductors of GMR D_s spaced a distance d apart, the bundle GMR is

(D_s d)^1/2 for two subconductors, (D_s d²)^1/3 for three in a triangle, and 1.09 (D_s d³)^1/4 for four in a square

A bundle's GMR is several times that of a single subconductor, so bundling lowers series inductance substantially. For the same geometric reason, it raises capacitance and lowers the surface gradient that causes corona.

These expressions give the positive-sequence impedance that governs balanced operation. Current returning through the earth and shield wires, as ground-fault current does, sees a substantially larger impedance, calculated with the ground-return corrections that John R. Carson published in 1926; that zero-sequence impedance matters mainly in fault calculations with symmetrical components.

Charging Current and Reactive Power

Shunt capacitance draws charging current whether or not the line carries load. Per unit length, each phase draws I_c = ωC_nV_ph, and the three phases generate reactive power Q_c = ωC_nV_LL², where V_ph and V_LL are the phase-to-neutral and line-to-line voltages. The series inductance absorbs 3I²ωL per unit length, which depends on load. Whichever is larger decides whether the line raises or lowers the voltage around it.

The product of the inductance and capacitance formulas is nearly the same for every overhead line: LC_n = μ₀ε₀ ln(GMD/GMR) / ln(GMD/r), only slightly more than 1/c². Waves on an overhead line therefore travel at nearly the speed of light.

Worked Example: Parameters of a 345 kV Line

A transposed 345 kV line has its phases in a flat row, 8 m apart, so D_ab = D_bc = 8 m and D_ca = 16 m. Each phase is a bundle of two subconductors spaced 0.45 m apart. Each ACSR subconductor has an outside diameter of 28.1 mm and a GMR of 11.4 mm; these are illustrative values.

1. Geometric mean distance: GMD = (8 × 8 × 16)^1/3 = 10.08 m.
2. Bundle radii: the GMR for inductance is (0.0114 × 0.45)^1/2 = 0.0716 m, and the capacitive radius is (0.01405 × 0.45)^1/2 = 0.0795 m.
3. Inductance: L = 2 × 10⁻⁷ ln(10.08/0.0716) H/m = 0.989 mH/km, so X = 0.373 Ω/km at 60 Hz.
4. Capacitance: C_n = 55.6/ln(10.08/0.0795) = 11.49 nF/km, so the shunt susceptance is ωC_n = 4.33 µS/km.
5. Charging: at 345 kV, the line generates ωC_nV_LL² = 0.516 Mvar/km, or about 155 Mvar over 300 km.

Used singly, one per phase, the same conductors would give 0.512 Ω/km and 8.46 nF/km, so bundling cuts the series reactance by 27 percent and raises the capacitance by 36 percent. The propagation velocity, 1/√(LC_n), is 0.989 times the speed of light.

Short Line

When the shunt admittance is negligible, the line is a series impedance Z, so A = D = 1, B = Z, and C = 0. The model ignores charging current entirely.

Medium Line: The Nominal π Circuit

The nominal π circuit places half the total shunt admittance at each end of the series impedance. The current through Z is I_R + V_RY/2, so V_S = V_R + Z(I_R + V_RY/2). The sending-end current adds the charging current of the sending-end admittance, V_SY/2. Collecting terms gives

A = D = 1 + ZY/2, B = Z, C = Y(1 + ZY/4)

Long Line: Distributed Parameters

A long line must be treated as distributed. Measure distance x from the receiving end. Over a length dx, the voltage rises by the series drop zI dx and the current rises by the shunt current yV dx:

dV/dx = zI, dI/dx = yV

Differentiating again gives d²V/dx² = zyV, whose solutions are waves traveling in both directions with propagation constant γ = √(zy) = α + jβ. The ratio of voltage to current in either wave is the characteristic impedance, Z_c = √(z/y). Power engineers usually write Z_c, keeping Z₀ for zero-sequence impedance. Applying V = V_R and I = I_R at x = 0 and evaluating the solution at x = l gives

A = D = cosh γl, B = Z_c sinh γl, C = (sinh γl)/Z_c

The exact line can also be drawn as an equivalent π circuit whose elements are the nominal values multiplied by correction factors:

Z′ = Z (sinh γl)/(γl), Y′/2 = (Y/2) tanh(γl/2)/(γl/2)

Both factors approach 1 as γl approaches zero, which is why the nominal π circuit converges on the exact result as a line gets shorter.

The Lossless Line

Neglecting resistance and conductance makes γ = jβ purely imaginary and Z_c real, with L and C_n per unit length:

β = ω√(LC_n), Z_c = √(L/C_n), A = D = cos βl, B = jZ_c sin βl, C = j(sin βl)/Z_c

High-voltage lines have series reactance many times their resistance, so the lossless form describes them well. Substituting the overhead-line formulas gives a quick estimate: Z_c ≈ 60 √(ln(GMD/GMR) × ln(GMD/r)) Ω.

Choosing a Model

Textbooks sort lines into short, medium, and long classes by length, with differing boundaries, but the real criterion is electrical length, βl. For the 345 kV example line carrying 400 MW at unity power factor, the short-line model overstates the sending-end voltage by 0.5 percent at 80 km and 4.7 percent at 250 km, and even at 80 km it misstates the sending-end reactive power by about 42 Mvar because it ignores charging. The nominal π model stays within 0.25 percent out to 250 km and is off by about 1.1 percent at 400 km. At 50 Hz, β is five-sixths of its 60 Hz value, so each model holds its accuracy over a line about one-fifth longer.

The St. Clair Curve

SIL is a natural yardstick for practical loading. In a paper presented at an AIEE meeting in Vancouver in September 1953, H. P. St. Clair of American Gas and Electric, later American Electric Power, gave curves of loadability, in multiples of SIL, against line length, drawn from experience rather than analysis. His benchmarks were 3.0 SIL for a 50-mile line, where thermal limits govern, and 1.0 SIL for a 300-mile line, a length at which lines were known to run with little or no reactive power supplied from either end.

In 1979, R. D. Dunlop, R. Gutman, and P. P. Marchenko of the American Electric Power Service Corporation derived the curve analytically and extended it to extra-high and ultra-high voltages. Their criteria were a voltage drop across the line of no more than 5 percent and a steady-state stability margin of 30 percent, that is, operation at no more than 70 percent of the maximum transferable power. Their curves came out nearly identical to St. Clair's, although the strength of the terminal systems mattered more as the voltage class rose.

MISO applies the curve as a safe loading limit, the lesser of the load that produces a 5 percent voltage drop across the line and the load that produces an angular displacement of 44.5 degrees; voltage drop governs shorter lines, and angle governs longer ones. Its multipliers fall from 3.00 at 50 miles to 2.05 at 100, 1.60 at 150, 1.30 at 200, and 1.00 at 300. By that measure, the 300 km (186-mile) example line could carry roughly 1.4 times its SIL, about 560 MW.

Power Transfer Over a Lossless Line

The lossless constants also give the power a line transfers between two buses whose voltage magnitudes are held fixed:

P = V_SV_R sin δ / (Z_c sin βl)

Here V_S and V_R are line-to-line voltage magnitudes, δ is the angle by which V_S leads V_R, and P is three-phase power; for a short line, Z_c sin βl approaches the series reactance X. The dynamics near the theoretical maximum at δ = 90° belong to power system stability. For the example line with both ends at 345 kV, an angle of 30° transfers 545 MW, or 1.34 SIL.

Voltage Regulation

Voltage regulation measures how much the receiving-end voltage changes between no load and full load while the sending-end voltage is held constant:

Regulation (%) = (|V_R,NL| − |V_R,FL|) / |V_R,FL| × 100, where |V_R,NL| = |V_S|/|A|

On a long line, part of the swing between no load and full load is the Ferranti rise, which is why the 300 km example shows 11.7 percent regulation although its sending-end voltage is only 3.7 percent above its receiving-end voltage at full load.

The Approximate Voltage Drop

For a short line, or any series impedance R + jX, take the receiving-end voltage V_R as the phase reference. The load draws real power P and reactive power Q per phase, with Q positive for a lagging load, so the line current is I = (P − jQ)/V_R. Then

V_S = V_R + (R + jX)(P − jQ)/V_R = V_R + (RP + XQ)/V_R + j(XP − RQ)/V_R

When the quadrature term is small compared with V_R, which means that the angle across the impedance is small, the in-phase term accounts for nearly all of the change in magnitude:

ΔV = |V_S| − |V_R| ≈ (RP + XQ)/V_R

The same expression holds with three-phase P and Q and line-to-line voltage, and in per unit, with voltages near 1, it reduces to ΔV ≈ RP + XQ. Dropping the quadrature term understates the drop by about (XP − RQ)²/(2V_R³) in per-phase quantities. The approximation ignores shunt capacitance, so it suits short lines and feeders, and it treats a single sending end and receiving end; networks of many buses are the subject of power flow analysis.

The formula shows why reactive power controls voltage on transmission lines. The example line has X/R = 9.3, so where the approximation holds, a megavar of reactive flow moves its voltage about nine times as much as a megawatt of real flow. Distribution conductors have much lower X/R ratios, so real power matters there too, and reversing P, as local generation does, can turn a drop into a rise, the subject of hosting capacity and distribution impacts.

Losses and Efficiency

The real power lost in a three-phase line is 3|I|²R, and its series reactance absorbs 3|I|²X of reactive power, partly offset by its charging. Because |I| = S/(√3 V_LL), a short line loses

P_loss = R(P² + Q²)/V_LL²

where P and Q are three-phase quantities and R is the total series resistance. Loss grows with the square of apparent power and falls with the square of voltage: delivering the same real power at a power factor of 0.8 instead of 1.0 multiplies it by (1/0.8)² = 1.56, and carrying the same load on the same conductor at 345 kV instead of 138 kV divides it by (345/138)² = 6.25. Efficiency is P_R/P_S, 96.1 percent in the 300 km example.

Shunt Reactors

A shunt reactor absorbs reactive power and cancels part of a line's charging. Extra-high-voltage lines often carry reactors at their terminals, some switched so that they can be removed at heavy load. Reactors limit the Ferranti rise at light load and the overvoltage that follows energizing a line or losing its load.

Shunt Capacitors

Shunt capacitor banks supply reactive power at heavy load, raising voltage and reducing the reactive current in the line, and with it the resistive loss. Their output falls with the square of voltage, so they help least when voltage is lowest. Transmission systems install them at substations, and distribution systems along feeders.

Series Capacitors

A series capacitor cancels part of a line's inductive reactance. With a degree of compensation k_se = X_C/X_L, the net series reactance becomes X_L(1 − k_se). For a short line, the power transferred at a given angle rises by the factor 1/(1 − k_se), so 50 percent compensation doubles it. A series capacitor regulates itself, because its reactive output grows with the square of the line current.

Series capacitors bring three complications. Fault current would overstress the capacitor, so a modern bank has a metal-oxide varistor, a bypass gap or fast switch, and a bypass breaker. The capacitor alters the impedance that a distance relay measures, which power system protection must allow for. And its resonance with the network inductance falls below the power frequency, where it can interact with the torsional modes of turbine-generator shafts, a hazard called subsynchronous resonance.

Compensation and Natural Load

Compensation spread along a line changes its effective constants. Shunt reactors that cancel a fraction k_sh of the charging reduce the effective capacitance to C_n(1 − k_sh), and series capacitors reduce the effective inductance to L(1 − k_se). The compensated natural load is

SIL′ = SIL √((1 − k_sh)/(1 − k_se))

For the 406 MW example line, 50 percent series compensation raises the natural load to 574 MW, and 60 percent shunt compensation lowers it to 257 MW. The formula treats compensation as evenly distributed, an idealization of equipment installed at a few locations.

Dynamic Compensation

Mechanically switched reactors and capacitors act slowly and in steps. Static VAR compensators, built from thyristor-controlled reactors and thyristor-switched capacitors, and STATCOMs, built on voltage-source converters, adjust continuously within a few cycles; static VAR compensators covers both, including midpoint compensation of long lines. Thyristor-controlled series capacitors bring the same speed to series compensation, and synchronous condensers, which supply both reactive power and fault current, are again being installed on grids that are losing synchronous generation.

Feeders, Laterals, and Secondaries

A feeder's three-phase main line, the backbone or trunk, runs through its service area. Laterals branch from it, most of them single phase, each protected by a fuse so that a fault interrupts only that lateral's customers. Distribution transformers on poles or pads step down to service voltage. North American residential service is usually 120/240 V single phase from a transformer that serves a handful of customers, on primaries that are mostly four-wire, multigrounded wye systems. European practice uses larger transformers that feed long 230/400 V three-phase networks with many more customers on each.

Radial, Loop, and Network Systems

Most feeders are radial: power flows one way, which keeps protection simple and inexpensive, but a fault interrupts every customer downstream of the device that clears it. Open-loop arrangements join two feeders through a normally open switch, so that healthy sections can be restored from the other side, manually or by automatic fault location, isolation, and service restoration. Underground residential systems often loop pad-mounted transformers this way.

Secondary networks go further. In a grid network, several feeders supply transformers whose secondaries join in a low-voltage mesh, so the loss of one feeder interrupts no customer, and a network protector on each secondary opens when power flows backward into a faulted feeder. Spot networks apply the idea to a single large building. Both serve dense downtown areas, where load density justifies their cost.

Modeling Distribution Lines

Overhead distribution lines are short enough that their shunt capacitance is usually neglected, but they are rarely balanced. Single-phase laterals, uneven loading, and untransposed construction defeat per-phase analysis, so distribution studies use a matrix of phase impedances derived from Carson's equations and solve each phase of the unbalanced feeder, the approach that William H. Kersting develops in Distribution System Modeling and Analysis.

The Voltage Budget

In North America, service voltage is governed by ANSI C84.1. Its current edition, ANSI C84.1-2020, was approved on March 10, 2020, and covers 60 Hz systems above 100 V, with preferred ratings up to 1,200 kV maximum system voltage. It defines Range A, within which service and utilization voltages should normally fall, and a wider Range B for less frequent excursions. For service voltage on a 120 V system, Range A spans 114 V to 126 V, plus or minus 5 percent.

A distribution engineer divides the width of Range A among the primary feeder, the distribution transformer, the secondary and service conductors, and the bandwidth of the regulating equipment, so that the first customer stays below the upper limit at light load and the last stays above the lower limit at peak load.

Distributed Load

If load is uniform along a feeder of length l, the current falls linearly from the substation to the far end. Integrating the approximate drop gives the same total as the whole load lumped at l/2, and integrating the I²R loss gives the same total as the whole load lumped at l/3. A two-load equivalent, with two-thirds of the load at l/4 and one-third at the far end, reproduces both results.

Worked Example: A 12.47 kV Feeder

An 8 km, 12.47 kV feeder with an illustrative positive-sequence impedance of 0.19 + j0.39 Ω/km serves 6 MW at 0.90 power factor lagging, spread uniformly along its length. Conditions are balanced.

1. Load: Q = 6 × tan(cos⁻¹ 0.90) = 2.906 Mvar, so the substation supplies 6.667 MVA, or 308.7 A.
2. Drop, with r and x the resistance and reactance per kilometer: ΔV ≈ (rP + xQ)(l/2)/V_LL = (0.19 × 6 + 0.39 × 2.906) × 4/12.47 = 0.729 kV, or 5.85 percent, which is 7.0 V on a 120 V base. The same load lumped at the end would drop twice as much. A phasor calculation of the midpoint equivalent at constant current gives 5.95 percent, so the approximation runs 0.1 percentage point low.
3. Loss: rlS²/(3V_LL²) = 0.19 × 8 × 6.667²/(3 × 12.47²) = 0.1448 MW, or 2.4 percent of the load.
4. Capacitor bank: a 1,800 kvar bank two-thirds of the way out, at 5.33 km, reduces the end-of-feeder drop to 3.44 percent, or 4.1 V on a 120 V base, and cuts the loss by 24.3 kW, or 16.8 percent.

The location follows the two-thirds rule: for a uniformly distributed reactive load, one bank rated at two-thirds of the reactive load and placed two-thirds of the way out removes eight-ninths of the loss caused by reactive current, here from 27.5 kW to 3.1 kW with a 1,937 kvar bank. The rule assumes a uniform, constant load on a uniform conductor, so placement studies start from it rather than end with it. At the end of the feeder, the 1,800 kvar bank would leave a smaller drop, 2.24 percent, but save only 19.5 kW, because its output would flow back toward the substation over much of the feeder.

Regulating Equipment

Utilities regulate feeder voltage with three kinds of equipment. A load tap changer on the substation transformer adjusts its ratio under load; the mechanism belongs to the study of power transformers. A step-voltage regulator, an autotransformer with its own tap changer installed on the feeder, usually provides a range of plus or minus 10 percent in 32 steps of 0.625 percent, or 0.75 V on a 120 V base. Its controller holds voltage within a bandwidth around a set point and waits out a time delay before changing taps so that it does not chase brief fluctuations. With line-drop compensation, it regulates the estimated voltage at a downstream load center, computed from the measured current and from resistance and reactance settings that model the feeder. Cascaded regulators are usually given longer delays downstream, so that the upstream unit acts first.

The third kind is the capacitor bank. Fixed banks supply the base reactive load, and switched banks add capacity at peak under controls that respond to time, temperature, voltage, reactive power, or current. Volt/VAR optimization coordinates all three, often holding voltage in the lower part of the permitted band so that voltage-sensitive loads draw less energy, a practice called conservation voltage reduction. Distributed generation complicates each of these controls by reversing power flow.

Charging Current

A single-core cable is a coaxial capacitor. With the conductor screen at diameter d and the outer surface of the insulation at diameter D, its capacitance per unit length is

C = 2πε₀ε_r / ln(D/d)

and each phase draws a charging current of ωCV_ph per unit length. Consider an illustrative 230 kV XLPE cable at 60 Hz, with d = 50 mm, D = 96 mm, and ε_r = 2.3:

1. Capacitance: C = 2π × 8.854 × 10⁻¹² × 2.3/ln(96/50) F/m = 0.1962 µF/km, about 22 times that of a 230 kV overhead line built with the conductor of the earlier example on a flat 6 m spacing.
2. Charging current: I_c = ωCV_ph = 2π × 60 × 0.1962 × 10⁻⁶ × 132,790 = 9.82 A/km.
3. Reactive power: Q = ωCV_LL² = 3.91 Mvar/km, or about 156 Mvar for a 40 km circuit.
4. Critical length: if the conductor is rated at an illustrative 1,000 A and all of the charging is supplied from one end, charging alone uses the whole rating at 1,000/9.82 = 102 km; supplying half of the charging from each end doubles that length to 204 km.
5. Remaining capacity: a 40 km cable charged from one end carries 393 A of charging current at its sending terminal, nearly in quadrature with the load current, so √(1,000² − 393²) = 920 A, about 92 percent of the rating, remains for load.

As a coaxial line with the return current in its screen, the cable has a surge impedance of about 26 Ω and an SIL near 2,050 MW, against about 400 MVA for its 1,000 A rating. Cross-bonded circuits have more inductance, set by phase spacing, but a natural load still well above their rating. A cable therefore generates net reactive power at any practical load, which is why long AC cable circuits need shunt reactors and long submarine links use direct current.

Ampacity of Overhead Conductors

Ampacity is the current a conductor can carry continuously without exceeding a temperature limit. For a bare overhead conductor, the limit guards against annealing, which permanently weakens aluminum held too hot for too long, and against sag that brings the conductor too close to the ground. IEEE 738-2023 gives a numerical method that relates a bare conductor's core and surface temperatures to its steady or time-varying current and to the weather, through a heat balance of Joule and solar heating against convective and radiative cooling. It leaves the choice of weather and conductor parameters to the user. Because wind cooling dominates, a rating built on conservative weather assumptions can sit well below what a conductor could carry on a cool, breezy day, which motivates ambient-adjusted and dynamic line ratings.

Ampacity of Underground Cables

A buried cable sheds its heat through a chain of thermal resistances: its insulation, its coverings, and the soil, which usually dominates. IEC 60287-1-1:2023 gives current rating equations and loss calculations for steady-state operation at 100 percent load factor, for cables buried directly, in ducts, troughs, or steel pipes, and in air. For AC cables where the soil does not dry out, the permissible current is

I = [(Δθ − W_d(0.5T₁ + n(T₂ + T₃ + T₄))) / (RT₁ + nR(1 + λ₁)T₂ + nR(1 + λ₁ + λ₂)(T₃ + T₄))]^1/2

where Δθ is the permissible rise of conductor temperature above ambient, R is the AC resistance per unit length at maximum operating temperature, W_d is the dielectric loss per unit length, T₁ through T₄ are the thermal resistances per unit length of the insulation, bedding, outer covering, and surroundings, n is the number of load-carrying conductors in the cable, and λ₁ and λ₂ are the sheath and armor losses as fractions of the conductor loss. North American practice traces the same thermal-circuit approach to a 1957 AIEE paper by J. H. Neher and M. H. McGrath, and the U.S. National Electrical Code permits ampacities calculated this way under engineering supervision.

Three practical factors often decide the rating. Soil around a hot cable can dry out, which raises its thermal resistivity and traps more heat, a runaway that engineered backfill is used to prevent. Neighboring circuits heat one another. And the sheaths of single-core cables carry induced voltage: bonding both ends lets circulating current flow and adds loss, single-point bonding leaves a standing voltage at the open end, and cross-bonding transposes the sheath connections at joints so that induced voltages cancel over each major section. XLPE insulation is commonly rated for a continuous conductor temperature of 90 °C.