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Conductor selection is an economic decision as well as an electrical one. A larger cross-section uses more conductor material, so its purchase and installation cost rises. However, the resistance falls, reducing I²R energy losses during operation. The selected size balances these costs over the expected service life.
Capital cost rises with area; resistance and loss generally fall as area rises.
Compare life-cycle costs, not just the initial purchase price.
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Kelvin’s law is used to estimate the economical cross-sectional area of a conductor by balancing the annual cost of energy lost in the conductor against the annual fixed charges associated with the conductor investment. It is an idealised economic rule and practical constraints such as current rating, voltage drop, mechanical strength and standard sizes must also be considered.
At the economic point: annual cost of energy losses ≈ annual fixed charges on the conductor.
Kelvin’s law concerns economical conductor area, not insulation or transformer design.
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At the economic conductor size, Kelvin’s law states that the annual cost of energy losses is approximately equal to the annual fixed charges on the conductor. Fixed charges can include interest on capital, depreciation and other ownership costs. If the conductor is too small, losses are costly; if it is too large, the extra capital cost may not be justified.
Annual loss cost ≈ annual fixed charges.
Remember the balance is between two annual costs, not total revenue or total equipment cost.
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Increasing conductor cross-sectional area requires more metal and therefore normally increases capital cost. Since conductor resistance is approximately inversely proportional to area, a larger conductor has lower resistance and lower I²R loss for the same current. The total economic cost therefore does not move in only one direction.
R = ρL/A and P_loss = I²R.
Larger area: higher investment, lower resistive losses (for the same material and length).
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The economical size is the compromise that minimises the sum of annualised ownership cost and operating energy-loss cost. A small conductor is cheaper to install but has greater resistance and losses. A very large conductor reduces losses but costs more to purchase and install.
Total annual cost = annual fixed charge + annual energy-loss cost.
Choose the size that minimises total cost while meeting technical requirements.
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Annual fixed charges are costs associated with owning the conductor rather than the energy it wastes. They commonly include interest or cost of capital, depreciation and other fixed ownership charges. Copper loss is an operating energy cost, so it is treated separately in Kelvin’s-law comparisons.
Annual fixed charges ≈ interest + depreciation + other fixed ownership costs.
Do not confuse fixed charges with variable I²R energy-loss costs.
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Underground cables need substantial electrical insulation around the conductor and often require trenching, ducts, jointing, terminations, protective layers and specialised installation. These requirements generally make them more expensive per route length than overhead lines, although underground systems may offer benefits such as reduced visual impact and less exposure to weather.
Cable cost includes conductor + insulation + sheath/armour + installation and accessories.
The cost difference is not because underground cables lack conductors or protection.
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The conductor must be electrically separated from the surrounding earth, other cores and metallic sheath. Insulating materials such as XLPE or suitable polymeric/paper insulation surround the conductor and withstand the operating electric field. The exact cable construction depends on its voltage rating and application.
Insulation separates the live conductor from the sheath/earth and other conductors.
Air alone is not the insulation system of a typical buried power cable.
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Cable insulation limits leakage current and prevents breakdown between the conductor and surrounding grounded parts or other cores. It must withstand the electric stress created by the operating voltage and maintain suitable dielectric strength over the cable’s life. Defective, wet or aged insulation can lead to leakage, partial discharge or failure.
Dielectric stress is electric field strength, commonly expressed in V/m or kV/mm.
Insulation is selected to withstand electrical stress, not to deliberately increase conductor resistance.
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A single-core cable’s central conductor and surrounding metallic sheath are separated by dielectric insulation. These two conductive surfaces behave like the plates of a cylindrical capacitor, with the insulation acting as the dielectric. Therefore, the cable stores electric charge when a voltage is applied between conductor and sheath.
For a coaxial cable, C per unit length = 2πε / ln(R/r), where r is conductor radius and R is inner sheath radius.
Capacitance is caused by conductor geometry and the dielectric, not by cable weight or mechanical tension.
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The central conductor is one electrode of the cable’s cylindrical capacitor. When voltage is applied between the conductor and metallic sheath, opposite charges appear on the two electrodes and an electric field forms through the insulation. The conductor’s main operating role is still to carry load current.
The conductor and sheath form the two capacitor electrodes.
The conductor is not an insulator or merely a mechanical support.
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The metallic sheath surrounds the insulation and forms the outer electrode of the cable’s conductor-to-sheath capacitance. It also commonly provides a screen and can provide a path for fault current or earthing, depending on the cable design and bonding arrangement. It does not normally carry all of the load current.
C′ = 2πε / ln(R/r) for an ideal coaxial geometry.
Distinguish the normal load-current conductor from the metallic sheath/screen.
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For a single-core cable, capacitance per unit length is C′ = 2πε/ln(R/r), where r is conductor radius and R is the inner radius of the sheath. With the sheath radius and dielectric unchanged, increasing conductor radius reduces ln(R/r), so capacitance increases. The conclusion assumes the other geometric and dielectric factors remain fixed.
C′ = 2πε / ln(R/r). Increasing r while R stays fixed increases C′.
Always state which dimensions are held constant when reasoning about cable capacitance.
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Capacitance is directly proportional to the dielectric permittivity ε of the insulation. A material with higher relative permittivity allows more charge to be stored for the same voltage and geometry. Since ε = ε₀εᵣ, increasing εᵣ increases capacitance if the cable dimensions remain unchanged.
C′ = 2π ε₀ εᵣ / ln(R/r).
Higher dielectric permittivity means higher capacitance for fixed geometry.
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For a cylindrical cable, increasing insulation thickness increases the distance between the conductor and the sheath, assuming the conductor radius remains fixed. This increases the logarithmic denominator ln(R/r) in the capacitance expression, so capacitance per unit length decreases.
C′ = 2πε / ln(R/r); increasing R with r fixed decreases C′.
Thicker insulation generally lowers capacitance per unit length if other properties are unchanged.
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The electric field in a single-core cable is not uniform across the insulation. For a coaxial cylindrical structure, field strength is inversely proportional to radius, E(r) = V/[r ln(R/r₀)], where r₀ is conductor radius and R is the sheath’s inner radius. The smallest radius is at the conductor surface, so the field is maximum there.
E(r) = V/[r ln(R/r₀)] for an ideal single-core cable.
Maximum dielectric stress occurs at the conductor surface.
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Electric field strength falls as radial distance from the conductor increases. In the ideal coaxial cable model, E(r) is proportional to 1/r. Thus, the insulation nearest the conductor experiences the highest stress, while the field at the outer sheath is lower.
E(r) ∝ 1/r.
Cable stress is non-uniform; it is highest at the inner radius.
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This repeats the standard result for a single-core cable: the conductor surface has the smallest radius and therefore the greatest electric field. The insulation must be designed for this maximum stress, because local breakdown is most likely where the field is strongest.
E_max = V/[r₀ ln(R/r₀)] in the ideal coaxial model.
Insulation design is governed by maximum stress, not merely average voltage divided by thickness.
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Dielectric stress is electric field strength: the voltage change per unit distance in the insulating material. Its SI unit is volt per metre (V/m); practical high-voltage cable calculations often use kV/mm. Ampere per metre is associated with magnetic field strength, not electric stress.
E = V/d is a simple uniform-field approximation; E is measured in V/m.
Do not confuse electric field units with current density or resistance per unit length.
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Cable grading is intended to reduce the concentration of electric stress near the conductor and make the stress distribution more uniform. This helps the insulation use its dielectric strength more effectively and can reduce the risk of breakdown. Two classic methods are capacitance grading and intersheath grading.
Goal: reduce E_max and improve utilisation of insulation.
Grading redistributes stress; it does not eliminate capacitance or insulation.
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Capacitance grading uses layers of dielectric materials with different relative permittivities. The permittivities and layer dimensions are selected so the electric field is better distributed across the insulation. This differs from intersheath grading, which introduces an intermediate-potential metallic sheath.
For each layer, electric displacement is related to permittivity and electric field: D = εE.
Different dielectric permittivities are the basis of capacitance grading.
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The dielectric closest to the conductor should have the highest permittivity. The electric field is stronger near the conductor, and a higher-permittivity inner layer helps reduce that peak stress. The outer layers can use progressively lower permittivities, subject to the design and material constraints.
For a given electric displacement, E = D/ε; larger ε gives smaller E.
Highest permittivity is placed nearest the conductor in capacitance grading.
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Capacitance grading seeks to reduce the maximum electric stress by choosing insulation layers with suitable permittivities. Since the inner region naturally experiences the greatest field, the material arrangement is chosen to lower that peak and distribute voltage stress more evenly. It cannot remove capacitance entirely.
E = D/ε; higher ε in the high-stress region reduces field for a given D.
The objective is lower peak stress, not higher stress.
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In intersheath grading, one or more extra metallic sheaths are inserted between the conductor and outer sheath. Each additional sheath is maintained at a selected intermediate potential, creating separate insulation sections with more controlled voltage drops. The potentials must be carefully designed and maintained.
Intermediate sheaths divide the total voltage into selected portions.
An intersheath is not necessarily at earth potential; it is held at a suitable intermediate potential.
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Intersheath grading controls how voltage is distributed across cable insulation by adding metallic sheaths at selected potentials. With suitable design, the voltage drop across each insulation section can be arranged to improve stress distribution and reduce the peak field. This is a different method from using dielectric layers of different permittivity.
Objective: control the electric-field distribution and reduce maximum stress.
Capacitance grading changes dielectric properties; intersheath grading adds intermediate-potential sheaths.
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A multi-core cable contains two or more insulated conductors within a common protective construction. Three-core cables are commonly used in three-phase systems because each core can carry one phase, with insulation between cores and suitable overall protection. The exact number of cores depends on the circuit and earthing arrangement.
Three-phase cable: typically one insulated core per phase.
Cable cores are electrical conductors, not mechanical shafts or cooling passages.
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The metallic sheath can provide a barrier against moisture and environmental effects and acts as an electrical screen around the insulated core. When properly bonded and earthed, it can also provide a path for earth-fault current. Its exact mechanical and electrical functions depend on construction; armouring is the layer most specifically associated with mechanical protection.
Sheath/screen function depends on cable design and bonding.
Do not confuse metallic sheath with the main load conductor or with armour alone.
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Armouring is a protective layer, often made of steel wires or strips, intended primarily to withstand external mechanical forces during installation and service. It helps protect the cable from crushing, impact and other damage. Armour design depends on the installation environment and cable type.
Armour provides mechanical protection; it is not added to increase capacitance.
Steel-wire or steel-tape armour is common where mechanical protection is required.
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Steel is commonly used for cable armour because it offers good mechanical strength and is practical to form into wires or tapes. Depending on the cable design, galvanised steel wires or steel strips may be used. Non-magnetic armour choices may be needed in some single-core AC cable arrangements to limit induced losses.
Armour material is chosen mainly for mechanical strength and installation conditions.
Steel is a common answer, but armour construction depends on cable type and AC magnetic effects.
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The bedding layer lies between the metallic sheath and the armour in many traditional cable constructions. It cushions the sheath and other cable components from direct contact with the armour, reducing the chance of abrasion or mechanical damage. The precise layers vary with modern cable designs.
Typical traditional sequence: conductor → insulation → sheath → bedding → armour → serving.
Bedding is a protective/cushioning layer, not a load-current conductor.
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Serving is the outer protective covering in many traditional cable constructions. It helps protect the cable against moisture, corrosion, abrasion and environmental exposure, according to the serving material and installation conditions. Modern cables may use different outer sheaths or jackets instead of traditional serving.
Outer covering protects against environmental and surface damage.
Serving is external protection, not a power-factor correction component.
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A cable joint joins two cable lengths while maintaining electrical continuity, insulation integrity and appropriate mechanical protection. In medium- and high-voltage systems, jointing must carefully manage the electric field and restore the insulation and screen layers; poor workmanship can cause partial discharge or failure.
A sound joint restores conductor continuity and the cable’s insulation/screen system.
Joints connect cable lengths; terminations connect a cable to equipment or another type of conductor.
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A termination prepares the end of a cable for safe connection to equipment, switchgear, a transformer, or an overhead line. It controls the electric field at the cable end and restores insulation, screening and sealing as required. A poorly designed termination can suffer tracking, partial discharge or flashover.
Termination = safe cable end connection plus electric-field control and sealing.
A joint connects cable-to-cable; a termination connects cable-to-equipment or overhead conductors.
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Underground cables generally have much higher capacitance than comparable overhead lines because the conductors are closer together and surrounded by a dielectric with higher permittivity than air. On AC, this capacitance draws charging current even when the cable has little or no useful load current.
I_c = ωCV = 2πfCV (for RMS voltage and capacitance under the usual sinusoidal model).
High cable capacitance leads to higher AC charging current.
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A long AC cable has substantial shunt capacitance, which draws charging current and produces reactive-power effects. These can limit the usable current capacity and complicate voltage/reactive-power management over long distances. HVDC avoids continuous steady-state AC charging current along the line, although converter stations and other system considerations are required.
Charging current: I_c = 2πfCV.
For very long-distance bulk transfer where AC cable charging is a major concern, HVDC may be suitable.
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An underground cable fault is hidden below ground or inside the cable structure, so locating the fault may require specialised tests and equipment. Repair may involve excavation, faulted-section removal, jointing and testing before re-energisation. Overhead faults are often easier to inspect visually and may be cleared by protection or weather recovery.
Common fault-location approaches include insulation tests, time-domain methods and bridge methods, selected for the cable and fault.
Underground faults are not impossible to detect; they are usually harder to locate and repair.
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A megger (insulation-resistance tester) applies a specified DC test voltage and measures the resistance of insulation between conductors or between a conductor and earth/sheath. It is commonly used during maintenance and commissioning, following the relevant safety procedure and equipment guidance.
Insulation resistance is measured in ohms, commonly MΩ or GΩ.
A megger measures insulation resistance, not sag, tower height or generator power factor.
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Healthy cable insulation should oppose leakage current strongly, so its insulation resistance should generally be high when measured under appropriate test conditions. The acceptable value depends on cable type, length, temperature, test voltage, age and applicable standards; a single universal threshold should not be assumed.
R_ins = V_test / I_leak (conceptually, during a DC insulation test).
Trend measurements and compare with manufacturer/standard guidance rather than relying on one universal number.
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A reduction in insulation resistance means more leakage current flows for the same test voltage. Moisture ingress, contamination, physical damage, thermal ageing or insulation deterioration can cause this change. Results should be compared with prior readings and corrected for temperature when appropriate.
R_ins = V_test / I_leak; if leakage current rises at fixed test voltage, measured resistance falls.
A declining insulation-resistance trend is a warning sign that needs investigation.
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Dielectric loss is energy converted to heat within insulating material under alternating electric stress. Real dielectrics have a small in-phase current component due to losses, unlike an ideal lossless capacitor. Excessive dielectric loss can indicate material characteristics, ageing, moisture or deterioration.
For sinusoidal voltage, dielectric loss is commonly expressed as P_d = ω C V² tanδ (using RMS voltage and the relevant capacitance).
Dielectric loss occurs in insulation, distinct from conductor I²R loss.
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An ideal dielectric stores and returns energy without dissipating any as heat. Its loss angle is zero and its dissipation factor tanδ is therefore zero. Real insulating materials have non-zero losses, although good cable insulation is designed to keep them small.
Dissipation factor = tanδ; ideal dielectric: tanδ = 0.
A zero ideal dissipation factor means no dielectric energy loss in the ideal model.
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The dielectric loss angle is conventionally denoted by the Greek letter delta, δ. It describes the deviation from ideal capacitive behaviour associated with dielectric losses. The dissipation factor is tanδ, often measured during insulation diagnostic tests.
Dissipation factor = tanδ.
Do not confuse δ (dielectric loss angle) with other phase-angle symbols used for circuit quantities.
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For a given voltage, frequency and capacitance, dielectric loss increases as the dissipation factor tanδ increases. The loss factor reflects how much energy the dielectric dissipates relative to the energy it stores. Increased loss can produce extra heating and may be a sign of degraded insulation.
P_d = ω C V² tanδ, for sinusoidal RMS voltage.
Higher tanδ generally means higher dielectric loss if ω, C and V remain fixed.
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For sinusoidal AC, the cable capacitance draws charging current. Its magnitude depends on angular frequency ω, capacitance C and voltage V. Higher frequency, larger capacitance or higher voltage increases the charging current proportionally, provided the other quantities remain fixed.
I_c = ωCV = 2πfCV.
Charging current is capacitive current; it is not determined by cable weight or mechanical tension.
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For an ideal cable capacitance under sinusoidal AC, capacitive current magnitude is I_c = ωCV, where ω = 2πf, C is capacitance and V is the RMS voltage across the capacitance. The formula shows direct proportionality to frequency, capacitance and voltage.
I_c = ωCV = 2πfCV.
Use consistent RMS voltage and capacitance units; ω is measured in rad/s.
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From I_c = ωCV, if voltage and frequency (hence ω) remain unchanged, charging current is directly proportional to capacitance. Doubling C therefore doubles I_c. This is a direct-proportion relationship, not a square-law relationship.
I_c ∝ C when ω and V are constant.
If one factor doubles and all other factors stay fixed, charging current doubles.
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Since angular frequency is ω = 2πf, the charging-current equation can also be written I_c = 2πfCV. If frequency doubles while voltage and capacitance remain constant, the charging current doubles. This conclusion assumes a sinusoidal steady-state model.
I_c ∝ f when C and V are constant.
Charging current is frequency-dependent on AC; steady-state DC has no continuous sinusoidal charging current after transients settle.
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Charging current is directly proportional to the voltage across the cable capacitance. If frequency and capacitance remain fixed, doubling voltage doubles the charging current. In practice, cable voltage must remain within its insulation rating and operating limits.
I_c ∝ V when ω and C are constant.
Doubling voltage doubles charging current, not four-times current.
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HVDC can be advantageous for very long-distance bulk power transfer when the charging current of an AC cable is a major limitation. With steady DC, the cable capacitance charges during energisation and then does not draw continuous capacitive charging current once the transient has settled. HVDC still requires converter stations and careful system-level economic evaluation.
For steady DC after the charging transient, dV/dt = 0, so i_C = C(dV/dt) = 0.
HVDC avoids continuous steady-state capacitive charging current, but does not mean the complete HVDC system has no losses.
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A capacitor current is i_C = C(dv/dt). Under steady DC conditions, voltage is constant after the charging transient, so dv/dt = 0 and the continuous capacitive current is zero. This is an important reason HVDC can be attractive for long cable routes; however, converter stations, cable losses and system economics still matter.
i_C = C(dv/dt); for steady DC, dv/dt = 0 ⇒ i_C = 0.
Do not interpret this as zero total transmission loss or zero converter requirements.
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