Skin & Proximity Effect on Cables — AC Resistance Explained

Every electrical engineer learns Ohm’s law: V = IR. The resistance R of a conductor is straightforward to calculate from its length, cross-sectional area, and material resistivity. But this calculation gives you the DC resistance — and in an AC system, the actual effective resistance is always higher.

Two electromagnetic phenomena cause this: skin effect and proximity effect. For small cables (up to about 25 mm²), the difference is negligible. But for large power cables — 185 mm², 300 mm², 630 mm² — the AC resistance can be 5% to 20% higher than the DC resistance. Over kilometres of cable run and decades of operation, this translates to enormous wasted energy and, more immediately, to cables running hotter than their DC resistance would predict.

Understanding these effects is essential for accurate voltage drop calculations, correct cable sizing, and optimising large installations where parallel smaller cables might outperform a single large conductor.

At DC (or theoretically at 0 Hz), current distributes itself uniformly across the entire cross-section of a conductor. Every square millimetre of copper carries the same current density. The resistance depends only on geometry and material:

R_dc = ρ × L / A — (Eq. 1)

Where:

• ρ = resistivity (1.7241 × 10⁻⁸ Ω·m for copper at 20°C)
• L = length (metres)
• A = cross-sectional area (m²)

For a 100-metre run of 630 mm² copper at 20°C:

R_dc = 1.7241 × 10⁻⁸ × 100 / 630 × 10⁻⁶ = 0.002737 Ω (2.737 mΩ) — (Eq. 2)

This is the minimum possible resistance. AC effects can only increase it.

When alternating current flows through a conductor, it creates a time-varying magnetic field inside the conductor itself. By Faraday’s law, this changing magnetic field induces circulating eddy currents within the conductor. These eddy currents oppose the flow of current in the interior and reinforce it near the surface.

The net result: current density is highest at the conductor surface and decreases exponentially toward the centre. The conductor’s interior carries less current than its fair share, effectively reducing the useful cross-sectional area and increasing the resistance.

The characteristic dimension of skin effect is the skin depth (δ), defined as the distance from the surface at which the current density has fallen to 1/e (about 37%) of its surface value:

δ = √(ρ / (π × f × μ₀ × μ_r)) — (Eq. 3)

For copper at 50 Hz:

δ = √(1.7241 × 10⁻⁸ / (π × 50 × 4π × 10⁻⁷ × 1)) = 9.35 mm — (Eq. 4)

At 60 Hz, the skin depth is slightly smaller: approximately 8.53 mm.

For a cable whose conductor radius is much smaller than the skin depth, skin effect is negligible. A 4 mm² cable has a radius of approximately 1.13 mm — far smaller than the 9.35 mm skin depth. Current distribution is essentially uniform.

But a 630 mm² cable has an equivalent radius of approximately 14.2 mm — larger than the skin depth. The centre of this conductor carries significantly less current than the surface, and the effective resistance increases.

IEC 60287-1-1:2006, Section 2.1 provides the methodology for calculating skin effect factors.

Standards express skin effect as a multiplicative factor. The AC resistance due to skin effect alone is:

R_ac,skin = R_dc × (1 + k_s) — (Eq. 5)

For typical power cables at 50 Hz:

Below 25 mm², skin effect is essentially zero. Above 300 mm², it becomes a meaningful contributor to losses.

Skin effect occurs in an isolated conductor. But cables rarely exist in isolation — they run alongside other current-carrying conductors. The alternating magnetic field from an adjacent conductor distorts the current distribution in its neighbour, pushing current toward or away from the adjacent cable depending on the current direction.

This effect is strongest when cables are close together and carrying high currents. It depends on:

• The spacing between conductors relative to their diameter
• The magnitude of current in the adjacent conductor
• The frequency

Similar to skin effect, proximity effect is expressed as a factor k_p:

R_ac = R_dc × (1 + k_s + k_p) — (Eq. 6)

The proximity effect factor depends heavily on cable formation:

IEC 60287-1-1:2006, Section 2.2; AS/NZS 3008.1.1:2017, Appendix F.

A single-core 630 mm² copper cable with XLPE insulation is installed in trefoil formation. Calculate the total R_ac/R_dc ratio.

Given values (from IEC 60287 methodology):

• DC resistance at 90°C (operating temperature): R_dc = 0.0469 Ω/km
• Skin effect factor: k_s = 0.055
• Proximity effect factor (trefoil): k_p = 0.025

Step 1 — Calculate total AC resistance:

R_ac = R_dc × (1 + k_s + k_p) = 0.0469 × (1 + 0.055 + 0.025) = 0.0469 × 1.080 = 0.05065 Ω/km — (Eq. 7)

Step 2 — Interpret the result:

The R_ac/R_dc ratio is 1.080, meaning the cable has 8% more resistance when carrying AC current than its DC resistance suggests. For a 500-metre run carrying 800 A:

Over three phases, that is 3.6 kW of additional losses. Over 8,760 hours per year at average 60% loading, the annual wasted energy is approximately 11,350 kWh — roughly what three households consume.

An engineer needs to supply 1,200 A at 415 V over a 200-metre run. Compare two options:

Option A: Single 630 mm² cable per phase

• R_ac = 0.05065 Ω/km (from the previous example)
• But 630 mm² has a current rating of approximately 870 A in trefoil — insufficient! Two cables per phase needed.
• Parallel resistance of two: 0.02533 Ω/km
• Losses Option A: 3 × 1200² × 0.02533 × 0.2 = 21,880 W (total 3-phase)

Option B: Three parallel 240 mm² cables per phase

• R_dc at 90°C: 0.0983 Ω/km
• k_s = 0.018, k_p = 0.010 (much lower for smaller cables)
• R_ac = 0.0983 × 1.028 = 0.1011 Ω/km
• Parallel resistance of three: 0.1011 / 3 = 0.0337 Ω/km
• Current per cable: 400 A (well within 530 A trefoil rating)
• Losses Option B: 3 × 1200² × 0.0337 × 0.2 = 29,082 W (total 3-phase)

In this case, Option A has lower total losses because the total copper area is greater (1,260 mm² vs 720 mm² per phase). But the AC effects are proportionally much smaller for Option B. If we compare losses per unit of copper area, the smaller cables are more efficient at converting copper into current-carrying capacity.

The real advantage of parallel smaller cables appears when voltage drop rather than current rating is the constraint, or when the installation geometry makes trefoil formation impractical for large singles.

In trefoil (triangular) formation, the three phase conductors are symmetrically arranged. Each conductor “sees” its two neighbours at equal distances, and the vector sum of the proximity effects partially cancels due to the 120° phase relationship.

In flat formation, the centre conductor has two neighbours while the outer conductors have only one. The centre cable runs hotter, creates an asymmetric impedance (the centre phase has different impedance than the outer phases), and the overall proximity effect is higher.

For a 300 mm² cable system:

• Trefoil formation: k_p ≈ 0.015
• Flat touching: k_p ≈ 0.028
• Flat spaced (one diameter): k_p ≈ 0.035

The difference in total R_ac/R_dc might only be 1–2%, but applied to thousands of amperes over long distances, it is significant.

AS/NZS 3008.1.1:2017, Table 30 and Appendix F; BS 7671:2018, Appendix 4.

Both skin effect and proximity effect worsen at higher frequencies. The 60 Hz systems common in North America have approximately 10% worse AC effects than the 50 Hz systems used in most other countries. The skin depth at 60 Hz is 8.53 mm versus 9.35 mm at 50 Hz.

This becomes dramatically important for cables carrying harmonic currents. A cable sized for 50 Hz fundamental frequency will have significantly higher AC resistance at the 250 Hz fifth harmonic (skin depth drops to 4.2 mm) or the 350 Hz seventh harmonic (skin depth of 3.5 mm). This is one reason why harmonic-rich environments require cable derating — covered in detail in Harmonics and Cable Derating.

A practical threshold: skin and proximity effects become relevant above approximately 120 mm² for copper and 150 mm² for aluminium at 50/60 Hz. Below these sizes, the R_ac/R_dc ratio is within 1–2% of unity, and the DC resistance is sufficient for engineering calculations.

Above 300 mm², ignoring AC effects will lead to:

• Underestimated voltage drop (by 3–8%)
• Underestimated cable operating temperature
• Potential for cables running closer to their thermal limit than designed
• Inaccurate loss calculations for energy efficiency studies

1. Skin effect forces current to the outer surface of a conductor. The skin depth in copper at 50 Hz is 9.35 mm — any conductor with a radius larger than this is significantly affected.
2. Proximity effect distorts current distribution based on neighbouring cables. It is worse in flat formation and better in trefoil.
3. A 630 mm² cable has an R_ac/R_dc ratio of approximately 1.08 at 50 Hz — 8% more losses than DC resistance predicts. This matters for accurate voltage drop and thermal calculations.
4. Parallel smaller cables can be more copper-efficient than a single large cable because the AC effects are proportionally smaller in conductors whose radius is well below the skin depth.
5. Trefoil formation reduces proximity effect compared to flat formation because the symmetric arrangement allows partial cancellation of the magnetic field distortions.

Standards referenced: IEC 60287-1-1:2006, Sections 2.1 and 2.2; AS/NZS 3008.1.1:2017, Appendix F and Table 30; BS 7671:2018+A2, Appendix 4.