Q1Electrical Materials

What is skin effect: when alternating current flows in a conductor, the current density is not uniform over the cross-section — it is maximum at the surface and decays exponentially toward the interior, so most of the current flows in a thin surface 'skin'. The effective conducting area is thereby reduced and the AC resistance rises above the DC resistance, increasingly so at higher frequency. The severity is characterized by the skin depth δ — the depth at which current density falls to 1/e (≈37%) of its surface value:

Why the effect is absent at DC: skin effect originates in electromagnetic induction. The current's own alternating magnetic flux, linking the interior of the conductor more than the surface layers, induces eddy EMFs (Faraday's law) that oppose the current most strongly at the centre — redistributing it toward the surface. Induction requires dΦ/dt ≠ 0. For direct current the internal flux is constant in time, no internal EMFs are induced, every filament of the conductor presents the same impedance (pure resistance), and the current distributes uniformly over the whole cross-section — hence no skin effect exists at DC, and the effect grows from zero smoothly as frequency rises from zero.

Derivation outline from Maxwell's equations: inside a conductor obeying Ohm's law J = σE, combining Ampere's law (∇×H = J, displacement current negligible in a good conductor) with Faraday's law (∇×E = −∂B/∂t) and B = μH yields a diffusion equation for the field penetrating the conductor:

For a sinusoidal field of angular frequency ω = 2πf incident on a plane conductor surface, this diffusion equation has the solution E(x) = E₀e^{−x/δ}e^{j(ωt − x/δ)}, i.e. the field (and hence current density) decays exponentially with depth x while simultaneously lagging in phase — both governed by the same characteristic length δ = √(2/ωμσ) = 1/√(πfμσ), which is exactly the skin depth quoted above. This is why skin depth is properly understood as an electromagnetic penetration depth, not merely an empirical current-crowding parameter: it emerges directly from requiring Maxwell's equations to be satisfied inside a lossy conductor under time-harmonic excitation, with the exponential decay constant and the phase-lag constant necessarily equal to one another.

Numerical evaluation: interpreting the given sample conductivity as σ = 10⁴ S/cm = 10⁶ S/m (a metallic-order conductivity; the printed unit 'Ω-cm' is the customary resistivity notation, its reciprocal being intended here), and taking μ = μ₀ = 4π×10⁻⁷ H/m for a non-magnetic conductor:

The skin depth is approximately 11.25 mm at 2,000 Hz and 3.56 mm at 20,000 Hz. Note the ratio: increasing frequency tenfold reduces δ by exactly √10 ≈ 3.16, directly exhibiting the 1/√f law derived above. Practically, these numbers mean that at 20 kHz any solid conductor thicker than ~7 mm diameter wastes its core material (carrying negligible current), which is why high-frequency and induction-heating conductors are made as thin-walled tubes, laminated packs, or litz wire (many fine insulated strands, each thinner than δ, transposed so each strand occupies all radial positions equally along the length) — engineering responses that follow immediately from the skin-depth values computed here.

AC resistance ratio implied by these skin depths: for a conductor whose radius a is large compared with δ, the AC resistance is well approximated by treating the current as confined to an effective annular shell of thickness δ at the surface, so that Rac/Rdc ≈ a/(2δ) for a ≫ δ. Even for a moderate conductor of radius, say, 10 mm, this gives Rac/Rdc ≈ 10/(2×11.25) ≈ 0.44 at 2 kHz — i.e. the effective area is already noticeably less than the true cross-sectional area — worsening to Rac/Rdc ≈ 10/(2×3.56) ≈ 1.4 at 20 kHz, meaning the AC resistance is roughly 40% higher than DC resistance at that radius and frequency. This quantifies, in terms of extra I²R loss and heating, precisely why cable and busbar designers must size conductors using an effective (skin-limited) area rather than the full geometric cross-section whenever the operating frequency is high enough that δ becomes comparable to or smaller than the conductor dimension, and it directly motivates the litz-wire and tubular-conductor solutions noted above.

Related proximity effect: in bundled or closely spaced conductors carrying AC (e.g., adjacent phases of a busbar, or turns of a coil), a second, related redistribution called the proximity effect further concentrates current toward the sides of each conductor facing (or facing away from, depending on current direction) its neighbours, due to the neighbouring conductor's alternating field inducing additional eddy currents; this effect compounds the skin-effect resistance rise and must be accounted for separately in the design of high-frequency transformer windings and multi-conductor cable bundles, again vanishing entirely at DC for the same fundamental reason that skin effect vanishes at DC — the absence of any time-varying flux to induce redistributing eddy currents.

Skin effect in ferromagnetic conductors: the derivation above used μ = μ₀ for a non-magnetic conductor, but the same formula δ = 1/√(πfμσ) shows that skin effect is far more severe in ferromagnetic conductors such as steel, where the relative permeability μr can be several hundred to a few thousand. Since δ scales as 1/√μ, a steel conductor of the same conductivity and at the same frequency has a skin depth tens of times smaller than a copper conductor — which is why steel ground wires and rails carry AC current in an extremely thin surface layer, why AC resistance of steel conductors is dramatically higher than their DC resistance, and why, conversely, non-magnetic (aluminium or copper-clad) conductors are strongly preferred wherever high-frequency or high-current AC transmission efficiency matters. This same μ-dependence is exploited deliberately in induction heating of steel components, where a thin, intensely heated skin is precisely the desired outcome rather than a parasitic loss.

Summary of governing variables: the skin depth formula δ = 1/√(πfμσ) shows that skin effect worsens (δ shrinks) with increasing frequency f, increasing permeability μ, and increasing conductivity σ — so the three practical levers available to a designer wanting to minimize skin-effect loss at a fixed frequency are choosing a lower-conductivity, non-magnetic conductor material, subdividing the conductor into insulated strands (litz wire) so each individual strand remains thin compared with δ, or increasing the effective perimeter-to-area ratio via tubular or hollow-core conductor geometry — all three of which are standard responses used in power and RF engineering once the operating frequency and hence the relevant skin depth, as computed numerically above, is known.