Skin Effect

When alternating current flows through a conductor, it does not distribute itself uniformly across the cross-section. Instead, it concentrates toward the outer surface of the conductor. This phenomenon, known as the skin effect, has direct consequences on the effective resistance of conductors at high frequencies and is a foundational concept in RF, microwave, and power electronics engineering.

Core Concept Explanation

The skin effect arises from Faraday's law of electromagnetic induction. When an AC current flows through a conductor, it produces a time-varying magnetic field, which in turn induces an eddy current inside the conductor. These induced currents oppose the original current in the interior and reinforce it near the surface. The net result is that the interior current is suppressed while the surface current is enhanced.

Mathematically, the skin effect is a direct consequence of the wave equation applied inside a good conductor. The current density J inside the conductor decays exponentially from the surface as J(z) = J₀ e^(−z/δ), where z is measured inward from the surface and δ is the skin depth. The skin depth represents the depth at which the current density has fallen to 1/e, approximately 36.8%, of its surface value.

As frequency increases, δ decreases. At power frequencies (50 Hz), the skin depth in copper is about 9.3 mm, which is larger than most conductors, so the effect is minor. At 1 GHz, δ in copper shrinks to about 2.1 μm, meaning only a thin skin of the conductor actually carries current. The conductor's interior is effectively unutilized, leading to a dramatic increase in effective resistance.

The surface resistance Rs is defined as Rs = 1/(σδ) = √(ωμ/2σ) in ohms per square. This quantity determines the power dissipated per unit area of conductor surface and appears directly in microwave cavity Q-factor and transmission line attenuation calculations.

Mathematical Expression

The skin depth formula for a good conductor (σ >> ωε) is δ = √(2/ωμσ) = 1/√(πfμσ). This shows that δ is inversely proportional to the square root of frequency, permeability, and conductivity. Doubling frequency reduces skin depth by a factor of √2.

The AC resistance per unit length of a round conductor of radius a, when δ << a (high frequency), approaches Rac = 1/(2πa σδ) = Rs/(2πa). The ratio Rac/Rdc = a/(2δ), which grows with frequency. This increased resistance is central to skin effect losses in coils and inductors at RF frequencies.

The power dissipated per unit area of a conductor surface is P = (1/2)|Js|² Rs, where Js is the surface current density. This result, derived from Poynting's theorem applied at the conductor boundary, is used extensively in cavity resonator and waveguide wall loss analysis.

Practical Understanding

In RF and microwave transmission lines, skin effect is the primary source of conductor loss. For a coaxial cable, the attenuation due to conductor loss increases approximately as √f. Engineers compensate by using silver plating on the inner surface of waveguides because silver has the highest conductivity of any metal, thereby minimizing Rs and reducing wall losses.

Transformer and inductor winding design must account for the skin effect. At power frequencies, stranded wire known as Litz wire is used to reduce skin effect losses. Litz wire consists of many individually insulated thin strands woven so that each strand occupies the surface position equally, effectively increasing the usable cross-sectional area at elevated frequencies.

In printed circuit boards, the skin effect determines the minimum trace thickness that is electrically useful at a given frequency. Using traces much thicker than the skin depth wastes copper without improving performance, which is relevant in high-density RF layout design.

Given:
Conductor: Copper (σ = 5.8 × 10⁷ S/m, μr = 1), f = 100 MHz
Wire radius a = 1 mm

Why this formula applies:
Copper is a good conductor, skin depth and surface resistance formulas apply.

Formula:
δ = √(2 / ωμσ) = 1 / √(πfμσ)
Rs = 1 / (σδ)
Rac/Rdc = a / (2δ)

Substitution:
ω = 2π × 10⁸ rad/s
μ = 4π × 10⁻⁷ H/m
πfμσ = π × 10⁸ × 4π × 10⁻⁷ × 5.8 × 10⁷
       = π × 10⁸ × 4π × 10⁻⁷ × 5.8 × 10⁷
       = 4π² × 5.8 × 10⁸ = 2.29 × 10¹⁰

Calculation:
δ = 1 / √(2.29 × 10¹⁰) = 1 / 1.513 × 10⁵ = 6.61 × 10⁻⁶ m
δ ≈ 6.61 μm
Rs = 1 / (5.8 × 10⁷ × 6.61 × 10⁻⁶) = 1 / 383.4 = 2.6 × 10⁻³ Ω/sq
Rac/Rdc = 1 × 10⁻³ / (2 × 6.61 × 10⁻⁶) = 75.6

Final Answer with units:
δ ≈ 6.61 μm at 100 MHz in copper.
Rs ≈ 2.6 mΩ/square.
AC resistance is about 75.6 times the DC resistance — skin effect is severe.

Exam Tip: Remember δ ∝ 1/√f. If frequency increases 4 times, skin depth halves and surface resistance Rs doubles. Also, Rs = 1/(σδ) = √(ωμ/2σ). Both forms appear in GATE questions on waveguide losses.

Mechanism Summary

• Skin effect occurs because induced eddy currents inside the conductor oppose the primary current in the interior while reinforcing it at the surface.
• Current density decays as J(z) = J₀ e^(−z/δ) where δ = √(2/ωμσ) is the skin depth.
• Surface resistance Rs = 1/(σδ) = √(ωμ/2σ). It determines conductor loss in waveguides, cavities, and coaxial lines.
• AC resistance ratio Rac/Rdc ≈ a/(2δ) for a round wire of radius a when δ << a.
• Silver plating is used in microwave components because silver maximizes σ and therefore minimizes Rs and wall losses.

Quick Revision

• δ = √(2/ωμσ) = 1/√(πfμσ). Doubles when frequency reduces 4×.
• Rs = 1/(σδ) = √(ωμ/2σ) in Ω/square. Increases as √f.
• J(z) = J₀ e^(−z/δ). At depth δ, current is 36.8% of surface value.
• Rac/Rdc ≈ a/(2δ) for round wire at high frequency.
• Skin effect increases with f, μ, σ. Silver has lowest Rs among metals.
• Exam trap: Skin depth formula δ = √(2/ωμσ) is valid only for good conductors (σ >> ωε).
• Exam trap: Rs has units of Ω/square, not Ω/m. Power loss per area = (1/2)|Js|² Rs.