What is the formula for skin depth?

Skin depth is δ = 1/√(πfμ₀μᵣσ) meters, where f is frequency in Hz, μ₀ = 4π×10⁻⁷ H/m is the permeability of free space, μᵣ is relative permeability, and σ is electrical conductivity in S/m. For copper (σ = 5.8×10⁷ S/m, μᵣ = 1) at 50 Hz, δ ≈ 9.3 mm; at 1 MHz, δ ≈ 0.066 mm.

How does skin depth change with frequency?

Skin depth is inversely proportional to the square root of frequency (δ ∝ 1/√f). Quadrupling the frequency halves the skin depth. For copper: 50 Hz → δ ≈ 9.3 mm, 200 Hz → δ ≈ 4.7 mm, 50 kHz → δ ≈ 0.30 mm. This relationship is used to estimate high-frequency losses in coil conductors.

Why is the skin depth in iron much smaller than in copper?

Iron has a relative permeability of μᵣ ≈ 200 (copper has μᵣ = 1). Since δ ∝ 1/√(μᵣσ), higher μᵣ reduces skin depth significantly. At 50 Hz, iron (σ = 1×10⁷ S/m, μᵣ = 200) has δ ≈ 1.1 mm — about 1/8 that of copper. This is the primary cause of core losses in transformers and electric motors.

What is the relationship between skin depth and AC resistance?

For a round conductor of diameter d where d/δ ≫ 1, the AC-to-DC resistance ratio is approximated as R_AC/R_DC ≈ d/(4δ) + 3/4. Current density decays exponentially from the surface: J(d) = J₀ × e^(−d/δ). At a depth of 4.6δ the current density falls to about 1% of its surface value, effectively confining the current to a thin surface layer.

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ELECTROMAGNETICS

Skin Effect & Skin Depth Calculator

Calculate skin depth δ=√(2ρ/ωμ), effective cross-section, and AC resistance from frequency, material, and geometry in real time. Visualize high-frequency losses for electromagnetic design and EMC analysis.

Skin Depth (Skin Effect) Calculator

Material Presets

Electrical Properties

Conductivity σ 5.8×10⁷ S/m

10¹10⁴10⁸

Relative Permeability μ_r

Frequency

Frequency f 50 Hz

1 Hz10 kHz10 GHz

Display Options

Comparison Mode (overlay Cu/Al/Fe)

Formulas

Key Formulas

$\delta = \dfrac{1}{\sqrt{\pi f \mu_0 \mu_r \sigma}}$

$|J(d)| = J_0 \cdot e^{-d/\delta}$

Results

Skin Depth δ

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Penetration Depth

Frequency Regime

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Band Classification

R_AC/R_DC Ratio (dia.=2δ)

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AC Resistance Increase

1% J₀ Penetration Depth

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≈ 4.6δ

1. Current Density Distribution |J(d)|/J₀ vs Depth d

2. Skin Depth δ(f) vs Frequency (Log-Log)

3. Eddy Current Loss Ratio vs Thickness/δ Ratio (d/δ)

Skin Effect Animation — Current Density Decay with Depth

// Skin depth current animation (function() { const el = document.getElementById('skinDepthAnimCanvas'); const ctx = el.getContext('2d'); let animT = 0; function getParams() { const sigma = getSigma(); const mur = getMur(); const freq = getFreq(); const MU0 = 4*Math.PI*1e-7; const delta = Math.sqrt(1 / (Math.PI * freq * mur * MU0 * sigma)); return { sigma, mur, freq, delta }; } function resize() { const dpr = window.devicePixelRatio || 1; const w = el.parentElement.clientWidth - 36; const H = Math.round(w * 0.38); if (Math.abs(el.width - w * dpr) > 2 || el.height !== H * dpr) { el.width = w * dpr; el.height = H * dpr; el.style.height = H + 'px'; ctx.setTransform(dpr, 0, 0, dpr, 0, 0); } return { W: w, H }; } function frame() { const { W, H } = resize(); const { delta, freq } = getParams(); animT += 1/60; ctx.clearRect(0, 0, W, H); ctx.fillStyle = '#001F3F'; ctx.fillRect(0, 0, W, H); const padL = 20, padR = 60, padT = 20, padB = 30; const drawW = W - padL - padR; const drawH = H - padT - padB; const N = 60; const maxDepth = delta * 4; // AC current oscillation at surface const acPhase = animT * 2 * Math.PI * Math.min(freq, 2); // visual freq capped const J0 = Math.sin(acPhase); // Draw current density distribution for (let i = 0; i < N; i++) { const depthFrac = i / N; const depth = depthFrac * drawW; const depthM = depthFrac * drawW / drawW * maxDepth; const J_mag = Math.exp(-depthM / delta); const J_val = J0 * J_mag; const x = padL + depth; const barH = Math.abs(J_val) * drawH * 0.45; const cy = padT + drawH / 2; // Color: bright blue at high density, dark at low const b = Math.round(100 + J_mag * 155); const g = Math.round(50 + J_mag * 80); const alpha = 0.3 + J_mag * 0.7; ctx.fillStyle = `rgba(0,${g},${b},${alpha})`; if (J_val >= 0) { ctx.fillRect(x, cy - barH, drawW/N + 0.5, barH); } else { ctx.fillRect(x, cy, drawW/N + 0.5, barH); } } // Skin depth marker line const deltaPixel = (delta / maxDepth) * drawW; ctx.strokeStyle = '#e17055'; ctx.lineWidth = 1.5; ctx.setLineDash([4, 3]); ctx.beginPath(); ctx.moveTo(padL + deltaPixel, padT); ctx.lineTo(padL + deltaPixel, padT + drawH); ctx.stroke(); ctx.setLineDash([]); ctx.fillStyle = '#e17055'; ctx.font = '10px Roboto Mono'; ctx.textAlign = 'center'; ctx.fillText('δ', padL + deltaPixel, padT + drawH + 16); ctx.fillText(delta.toFixed(3) + ' mm', padL + deltaPixel, padT - 5); // Axes ctx.strokeStyle = 'rgba(255,255,255,0.3)'; ctx.lineWidth = 1; ctx.beginPath(); ctx.moveTo(padL, padT + drawH/2); ctx.lineTo(padL + drawW, padT + drawH/2); ctx.stroke(); // Labels ctx.fillStyle = 'rgba(255,255,255,0.6)'; ctx.font = '9px Roboto Mono'; ctx.textAlign = 'left'; ctx.fillText('Surface', padL, padT + drawH/2 - 5); ctx.textAlign = 'right'; ctx.fillText('Depth ->', W - padR + 35, padT + drawH/2 - 5); ctx.fillStyle = '#00B4D8'; ctx.font = '10px Roboto Mono'; ctx.textAlign = 'left'; ctx.fillText('J(d)/J₀', padL + 2, padT + 14); requestAnimationFrame(frame); } frame(); })();

What is Skin Depth?

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What exactly is "skin depth"? I've heard high-frequency electricity doesn't use the whole wire.

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That's right! Basically, it's the "skin effect." At high frequencies, alternating current gets pushed towards the outer surface of a conductor. The skin depth, δ, is the depth where the current density has fallen to about 37% of its value at the surface. Try moving the frequency slider in the simulator above from 50 Hz to 1 MHz and watch the skin depth shrink dramatically.

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Wait, really? So the whole center of the wire is just... unused? What determines how thin this "skin" layer is?

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In practice, four key properties: frequency, conductivity, and magnetic permeability. The higher the frequency, the thinner the skin. A great example is copper vs. iron. Iron has higher magnetic permeability, which confines current even more. You can test this in the simulator by changing the material's relative permeability (μᵣ) from 1 (like copper) to 1000 (like steel).

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So if the useful cross-section gets smaller at high frequencies, does that mean the wire's effective resistance goes up? That seems important for design.

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Exactly! That's the major engineering consequence. The AC resistance becomes much higher than the DC resistance. For instance, in a radio antenna or a high-speed circuit board trace, you can't just use a thick wire and expect low loss. The simulator shows you the raw skin depth number; designers then use that to calculate the increased resistance and power loss.

Physical Model & Key Equations

The skin depth (δ) is derived from the diffusion of electromagnetic waves into a conductive material. The governing equation shows it depends on the square root of frequency and material properties.

$$ \delta = \sqrt{\frac{2}{\omega \mu \sigma}}= \frac{1}{\sqrt{\pi f \mu_0 \mu_r \sigma}}$$

Where:
δ = Skin depth (meters)
f = Frequency (Hz)
ω = Angular frequency, ω = 2πf (rad/s)
μ₀ = Permeability of free space (4π × 10⁻⁷ H/m)
μᵣ = Material's relative permeability (dimensionless)
σ = Electrical conductivity (Siemens/meter, S/m)
The inverse square-root relationship with frequency is the key takeaway.

The increased AC resistance due to the skin effect can be approximated for a round wire. When the skin depth is much smaller than the wire radius, the effective cross-sectional area is just a thin cylindrical shell.

$$ R_{ac}\approx \frac{l}{\sigma (2 \pi a \delta)}\quad \text{for} \quad \delta \ll a $$

Where:
R_ac = AC resistance (Ohms)
l = Length of the conductor (m)
a = Radius of the wire (m)
This shows why, at high frequencies, using a thick solid conductor is wasteful—the center doesn't carry current.

Real-World Applications

Power Transmission: At mains frequency (50/60 Hz), skin depth in aluminum is over 1 cm, so solid cores are fine. For extra-high-voltage lines, hollow or segmented conductors are sometimes used to mitigate the effect and reduce material cost, as the center contributes little to current carrying.

RF Engineering & Antennas: At radio frequencies (MHz to GHz), skin depth is fractions of a millimeter. Antennas and coaxial cables often use a thin silver or gold plating over a cheaper, stronger base metal (like steel or brass) because the current flows only in the plating, achieving optimal performance at lower cost.

High-Speed Electronics (PCBs): In motherboards and communication devices, signal traces operate at multi-gigahertz frequencies. Here, skin depth is microns thin. This influences PCB stack-up design and plating choices, and is a critical factor in signal integrity, causing frequency-dependent attenuation.

Induction Heating & Non-Destructive Testing: The skin effect is exploited intentionally. In induction heating, high-frequency currents are induced in a metal workpiece, and the heating is concentrated in a thin surface layer, allowing for surface hardening. In eddy-current testing, the depth of flaw detection is directly controlled by the test frequency's skin depth.

Common Misconceptions and Points of Attention

First, upon seeing the skin depth formula $\delta = \frac{1}{\sqrt{\pi f \mu_r \mu_0 \sigma}}$, some people hastily conclude, "Higher frequency means shallower depth, so a thinner shield is fine." While this is technically correct, in practice, a thickness of "several $\delta$" is required. For example, for a copper shield at 1GHz ($\delta \approx 2.1 \mu m$), making the thickness $3\delta$ (approx. 6.3μm) will attenuate the current density to about 5% at a depth of $3\delta$ from the surface. However, whether this provides sufficient shielding effectiveness is another matter. In actual case design, considering mechanical strength and manufacturability, thicknesses around 0.1mm to 0.3mm are typically used, which is tens to hundreds of times the skin depth. Do not blindly trust the formula alone; design with actual EMC testing in mind is essential.

Next, a pitfall in simulation settings. While finely meshing the conductor "surface" is correct, making the internal mesh too coarse prevents you from correctly capturing the attenuation gradient. Especially for materials with high permeability (e.g., iron), where the skin depth becomes extremely shallow, you need to set the mesh size of the first surface layer sufficiently finer than $\delta$ (e.g., $\delta/3$ or less). Otherwise, the calculated resistance may be lower than the actual value, risking an underestimation of heat generation.

Finally, handling material constants. In simulators, you input "relative permeability $\mu_r$" and "electrical conductivity $\sigma$" as constants, but these vary with frequency and temperature. For instance, the $\mu_r$ of iron-based materials decreases as frequency increases, and $\sigma$ decreases with rising temperature. For accurate loss estimation, it's crucial to use values based on actual measurements within the target frequency band or manufacturer datasheets. Avoid assuming "copper is fixed at $\mu_r=1$, $\sigma=5.8\times10^7$ S/m".