Skin Effect Simulator — AC Current Surface Concentration
Compute the skin depth delta and the AC/DC resistance ratio of a round conductor in real time from frequency f, electrical conductivity sigma, relative permeability mu_r and wire radius R. A cross-section current-density map and a log-log delta vs frequency plot make the physics of high-frequency resistance increase intuitive.
Parameters
log10 frequency f
log Hz
f = 1.00 MHz
Conductivity sigma
×10⁷ S/m
Relative permeability mu_r
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Wire radius R
mm
Defaults: f = 1 MHz, sigma = 5.96e7 S/m (copper), mu_r = 1, R = 1.00 mm. Vacuum permeability mu_0 = 4 pi × 1e-7 H/m. Uses the high-frequency approximation R_ac/R_dc ~ R/(2 delta), valid for R much larger than delta.
Results
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Skin depth delta
—
delta / R
—
AC/DC resistance ratio
—
Frequency where delta = R
Current density across a round conductor
Circle = round conductor cross-section of radius R. Color = current density |J| proportional to exp(-(R-r)/delta) (red = high, blue = low). Dashed circle = skin depth delta position. Arrows label R and delta. For delta much smaller than R, current is confined near the surface.
Frequency vs skin depth (log-log)
Horizontal axis = log10 frequency from 1 kHz to 1 GHz. Vertical axis = log10 skin depth in micrometers. Orange line = the delta proportional to 1/sqrt(f) relation. Yellow circle = current operating point. Green horizontal line = wire radius R; its intersection is the critical frequency f_R.
Theory & Key Formulas
The skin effect is the concentration of AC current near a conductor surface. It is derived from Maxwell's equations (Faraday's law in particular) and is described by the following formulas.
Skin depth for a good conductor:
$$\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}$$
AC/DC resistance ratio for a round conductor (high-frequency approximation, R much larger than delta):
Critical frequency at which delta equals the wire radius:
$$f_R = \frac{1}{\pi \mu \sigma R^2}$$
$\mu = \mu_0 \mu_r$ is the permeability with $\mu_0 = 4\pi\times10^{-7}$ H/m, $\sigma$ is the electrical conductivity in S/m, $f$ is the frequency in Hz and $R$ is the wire radius in m. $\delta$ is the depth at which the current density falls to 1/e of its surface value, and the skin-effect resistance rise becomes significant above $f_R$.
What is the Skin Effect Simulator?
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My power-engineering book says that at high frequencies even a thick copper wire becomes more resistive. That seems counter-intuitive — a thicker wire should give more area for current and less resistance. Why does it flip at high frequencies?
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Great question. This is the skin effect: as the frequency rises, AC current concentrates near the surface of the conductor. The reason is the magnetic flux inside the conductor. Time-varying current induces eddy currents through Faraday's law, and they push the current outward from the center. The current density decays from the surface as exp(-x/delta), and the characteristic decay length delta = 1/sqrt(pi f mu sigma) is the skin depth. For copper at 1 MHz, delta is only about 65 micrometers, so even a 1 mm radius wire ends up using only a thin outer shell.
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With the defaults of this tool (f = 1 MHz, copper, R = 1 mm) I see R_ac/R_dc ~ 7.67. Is that a normal number for an RF circuit?
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It is exactly typical of RF reality. At AM radio frequencies (around 1 MHz) an AWG 18 copper wire (radius 0.5 mm) gives almost the same 7 to 8 fold loss increase. This is the main culprit that lowers antenna Q, and to improve efficiency we use Litz wire — many fine insulated strands twisted together so each strand radius stays below delta. Shrinking R toward 0.5 mm in this tool reduces R_ac/R_dc; in practice you bundle 100 to 400 such strands in parallel to carry the desired current.
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When I push mu_r up, delta shrinks dramatically. What physical situation does that correspond to?
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Raising mu_r from 1 (copper, non-magnetic) to 1000 (iron) shrinks delta by a factor of about sqrt(1000) ~ 32. This applies when current flows inside iron or electrical steel, which matters in transformer cores and motor laminations. Iron has sigma about 1e7 S/m, lower than copper, but mu_r around 500 to 2000, so even at 50 Hz delta is only 0.5 to 1 mm. That is why cores are stacked from 0.35 mm thick electrical-steel sheets — making the sheet roughly equal to delta minimizes eddy-current loss. Try sigma = 1.0e7, mu_r = 1000 and f = 50 Hz (log F = 1.7) in this tool to verify.
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Does the skin effect matter at 50 or 60 Hz power frequencies? It feels far too low to be relevant.
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It is relevant once the conductor gets thick. A power busbar of radius 10 mm gives f_R = 1/(pi mu sigma R^2) ~ 42 Hz, so the power frequency is already in the delta ~ R regime. Try R = 10 mm and f = 50 Hz in this tool — you get R_ac/R_dc around 1.5 to 2. That is why 154 kV busbars and above use hollow copper/aluminum pipes or ACSR (steel-cored aluminum cables) with fine outer strands. The same physics is absent in HVDC subsea cables (DC gives delta = infinity), so only AC subsea cables suffer from the diameter constraint.
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Modern switching-mode power supplies run PWM up to a few MHz. How are their windings designed?
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Excellent. At 1 MHz copper has delta ~ 65 micrometers, so standard practice is Litz wire built from 50 to 200 AWG 38 strands (diameter ~ 100 micrometers). For 5 to 10 MHz GaN/SiC converters delta shrinks to 20 to 30 micrometers, so engineers move to ultra-fine AWG 44 (50 micrometer) Litz, or stack many PCB copper layers (35 micrometer foil) in parallel. Try f = 5e6 Hz (log F = 6.7) in this tool to check delta and pick strand diameters. The skin effect remains a central design challenge of modern power electronics.
Frequently Asked Questions
Starting from Maxwell's equations (Ampere and Faraday) and the good-conductor approximation sigma much larger than omega epsilon, the wave equation for E inside the conductor gives the solution E(x,t) = E_0 exp(-x/delta) cos(omega t - x/delta), where delta = sqrt(2 / (omega mu sigma)) = 1/sqrt(pi f mu sigma) with omega = 2 pi f. delta is the depth at which both amplitude and phase change by the same characteristic length, and the half-wavelength inside the conductor is 2 pi delta. The skin-layer current distribution is thus not a pure exponential decay but the combination of amplitude exp(-x/delta) and phase cos(x/delta - omega t). This tool visualizes only the amplitude; strictly, the phase information also matters for current direction.
It is accurate only in the high-frequency limit R much greater than delta (roughly R/delta greater than 3). At low frequencies R < delta the current is essentially uniform and R_ac/R_dc approaches 1, so R/(2 delta) would give a value much less than 1, which is unphysical. The rigorous solution uses Kelvin functions ber(x), bei(x), ber'(x), bei'(x): R_ac/R_dc = (kR/2) [ber(kR) bei'(kR) - bei(kR) ber'(kR)] / [ber'(kR)^2 + bei'(kR)^2], with k = sqrt(2)/delta. This tool only displays the high-frequency approximation, so the value is under-predicted in the very low-frequency, thin-wire region delta > R/3. For accurate analysis there, use MATLAB Bessel functions or finite-element software (COMSOL, ANSYS Maxwell) with frequency-domain solvers.
The skin effect is the self-induced concentration of current near the surface of an isolated conductor, while the proximity effect is the redistribution of current caused by the magnetic field of nearby AC conductors. Both increase loss with frequency, but have different remedies: skin-effect mitigation means using strands thinner than delta (Litz wire), while proximity mitigation means optimizing conductor spacing or reducing winding-layer count. Both occur simultaneously in transformer windings, so Dowell's 1966 one-dimensional layer analysis is commonly used to compute frequency-dependent winding resistance. Simply bundling untwisted strands actually worsens the proximity effect even as it suppresses the skin effect; only a properly twisted Litz pattern suppresses both.
The skin effect connects directly to other electromagnetic applications. (1) EM shielding: a conducting sheet of thickness t in the range 3 to 5 delta attenuates incident magnetic waves by exp(-t/delta) ~ 5% or less. Low-frequency 50 Hz magnetic shielding uses thick iron or permalloy (delta = 1 to 10 mm), while GHz RF shielding requires only 35 micrometer copper foil (delta ~ 2 micrometers). (2) Induction heating (IH cookers, surface hardening): the workpiece is heated by eddy currents within the delta layer. An iron pan at 30 kHz has delta ~ 0.05 mm and heats only the surface — the same principle as steel surface hardening. Try sigma = 1e7, mu_r = 500, f = 30 kHz in this tool to estimate the IH penetration depth. Electromagnetic induction itself is an application of the skin effect.
Real-World Applications
Power-system busbars and transmission lines: Even at 50/60 Hz power frequency, thick copper or aluminum conductors with radius above about 10 mm reach delta ~ R and the skin-effect resistance rise becomes noticeable. For 154 kV and higher busbars, hollow-pipe conductors or ACSR stranded designs preserve effective cross-section while suppressing the skin effect. HVDC subsea cables, by contrast, suffer no skin effect (delta is infinite at DC) and reach the same current capacity in much thinner conductors — a key reason for their adoption in large European and Asian transmission projects. Try R = 10 mm and f = 50 Hz in this tool to confirm that the power frequency is already on the threshold.
Switching-mode power supplies and power electronics: Modern GaN and SiC devices push switching frequency from 100 kHz toward 1 to 10 MHz. At 1 MHz copper has a 65 micrometer skin depth, so transformer and inductor windings are typically made of Litz wire with 25 to 50 micrometer strand diameter (AWG 40 to 45). Above 5 MHz, parallel multi-layer PCB-foil windings (35 micrometer thickness) or copper braids are also used. Sweep f from 1 to 10 MHz in this tool to see why power electronics depend heavily on ultra-fine Litz. This directly impacts EV fast-charging and solar-PV inverter efficiency.
RF and microwave engineering: In the UHF to microwave range (100 MHz to tens of GHz), copper's skin depth shrinks to 1 to 10 micrometers, so coaxial cables, waveguides and connectors are polished and silver-plated (sigma_Ag = 6.30e7 S/m) to minimize effective resistance. Surface roughness larger than delta lengthens the current path and increases insertion loss, so Ra < 0.2 micrometer ultra-fine polishing is required for 5G base-station antennas and satellite high-band filters. Try sigma = 6.30e7 in this tool near the highest frequencies to see the micrometer-scale delta that drives these surface-finish specifications.
Induction heating and IH cookers: IH cookers use 20 to 100 kHz magnetic fields to induce eddy currents in iron pans, dissipating energy as heat in the skin layer. For iron (sigma ~ 1e7 S/m, mu_r ~ 500) at 30 kHz, delta ~ 50 micrometers, so heating concentrates in the thin pan base and achieves over 90% thermal efficiency. The same principle drives industrial surface hardening (induction hardening) at 10 to 500 kHz to harden only the surface of automotive shafts and gear teeth. Set sigma = 1.0e7, mu_r = 500, f = 50 kHz in this tool to see how the induction-heating penetration depth is determined.
Common Pitfalls and Notes
The most common misconception is to assume that a conductor with low DC resistance must also have low AC resistance. In reality, skin-effect resistance grows as the square root of frequency, and in the R much greater than delta regime it routinely reaches a factor of several to several tens above the DC value. At 50 Hz a 20 mm radius copper busbar already has R_ac/R_dc ~ 1.5, and a 1 mm radius copper wire at 1 MHz reaches 7.7. Always compute AC resistance at the expected frequency and consider Litz wire, hollow conductors or silver plating where needed. The DC intuition that "thicker is always safer" can mislead at high AC frequencies.
The second pitfall is to confuse skin depth with semiconductor doping or oxide thicknesses. The skin depth is the characteristic length of electromagnetic-wave penetration in a conductor and is independent of semiconductor device dimensions. Power-MOSFET n+ drain layers (10 to 100 nm) or gate oxides (10 to 100 nm) are far thinner than any practical delta, so internal skin effect is negligible there. On the other hand, bonding wires and lead frames (100 micrometers to 1 mm thick) carry significant skin-effect resistance above 100 MHz, appearing as frequency-dependent package parasitics. Keep "device dimension" and "conductor dimension" conceptually separate.
The third pitfall is the over-confident belief that Litz wire completely eliminates the skin effect. Litz works only when each strand radius is well below delta at the design frequency. AWG 38 (100 micrometer diameter, 50 micrometer radius) strands are fine at 1 MHz (delta = 65 micrometers > 50 micrometers) but fall short at 5 MHz (delta = 29 micrometers < 50 micrometers). Worse, an inappropriate twist pitch can cause resonance between twist length and wavelength, leading to anomalous internal-strand parallel-series switching losses. Always specify the maximum operating frequency and co-optimize strand diameter and twist pitch.