Free online Skin Depth Calculator – Skin Effect & AC Resistance

Step 1

Skin depth

Pick a conductor or enter its conductivity and permeability. The calculator solves the full lossy-medium equation, so it also covers seawater, soil and metals far above the good-conductor range.

Conductor

Material · typical bulk values at 20 °C; replace with your own

Conductivity at 20 °C

Temperature coefficient

1/K

Conductor temperature

°C

Relative permeability μr

Relative permittivity εr

Frequency

Common:

Skin depth against frequency log–log

The dashed line marks your frequency. Media with low conductivity (seawater, soil) level off once σ/ωε drops below 1 and the medium behaves as a lossy dielectric.

Surface resistance against frequency Ω per square, log–log

For a good conductor R_s = 1/(σδ) = √(πfμ/σ) and rises by √10 per decade. It is the resistance of a square of thick conductor surface, whatever the size of the square.

Inside the conductor

Current in depth

Inside a thick conductor the current density decays as e^−z/δ and lags in phase by z/δ radians. Enter a conductor or foil thickness to see how much of the current and the loss it holds.

Conductor or foil thickness

µm

Thickness preset

Current flows on

Current density against depth relative to the surface

The in-phase part shows the current at one instant: it changes sign below about 1.6δ, so deeper layers carry current flowing backwards at that moment. Hover for the phase.

Foil resistance against frequency one face, Ω per square

Thickness needed share of the loss carried within a given depth

How to read it. The share of the loss within depth t of a thick conductor is 1 − e^−2t/δ; the share of the current magnitude is 1 − e^−t/δ. A layer 3δ thick carries 99.75 % of the loss, so the material under it hardly matters. Thinner layers make the material underneath, or the lack of it, count.

Other conductors

Compare materials

Every conductor in the library at your frequency and temperature, sorted by surface resistance. The bar shows R_s relative to copper; the highlighted row is your material.

Materials at

Conductivities are typical bulk values; alloys, platings and thin films vary with composition and process, often by 10–30 %. Ferromagnetic permeabilities fall with frequency and field strength, so their rows are indicative only.

Step 3

Wire & trace

AC resistance of a round wire, from the exact Bessel-function solution, and of a PCB trace with copper roughness. The conductor, temperature and frequency are taken from step 1.

Round wire isolated, straight

Wire size

Diameter

Length

AC resistance and internal inductance relative to DC, against frequency

Current density across the wire

Magnitude of the current density relative to the surface. The dashed circle sits one skin depth below the surface.

Printed circuit boards

PCB trace & roughness

AC resistance of a rectangular trace, with the increase caused by copper surface roughness and the conductor loss it adds to a line of the impedance you enter.

Trace width

Copper weight

Thickness

µm

Length

Line impedance Z₀

Copper foil · typical RMS roughness; check the laminate datasheet

RMS roughness Rq

µm

Roughness model

Trace resistance against frequency whole length, log–log

Current is assumed to flow in a skin layer around the whole perimeter. On a microstrip it crowds onto the face toward the ground plane and the edges, so the real resistance is higher; treat this as a lower bound and use a field solver for the line.

Roughness correction factor K against frequency

Both models saturate at K = 2 once δ is much smaller than the roughness. Measured loss on very rough foil can exceed this; the Huray snowball model, which needs nodule data from the foil maker, covers that case.

About this skin depth calculator

This free calculator from NextAntennas works out how deep alternating current penetrates a conductor and what that does to its resistance. It computes the skin depth, surface resistance and surface impedance of 17 conductors and lossy media from 1 Hz to 1 THz, with temperature correction and editable permeability, and applies them to the cases RF and power engineers meet every day: plated finishes, round wires and PCB traces.

How to use it

Skin depth: pick a material or enter σ, μr and εr, set the temperature and frequency. Skin depth, surface resistance, wavelength in the conductor and the conductivity in % IACS are shown and plotted from 1 Hz to 1 THz, with the decay and phase of the current in depth, the thickness needed to carry a given share of the loss, the resistance of thin foils and a comparison of every conductor at your frequency.
Plating & finishes: the exact surface resistance of ENIG, immersion silver and tin, HASL, silver-plated brass, gold-over-nickel connector plating or your own stack, and where the loss goes.
Wire & trace: round-wire AC resistance and internal inductance from the Kelvin-function solution for AWG or metric sizes, and PCB trace resistance with Hammerstad or Groisse roughness correction and the resulting conductor loss in dB.

Skin depth equations

For a good conductor δ = 1/√(πfμσ) and R_s = 1/(σδ) = √(πfμ/σ). In general δ = 1/Re(γ) with γ = √(jωμ(σ + jωε)), and the surface impedance is Z_s = √(jωμ/(σ + jωε)). For a round wire of radius a, Rac/Rdc follows from the Kelvin functions of x = √2·a/δ and tends to a/(2δ) + 1/4 at high frequency.

Theory, data and FAQ

The derivations are in Theory Behind the Equations, the conductivities and their sources in Material Data, and answers to common questions in the FAQ below.

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Engineering Theory

Theory Behind the Equations

How the calculator works: from Maxwell's equations in a conductor to skin depth and surface impedance, and from there to layered finishes, round wires, traces and rough copper.

1. What Is the Skin Effect?

An alternating current sets up a changing magnetic field inside its own conductor. That field induces eddy currents that oppose the current in the centre and reinforce it near the surface, so the current crowds into a thin layer at the skin. The higher the frequency, the conductivity and the permeability, the thinner the layer.

The same physics describes a wave arriving at a metal surface: the field that enters the metal decays exponentially, and the energy it carries is turned into heat. The skin depth δ is the depth at which the amplitude has fallen to 1/e, about 37 % of its surface value.

2. Skin Depth of a Good Conductor

When the conduction current is much larger than the displacement current (σ ≫ ωε, true for all metals well into the terahertz range), the diffusion equation gives:

Skin depth, good conductor

δ = \frac{1}{\sqrt{π f μ_{0} μ_{r} σ}} = \sqrt{\frac{2}{ω μ σ}}

f in Hz · μ₀ = 4π × 10⁻⁷ H/m · σ in S/m · copper: δ ≈ 66 µm / √(f in MHz)

Halving δ takes four times the frequency. Inside the conductor the wavelength is λ_c = 2πδ and the phase velocity v_p = ωδ, a few metres per second for copper at power frequencies.

3. Any Lossy Medium

For seawater, soil or semiconductors the displacement current cannot be ignored. The calculator then uses the complex propagation constant directly:

Propagation constant and exact skin depth

γ = \sqrt{j ω μ (σ + j ω ε)}

δ = \frac{1}{Re (γ)} = \frac{1}{ω \sqrt{\frac{μ ε}{2}} {[\sqrt{1 + {(\frac{σ}{ω ε})}^{2}} - 1]}^{1 / 2}}

ε = ε₀ε_r · reduces to section 2 when σ/ωε ≫ 1

The ratio σ/ωε decides the regime. Above about 100 the good-conductor formula is within 1 %; near 1 the medium is a lossy dielectric and δ stops falling with frequency, tending to (2/σ)√(ε/μ).

4. Surface Impedance and Surface Resistance

The ratio of tangential E to tangential H at the surface of a thick conductor is its surface impedance. Its real part is the surface resistance, which multiplies the square of the surface current to give the loss per unit area:

Surface impedance

Z_{s} = \sqrt{\frac{j ω μ}{σ + j ω ε}} \approx \frac{1 + j}{σ δ}

R_{s} = \frac{1}{σ δ} = \sqrt{\frac{π f μ}{σ}}

P_{loss} = \frac{1}{2} R_{s} {| J_{s} |}^{2}

Ω per square · J_s = surface current density in A/m (peak) · copper at 1 GHz: R_s ≈ 8.25 mΩ

R_s is the DC resistance of a slab of the same conductor exactly one skin depth thick. The surface reactance X_s equals R_s for a good conductor and adds a small internal inductance.

5. Current Density in Depth

In a thick plane conductor the current density decays and rotates in phase with depth z:

Current distribution

J (z) = J_{0} e^{- z / δ} e^{- j z / δ}

amplitude falls 8.69 dB per skin depth · phase lags 57.3° per skin depth

A foil of thickness t carrying current on one face has Z_s = Z coth(γt); with current on both faces, as in an isolated strip, each face sees a half-thickness with Z coth(γt/2). When t ≪ δ both reduce to the DC sheet resistance 1/(σt).

6. Plated and Layered Conductors

Each layer acts as a short lossy transmission line terminated by the surface impedance of what lies beneath. Starting at the thick base Z_b and working up to the surface:

Surface impedance of a layer over a load

Z_{in} = Z_{k} \frac{Z_{L} + Z_{k} \tanh (γ_{k} t_{k})}{Z_{k} + Z_{L} \tanh (γ_{k} t_{k})}

Z_k, γ_k = intrinsic impedance and propagation constant of layer k · Z_L = surface impedance below it

The magnetic field falls through each layer by 1/[cosh(γt) + (Z_L/Z_k)sinh(γt)], and the loss in a layer is the difference between the power entering its top and leaving its bottom, Re(Z)|H|²/2 at each interface. This is how the share of loss by layer is found. A poor conductor under a good one, such as nickel under gold in ENIG, raises the loss most when its thickness is near one or two of its own skin depths.

7. Round Wire — the Exact Solution

In a round wire of radius a the current density follows a Bessel function of complex argument. Writing it with the Kelvin functions ber and bei of x = √2·a/δ:

AC resistance and internal inductance

\frac{R_{ac}}{R_{dc}} = \frac{x}{2} \frac{ber x {bei}^{'} x - bei x {ber}^{'} x}{{({ber}^{'} x)}^{2} + {({bei}^{'} x)}^{2}}

\frac{ω L_{i}}{R_{dc}} = \frac{x}{2} \frac{ber x {ber}^{'} x + bei x {bei}^{'} x}{{({ber}^{'} x)}^{2} + {({bei}^{'} x)}^{2}}

R_dc = ℓ/(σπa²) · L_i,dc = μℓ/(8π), 50 nH/m for a non-magnetic wire

The calculator sums the power series of ber and bei up to x = 20 and uses the asymptotic forms beyond, where R_ac/R_dc ≈ ξ/2 + 1/4 + 3/(32ξ) and ωL_i/R_dc ≈ ξ/2 − 3/(32ξ), with ξ = a/δ. At low frequency R_ac/R_dc ≈ 1 + x⁴/192, so the resistance doubles at a ≈ 3.45δ.

8. Rectangular Traces

There is no closed-form solution for a rectangle. The calculator uses the common perimeter model, in which the current fills a band one skin depth deep around the edge:

Effective area of a w × t trace

A_{eff} = w t - \max (0, w - 2 δ) \cdot \max (0, t - 2 δ) R_{ac} = \frac{ℓ}{σ A_{eff}}

tends to R_sℓ/(2(w + t)) at high frequency and to R_dc at low frequency

The conductor attenuation of a line of impedance Z₀ is then α_c ≈ R′/(2Z₀) Np/m, or 8.686·R′/(2Z₀) dB/m, with R′ the resistance per metre. For a microstrip the current also flows in the ground plane and crowds toward it, so this is a lower bound.

9. Surface Roughness

Copper foil is roughened to bond to the laminate. When δ approaches the RMS roughness R_q the current follows the teeth and the path lengthens. Two widely used correction factors multiply the smooth-conductor resistance:

Roughness correction

K_{H} = 1 + \frac{2}{π} \arctan [1.4 {(\frac{R_{q}}{δ})}^{2}]

K_{G} = 1 + \exp [- {(\frac{δ}{2 R_{q}})}^{1.6}]

Hammerstad–Jensen (1980) · Groisse et al. (1996) · both tend to 2

Laminate makers quote R_z or R_a more often than R_q; for typical foil profiles R_q is roughly 1.1–1.25 × R_a and well below R_z.

10. Temperature and Magnetic Metals

Conductivity is corrected linearly from its 20 °C value, σ(T) = σ₂₀/[1 + α(T − 20)], so δ and R_s each change by about half of α per kelvin. Ferromagnetic metals (iron, nickel, carbon steel) have large μr at low frequency, which shrinks δ and raises R_s by √μr. Their permeability falls with frequency, collapsing toward 1 above ferromagnetic resonance in the gigahertz range, and depends on the field level and on DC bias, so enter an effective μr for your case.

11. Validity and Limitations

The plane and wire results assume an isolated conductor. Proximity effect from nearby conductors, coil turns and ground planes raises the resistance further.
Bulk conductivity is used. Thin films, plated and sputtered layers often have 10–50 % lower conductivity than the bulk metal.
At very low temperature and very high frequency the anomalous skin effect limits how far R_s falls, as the electron mean free path exceeds δ.
Ferromagnetic results depend strongly on the μr you enter.

References

Ramo, S., Whinnery, J.R. and Van Duzer, T., Fields and Waves in Communication Electronics, 3rd ed., Wiley, 1994 (skin effect, Kelvin-function wire solution).
Pozar, D.M., Microwave Engineering, 4th ed., Wiley, 2012 (surface resistance, conductor loss).
Wheeler, H.A., “Formulas for the Skin Effect,” Proc. IRE, vol. 30, no. 9, pp. 412–424, 1942.
Hammerstad, E. and Jensen, Ø., “Accurate Models for Microstrip Computer-Aided Design,” IEEE MTT-S Int. Microwave Symp. Digest, pp. 407–409, 1980.
Groisse, S. et al., “Influence of the line characteristics on the transmission properties of microstrip lines,” Proc. 26th European Microwave Conf., 1996.
Huray, P.G. et al., “Impact of Copper Surface Texture on Loss: A Model that Works,” DesignCon, 2010.
Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, §9.9–9.10, Kelvin functions, 1964.
IEC 60028, International standard of resistance for copper (100 % IACS = 58.0 MS/m at 20 °C).
CRC Handbook of Chemistry and Physics, Electrical Resistivity of Pure Metals.

Reference Data

Material Data

The conductivity, temperature coefficient and permeability the calculator uses for each material, with the skin depth and surface resistance at common frequencies. Check your supplier's data for alloys, platings and films.

Conductor Library

Table D1. Typical bulk properties at 20 °C used in this calculator
Material	σ (S/m)	ρ (µΩ·cm)	% IACS	α (1/K)	μr	εr	R_s at 1 GHz
Silver	6.30×10⁷	1.587	109	0.0038	1	1	7.92 mΩ
Copper, annealed (100 % IACS)	5.80×10⁷	1.724	100	0.00393	1	1	8.25 mΩ
Gold	4.10×10⁷	2.439	70.7	0.0034	1	1	9.81 mΩ
Aluminium	3.77×10⁷	2.653	65	0.004	1	1	10.2 mΩ
Tungsten	1.79×10⁷	5.587	30.9	0.0045	1	1	14.9 mΩ
Brass, C26000 70/30	1.60×10⁷	6.25	27.6	0.0015	1	1	15.7 mΩ
Nickel, electrolytic (μr ≈ 100 at low f)	1.43×10⁷	6.993	24.7	0.006	100	1	166 mΩ
Tin	9.17×10⁶	10.91	15.8	0.0042	1	1	20.7 mΩ
Carbon steel, 1010 (μr ≈ 100 at low f)	6.99×10⁶	14.31	12.1	0.005	100	1	238 mΩ
Solder, Sn63Pb37	6.90×10⁶	14.49	11.9	0.004	1	1	23.9 mΩ
Titanium	2.38×10⁶	42.02	4.1	0.0035	1	1	40.7 mΩ
Electroless nickel, Ni-P (ENIG)	1.45×10⁶	68.97	2.5	0.001	1	1	52.2 mΩ
Stainless steel, 304	1.39×10⁶	71.94	2.4	0.001	1	1	53.3 mΩ
Graphite (in-plane, typical)	1.00×10⁵	1000	0.172	-0.0005	1	1	199 mΩ
Seawater, 35 ‰	4.80×10⁰	2.08e7	—	-0.02	1	81	31.8 Ω
Fresh water, river or lake	1.00e-2	1.00e10	—	-0.02	1	80	42.1 Ω
Wet soil, typical	1.00e-2	1.00e10	—	0	1	20	84.2 Ω

Pure metals from the CRC Handbook resistivity tables; copper per IACS (IEC 60028); brass, solder, steel and stainless steel from ASM and alloy-maker data; electroless nickel-phosphorus for mid-phosphorus deposits; seawater at 35 ‰ salinity and about 20 °C. Ferromagnetic μr values are representative low-frequency figures.

Skin Depth at Common Frequencies

Table D2. Skin depth δ at 20 °C
Material	50 Hz	1 kHz	1 MHz	100 MHz	1 GHz	10 GHz	100 GHz
Silver	8.97 mm	2.01 mm	63.4 µm	6.34 µm	2.01 µm	634 nm	201 nm
Copper	9.35 mm	2.09 mm	66.1 µm	6.61 µm	2.09 µm	661 nm	209 nm
Gold	11.1 mm	2.49 mm	78.6 µm	7.86 µm	2.49 µm	786 nm	249 nm
Aluminium	11.6 mm	2.59 mm	82 µm	8.2 µm	2.59 µm	820 nm	259 nm
Tungsten	16.8 mm	3.76 mm	119 µm	11.9 µm	3.76 µm	1.19 µm	376 nm
Brass	17.8 mm	3.98 mm	126 µm	12.6 µm	3.98 µm	1.26 µm	398 nm
Nickel	1.88 mm	421 µm	13.3 µm	1.33 µm	421 nm	133 nm	42.1 nm
Tin	23.5 mm	5.26 mm	166 µm	16.6 µm	5.26 µm	1.66 µm	526 nm
Carbon steel	2.69 mm	602 µm	19 µm	1.9 µm	602 nm	190 nm	60.2 nm
Sn63Pb37	27.1 mm	6.06 mm	192 µm	19.2 µm	6.06 µm	1.92 µm	606 nm
Titanium	46.1 mm	10.3 mm	326 µm	32.6 µm	10.3 µm	3.26 µm	1.03 µm
Ni-P	59.1 mm	13.2 mm	418 µm	41.8 µm	13.2 µm	4.18 µm	1.32 µm
SS 304	60.4 mm	13.5 mm	427 µm	42.7 µm	13.5 µm	4.27 µm	1.35 µm
Graphite	225 mm	50.3 mm	1.59 mm	159 µm	50.3 µm	15.9 µm	5.03 µm
Seawater	32.5 m	7.26 m	230 mm	24.1 mm	11 mm	9.97 mm	9.95 mm
Fresh water	712 m	159 m	6.24 m	4.75 m	4.75 m	4.75 m	4.75 m
Wet soil	712 m	159 m	5.32 m	2.38 m	2.37 m	2.37 m	2.37 m

FAQ

Frequently Asked Questions

Common questions about skin depth, surface resistance, plating and AC resistance.

What is skin depth?

Skin depth δ is the depth below the surface of a conductor at which an alternating current density, or a field entering the conductor, has fallen to 1/e (about 37 %) of its value at the surface. Most of the current flows in the first few skin depths, so δ sets the effective thickness of a conductor at high frequency.

How do you calculate skin depth?

For a good conductor δ = 1/√(π f μ σ), where f is the frequency in hertz, μ = μ0·μr the permeability and σ the conductivity in S/m. For copper this is about 66 µm divided by √(f in MHz): 2.09 µm at 1 GHz and 9.3 mm at 50 Hz. For media that are not good conductors, such as seawater at high frequency, the calculator uses the exact attenuation constant of a lossy medium.

What is the skin depth of copper at 1 GHz?

About 2.09 µm for annealed copper (σ = 5.80 × 10⁷ S/m) at 20 °C. At 10 GHz it is about 0.66 µm, at 100 MHz about 6.6 µm, at 1 MHz about 66 µm and at 60 Hz about 8.5 mm.

What is surface resistance?

Surface resistance Rs is the resistance of a square of conductor surface to a current that flows in the skin layer, in ohms per square. For a good conductor Rs = 1/(σδ) = √(π f μ / σ). It grows with the square root of frequency and sets the conductor loss of waveguides, cavities, transmission lines and antennas.

How thick should plating be for RF?

A plating three to five skin depths thick carries essentially all the current, so the base metal underneath no longer matters: 3δ holds 99.75 % of the loss and 5δ more than 99.99 %. Silver at 1 GHz has δ ≈ 2 µm, so about 6 to 10 µm of silver makes a brass part behave like solid silver. Step 2 computes the exact surface resistance of the stack.

Why does ENIG increase conductor loss?

ENIG puts a 3 to 6 µm layer of electroless nickel-phosphorus under a thin gold flash. Above a few hundred megahertz the skin depth becomes comparable to the nickel thickness, so the current flows mostly in the nickel, whose conductivity is roughly forty times lower than copper's. The surface resistance of the trace edges and top rises accordingly.

How do you calculate the AC resistance of a wire?

The exact solution for a round wire uses Bessel functions of complex argument, equivalently Kelvin functions ber and bei of x = √2·a/δ. When the radius a is much larger than δ, Rac/Rdc ≈ a/(2δ) + 1/4. Step 3 uses the exact series up to x = 20 and the asymptotic form beyond.

When does skin effect start to matter in a wire?

Rac rises 1 % above Rdc when the radius is about 0.83 skin depths and doubles when the radius is about 3.45 skin depths. For a 1 mm copper wire that is near 12 kHz and 210 kHz. Step 3 reports the frequency at which Rac = 2·Rdc for your wire.

How does surface roughness affect conductor loss?

When the skin depth becomes comparable to the RMS height of the copper roughness, the current follows the rough profile and travels a longer path, raising the resistance. The Hammerstad model caps the increase at a factor of 2; the Groisse model also saturates at 2. Measured losses on very rough foils can exceed this, for which the Huray model is used.

Does skin depth depend on temperature?

Yes, through the conductivity. For copper the resistivity rises about 0.39 % per kelvin, so δ grows by about 0.2 % per kelvin and Rs by the same amount. Set the temperature in step 1 and every result uses the corrected conductivity.

Why is the skin depth of steel and nickel so small?

Skin depth falls as 1/√μr. Ferromagnetic metals have μr of tens to thousands at low frequency, so their skin depth is many times smaller than copper's despite lower conductivity, and their surface resistance is much higher. Their permeability falls with frequency and with field strength, so enter an effective μr for your case.

What is the skin depth of seawater?

With σ ≈ 4.8 S/m, seawater has a skin depth of about 7 m at 1 kHz, 23 cm at 1 MHz and a few centimetres above 100 MHz, where it stops behaving as a good conductor. That is why submarines communicate at very low frequencies. The calculator uses the exact lossy-medium equation for such media.

What is Litz wire and when does it help?

Litz wire is made of many insulated strands, each thinner than a skin depth, twisted so that every strand takes every position in the bundle. It reduces both skin and proximity effects up to a few megahertz. Above that, strand capacitance and imperfect transposition limit the benefit.

Does the calculator include proximity effect?

No. The wire and trace results are for an isolated conductor. Nearby conductors, such as the turns of a coil or the ground plane under a microstrip, push the current to one side and raise the resistance further. Use a field solver or the microstrip calculator for those cases.