Skin depth calculator: skin effect and AC resistance

The skin depth of copper and other metals at a frequency, from their resistivity and permeability, and the AC resistance it gives a round wire or a PCB trace. The wire uses Kelvin's exact solution with Bessel functions, as Rosa and Grover's 1916 Bureau of Standards paper tabulates it, so it holds from DC to the point where the current is a thin skin, and shows the common approximations beside it. The trace uses the surface resistance ρ/δ on one face or two, and reports where that model stops.

Fig 1 — a 2.50 mm copper wire at 500 kHz: δ = 92.2 µm, 7.4 % of the radius. Left, the magnitude of the current density across the section, relative to the surface, from Kelvin's solution; dashed, the circle one δ below the surface. Below, the same against depth: it falls to 1/e near one δ only when δ is small beside r. Right, R_ac/R_dc against frequency: exact (solid), SLUP197's uniform shell of depth δ (thin) and Rayleigh's limit r/2δ (dashed). At this frequency the exact ratio is 7.036.

Conductor

Metal

Relative permeability μr

Frequency f (kHz)

Frequency unit

Wire diameter d (mm)

Wire length l (m)

Skin depth δ = √(ρ / (π·μ0·μr·f)): 92.2 µm
AC to DC resistance R_ac/R_dc, exact (Kelvin): 7.0356
Resistance per metre, DC · at 500 kHz: 3.42 mΩ/m · 24.1 mΩ/m
Resistance of 2.00 m, DC · at 500 kHz: 6.84 mΩ · 48.1 mΩ
Kelvin's argument x = √2·r/δ · δ as a share of the radius: 19.173 · 7.38 %
Rayleigh's limit r/(2δ) · uniform shell r²/(2rδ − δ²): 6.7787 · 7.0383
Internal inductance per metre, DC · at f: 50.0 nH/m · 7.37 nH/m
Frequency where δ equals the radius: 2.72 kHz
Frequency where R_ac is 10 % above R_dc: 6.21 kHz
Surface resistance ρ/δ: 182 µΩ per square
Resistivity ρ used: 1.6780 µΩ·cm, Matula 1979, Table 2, 293 K

How this is calculated

Standard: TI SLUP197 (Dixon), Eddy Current Losses in Transformer Windings and Circuit Wiring, Eq 5; Rosa and Grover, NBS Scientific Paper 169 (revised 1916), section 10, Eqs 143–152 and Table XXII; ADI Basic Linear Design ch. 12, p. 12.34; Matula, JPCRD 8, 1147 (1979) and Desai, JPCRD 13, 1131 (1984) for resistivity; BIPM SI Brochure, 9th edition, for μ0

$\delta = \sqrt{\frac{\rho}{\pi \mu_0 \mu_r f}}, \qquad \mu_0 = 4\pi \times 10^{-7}\ \text{H/m}$: SLUP197 Eq 5, "D_PEN = [ρ/(πμf)]^½ m": "the distance from the surface to where the current density is 1/e times the surface current density". It is "accurate for a flat conductor surface, or when the radius of curvature is much greater than the penetration depth". For pure copper at 20 °C this is 6.52/√f cm. The SI Brochure (p. 128): μ0 was exactly 4π × 10⁻⁷ H/m before 2019 and equal to it "with a relative standard uncertainty of 2.3 × 10⁻¹⁰" when the ampere was redefined.
$x = 2\rho\sqrt{\frac{\pi p \mu}{\sigma}} \;\;\text{(cgs)} \quad\Longrightarrow\quad x = r\sqrt{\frac{\omega \mu_0 \mu_r}{\rho}} = \frac{\sqrt{2}\,r}{\delta}$: Rosa and Grover's argument (p. 174), with the paper's ρ the radius, σ the specific resistance and p = 2πf. In SI it is √2 times the radius over the skin depth (derived). For their copper, 1.721 µΩ·cm, the paper writes x = 0.2142·ρ·√f with ρ in cm (p. 179); the calculator's own value of that constant is 0.2142.
$\frac{R'}{R} = \frac{x}{2}\,\frac{W}{Y}, \qquad W = \operatorname{ber} x \operatorname{bei}' x - \operatorname{bei} x \operatorname{ber}' x, \qquad Y = (\operatorname{ber}' x)^2 + (\operatorname{bei}' x)^2$: Kelvin's solution for a straight round wire, Rosa and Grover Eq 143 and 144a: the ratio of the resistance at frequency f to the DC resistance. ber and bei are "the real and imaginary parts of the ordinary Bessel function of order zero, J0, having for its argument xi√i" (Eq 148: ber x = 1 − x⁴/(2²4²) + x⁸/(2²4²6²8²) − …, bei x = x²/2² − x⁶/(2²4²6²) + …).
$\frac{R'}{R} = \frac{\sum_{n \ge 0} a_n u^n}{\sum_{n \ge 0} a_n u^n/(n+1)}, \qquad \frac{4Z}{xY} = \frac{\sum_{n \ge 0} a_n u^n/(n+1)^2}{\sum_{n \ge 0} a_n u^n/(n+1)}, \qquad a_n = \frac{1}{(n!)^2\,(2n+1)!}, \quad u = \left(\frac{x}{2}\right)^4$: How the calculator evaluates it: Rosa and Grover's Eq 149, their extension of Russell's series for W, Y and Z, whose terms are all positive, so nothing cancels. The coefficients are the printed ones (6/(3!)², 30/(5!)², 140/(7!)², … for W). Above x = 40 it uses Savidge's asymptotic forms, Eq 150, which the paper says "give results to one in one hundred thousand for x ≥ 10". The tests reproduce Table XXII from x = 0.5 to 100 to five decimals and the paper's Examples 82 and 83.
$L_{int} = \frac{\mu_0 \mu_r}{8\pi}\cdot\frac{4Z}{xY}, \qquad Z = \operatorname{ber} x \operatorname{ber}' x + \operatorname{bei} x \operatorname{bei}' x$: The internal inductance of the wire per metre: the μ/4 term of Rosa and Grover's Eq 144, which at DC is μ0·μr/8π, 50.0 nH/m for a non-magnetic wire, and falls as the current leaves the inside of the wire.
$\left(\frac{R'}{R}\right)_{x \to \infty} = \frac{x}{2\sqrt{2}} = \frac{r}{2\delta}, \qquad \frac{R_{ac}}{R_{dc}} \approx \frac{r^2}{2r\delta - \delta^2} \;\; (\delta < r)$: Left, Rayleigh's limit, Rosa and Grover Eq 152: "For x = 70 the error is about 1 per cent". Right, SLUP197's uniform shell, "as if the current density were constant from the surface to the penetration depth, then went abruptly to zero", applied to a round wire as an annulus of depth δ (derived). Both are shown for comparison; neither is used for the result.
$R_s = \frac{\rho}{\delta}, \qquad R'_{trace} = \max\!\left(\frac{R_s}{n\,w},\; \frac{\rho}{w\,t}\right), \quad n = 1 \text{ or } 2$: ADI Basic Linear Design ch. 12, p. 12.34: "the resistance for copper is 2.6 x 10⁻⁷ √f Ω/square ... This formula is invalid if the skin thickness is greater than the conductor thickness", and current "generally flows in both sides of the PC foil (this is not necessarily the case in microstrip lines ...), so the resistance per square of PC foil may be half the above value". ADI's 6.61/√f cm and 2.6 × 10⁻⁷ are ρ = 1.724 µΩ·cm in Eq 5: 6.608/√f cm and 2.609 × 10⁻⁷. n = 1 for a microstrip over its plane, 2 for a foil with no plane nearby; the floor at DC is where the formula stops, not a model of the transition.

Assumptions

A sinusoidal current at one frequency. SLUP197: "the concept of skin depth has no meaning in the time domain". For a switching waveform, run each harmonic.
An isolated conductor with its return far away: no proximity effect. Neighbouring conductors, windings and close return paths move the current to the facing surfaces and raise the resistance further.
A straight round wire (Kelvin) or a flat trace much wider than it is thick (ADI). Trace edges and corners are not modelled, and surfaces are taken as smooth.
Resistivity at 20 °C unless a custom value is entered, and a constant, uniform μr. Resistivity rises with temperature. For a magnetic conductor μr is whatever is entered, as Rosa and Grover assumed 100 for iron.
Pure annealed copper, 1.678 µΩ·cm (Matula), is the default; the site's wire and via calculators use the same value rounded to 1.68. The IACS standard, 1.7241 µΩ·cm, is 2.7 % higher and is offered separately.

What skin depth is

A direct current spreads evenly across a conductor. An alternating one does not: it crowds towards the surface, and the faster it alternates the thinner the layer it uses. TI's SLUP197, Lloyd Dixon's seminar paper on eddy current losses in transformer windings, gives the mechanism in one paragraph. The changing current changes the flux inside the wire, and "the changing flux induces a voltage loop, or eddy" which "reinforces the main current flow at the surface, but opposes it toward the center of the wire. The result is that as frequency rises, current density increases at the conductor surface and decreases toward zero at the center."

The skin depth, which SLUP197 calls the penetration depth, is the measure of that layer: "the distance from the surface to where the current density is 1/e times the surface current density (e is the natural log base)". Its Equation 5 gives it as δ = √(ρ/(πμf)), with ρ the resistivity, f the frequency and μ the permeability, μ₀·μ_r. The value of μ₀ is 4π × 10⁻⁷ H/m: exact under the ampere's definition until 2019, and since then a measured quantity that the SI Brochure records as equal to it "with a relative standard uncertainty of 2.3 × 10⁻¹⁰". So δ grows with the square root of the resistivity and shrinks with the square root of the frequency and of the permeability. Four times the frequency halves the skin depth; a metal with twice the resistivity has a skin depth √2 times larger.

SLUP197 then makes the observation that turns δ into a resistance: "Although the current density tails off exponentially from the surface, the high frequency resistance (and loss) is the same as if the current density were constant from the surface to the penetration depth, then went abruptly to zero." A layer of thickness δ and resistivity ρ has a sheet resistance of ρ/δ ohms per square, the surface resistance, and that is what a thick conductor's surface presents to a high-frequency current. Two warnings come with it on the same page. The formula is "accurate for a flat conductor surface, or when the radius of curvature is much greater than the penetration depth", and on the next: "Note that the concept of skin depth has no meaning in the time domain." It describes a sinusoid at one frequency.

Skin depth of copper, aluminium, silver and gold from 50 Hz to 10 GHz

Equation 5 at 20 °C for the four metals conductors are usually made or plated from, with μ_r = 1. The resistivities are the NIST reference values at 293 K: Matula's recommended figures for annealed 99.999 % copper, silver and gold, and Desai's for 99.99 % aluminium.

Frequency	Copper	Aluminium	Silver	Gold
50.0 Hz	9.22 mm	11.6 mm	8.97 mm	10.6 mm
60.0 Hz	8.42 mm	10.6 mm	8.19 mm	9.67 mm
1.00 kHz	2.06 mm	2.59 mm	2.00 mm	2.37 mm
10.0 kHz	652 µm	819 µm	634 µm	749 µm
100 kHz	206 µm	259 µm	200 µm	237 µm
1.00 MHz	65.2 µm	81.9 µm	63.4 µm	74.9 µm
10.0 MHz	20.6 µm	25.9 µm	20.0 µm	23.7 µm
100 MHz	6.52 µm	8.19 µm	6.34 µm	7.49 µm
1.00 GHz	2.06 µm	2.59 µm	2.00 µm	2.37 µm
10.0 GHz	652 nm	819 nm	634 nm	749 nm

For pure copper this is δ = 6.52/√f cm, f in hertz. The constant is often printed as 6.61, which is the annealed-copper standard's 1.7241 µΩ·cm and gives 6.61; the difference is 1.4 % in δ and comes entirely from which copper is meant. Aluminium's skin depth is 1.26 times copper's at every frequency, and silver's is 0.973 times: silver plating thins the skin by 2.7 %. At the mains the skin depth is around a centimetre, which is why it only matters for busbars and heavy cable. At 1 MHz it is tens of micrometres, comparable to PCB copper, and at 1 GHz it is a couple of micrometres.

AC resistance of a round wire: Kelvin's exact solution

A round wire is the one conductor whose AC resistance has an exact closed-form answer. Rosa and Grover's 1916 Bureau of Standards compendium, the same paper the air-core inductor calculator is built on, calls the straight cylindrical wire "the most important case of all, since the solution is rigorous", and gives Kelvin's result as its Equation 143: R′/R = (x/2)(W/Y), where R′ is the resistance at frequency f, R the DC resistance, and W and Y are combinations of ber and bei, "functions introduced by Lord Kelvin, being respectively the real and imaginary parts of the ordinary Bessel function of order zero". The argument x depends on the radius, the frequency, the permeability and the resistivity; written with the skin depth it is x = √2·r/δ. Everything about a round wire's AC resistance is a function of the ratio of its radius to the skin depth.

The paper prints ber and bei as power series (Equation 148) and, more usefully, an extension of Russell's series (Equation 149) that writes W, Y and Z directly as sums of positive terms. The calculator sums those, so there is no cancellation at any x, and switches to Savidge's asymptotic expansions (Equation 150) above x = 40; the paper says those "give results to one in one hundred thousand for x ≥ 10". The tests check the result against the ber and bei route, against the paper's Table XXII from x = 0.5 to x = 100 to its five decimal places, and against its worked examples.

Two limits bracket the exact curve. For small x the ratio rises as the fourth power of x, so as f², and Rayleigh's expansion (Equation 151) applies; the paper limits it "to the range of values of x less than about 2". The printed third term of that expansion for R′/R has the coefficient 11/(12·28·30); the exact series gives 11/(12·28·80), 4.0923e-4 against the printed 1.0913e-3. The first two terms and the whole of the companion series for the inductance agree, so the 30 looks like a misprint for 80. At x = 1 it moves the ratio by 1.7e-7. For large x the ratio tends to Rayleigh's limit, Equation 152, x/(2√2), which is r/(2δ): the wire's cross-section divided by a strip one skin depth deep around its circumference. The paper is blunt about it: "In some instances these formulas have been used, as though they were exact, over a considerable range of frequencies, without any statement being made as to the magnitude of the error involved." By the exact ratio the error is 1.0 % at x = 70, as the paper says, and 0.10 % at x = 707; the paper's "about 900" for a tenth of a per cent is on the safe side, 0.079 % there.

SLUP197's uniform shell, applied to a round wire, is an annulus of thickness δ: R_ac/R_dc = r²/(2rδ − δ²). At large x it is closer than Rayleigh's limit, because it carries the exact curve's next term, the +¼; as δ approaches the radius it overstates the ratio, and once δ is larger than r it has no meaning. The calculator shows both beside the exact figure, and the chart draws all three.

Wire	1.00 kHz	10.0 kHz	100 kHz	1.00 MHz	10.0 MHz	+10 % at
AWG 24, 511 µm	1.0000	1.0005	1.0471	2.2321	6.4487	149 kHz
AWG 18, 1.02 mm	1.0001	1.0079	1.4961	4.1873	12.6672	37.0 kHz
AWG 12, 2.05 mm	1.0013	1.1162	2.7571	8.1266	25.1412	9.21 kHz

R_ac/R_dc for pure copper, from the exact solution, with the diameters from the AWG definition the wire gauge calculator uses. The last column is the frequency at which the AC resistance is 10 % above DC. A single AWG 12 conductor at 1 MHz has 8.13 times its DC resistance; its skin depth there, 65.2 µm, is 6.4 % of its radius. This is the isolated wire: in a winding or a cable the neighbours make it worse, as the section on validity below explains.

Worked example: Rosa and Grover's 2.5 mm wire at 500 kHz

The paper's Example 82 takes a straight wire 200 cm long and 0.125 cm in radius, "the specific resistance of annealed copper at 20° as 1.721 microhms", and a frequency of 500 000 cycles per second, then repeats it for manganin, "for which the conductivity was one thirtieth of that of copper", and for iron, "conductivity one-seventh of that of copper" with "the permeability ... assumed as low as 100". Example 83 is the copper wire at 1000 cycles. The paper read its ratios from Table XXII by interpolation; the left column is the calculator's arithmetic, the right what the paper prints.

copper    δ = √(1.721 µΩ·cm / (π·μ0·500 kHz))      = 93.4 µm    
          x0 = √2 × 1 mm / δ   (Table XXIV)        = 15.146      paper: 15.146
          x = √2 × 1.25 mm / δ                     = 18.932      paper: 18.932
          R′/R = (x/2)(W/Y)                        = 6.95049     paper: 6.95035
          4Z/xY                                    = 0.14923     paper: 0.14923
          R_dc of 200 cm = ρl/πr²                  = 7.01 mΩ    
          R at 500 kHz                             = 48.7 mΩ    
manganin  x = 18.932 × √(1/30)                     = 3.4565      paper: 3.4566
          R′/R                                     = 1.47616     paper: 1.47620
iron      x = √100 × √(1/7) × 18.932               = 71.557      paper: 71.556
          R′/R                                     = 25.551      paper: 25.551
          Rayleigh x/(2√2)                         = 25.299     
copper    x at 1 kHz                               = 0.84667     paper: 0.84675
          R′/R                                     = 1.00267     paper: 1.00266

Every ratio agrees with the paper's to within 24 parts per million, and the paper says how it got its figures: "by interpolation, using second differences" in Table XXII. The paper's own conclusion on the copper: "the resistance at 500000 cycles per second is 6.95 times as great as with direct current". On the iron it notes that "The influence of this relatively low permeability is enormous. The resistance is more than twenty-five times its direct current value". Rayleigh's limit there would give 25.30, 1.0 % low even at x = 71.6.

The calculator's defaults are this wire, but in pure copper at 1.678 µΩ·cm rather than the paper's 1.721. The ratio at 500 kHz is then 7.0356, the skin depth 92.2 µm, and the wire's DC resistance 6.84 mΩ over its 2 m. Choose "Copper, Rosa and Grover" in the calculator to reproduce the paper exactly.

Skin effect in PCB traces

A PCB trace is a flat strip, and for flat conductors SLUP197's Equation 5 is accurate as it stands. Analog Devices' Basic Linear Design, chapter 12, gives the working version for copper foil: "a good approximation for copper is that the skin depth in centimeters is 6.61/√f", and "Where skin effect is important, the resistance for copper is 2.6 x 10⁻⁷ √f Ω/square, (f in Hz). This formula is invalid if the skin thickness is greater than the conductor thickness (i.e., at dc or LF)." Both constants are Equation 5 for ADI's copper, 1.724 µΩ·cm: that resistivity gives 6.608/√f cm and 2.609 × 10⁻⁷ √f Ω per square.

Which faces carry the current matters as much as the depth. ADI: "current generally flows in both sides of the PC foil (this is not necessarily the case in microstrip lines, see below), so the resistance per square of PC foil may be half the above value", and its microstrip drawing is labelled "HF current flows in one side of the conductor only", the side facing the plane. SLUP197 says the same of any strip with its return close by: "Note that current penetration is from one side only -- the side where the field is. This means that a strip thicker than the penetration depth is not fully utilized." The calculator's trace mode therefore takes the surface resistance ρ/δ on one face, R = R_s/w per metre, or on both, R_s/2w, and floors it at the DC resistance ρ/(w·t) once the skin reaches through the foil.

ADI's threshold assumes "that skin effects become important when the skin depth is less than 50% of the thickness of the conductor", and concludes that for a typical PC foil they matter "at frequencies above approximately 12 MHz." With ADI's own figures, 6.61/√f cm and the "0.036 mm" of 1 oz foil it gives earlier in the chapter, δ equals half the foil at 13.5 MHz, and the calculator, from the resistivity, gets 13.5 MHz. By its own numbers the criterion is met there, not at 12 MHz; the calculator works from the equation. At 10 MHz the skin depth, 20.9 µm, is still 58 % of the foil.

Worked example: ADI's 5 cm track at 100 MHz

Earlier in the same chapter ADI works the DC resistance of "5 cm of 0.25 mm wide 1 oz. PCB track", with "the resistivity of pure copper" at 25 °C as 1.724 × 10⁻⁶ Ω·cm and a sheet resistance of "0.48 mΩ/square": "The track resistance of nearly 0.1 Ω forms a divider with the 5 kΩ load". The same track at 100 MHz, in ADI's copper:

DC        R_sq = ρ/t = 1.724 µΩ·cm / 0.036 mm      = 479 µΩ/sq   paper: 0.48 mΩ/sq
          squares = 5 cm / 0.25 mm                 = 200        
          R_dc = R_sq × squares                    = 95.8 mΩ     paper: nearly 0.1 Ω
100 MHz   δ = √(ρ/(π·μ0·f))                        = 6.61 µm    
          R_s = ρ/δ                                = 2.61 mΩ/sq  paper: 2.6e-7·√f = 2.60 mΩ/sq
          one face: R_s × squares                  = 522 mΩ     
          two faces: R_s × squares / 2             = 261 mΩ     
          R_ac/R_dc, one face  ·  two faces        = 5.45  ·  2.72

Over a ground plane, where the current uses one face, the track has 5.4 times its DC resistance at 100 MHz; as a lone foil with current on both faces, 2.7 times. The ratio is simply t/δ or t/2δ, since R_s/w over ρ/(w·t) is t/δ: it depends on the copper thickness and the frequency and not on the width. The width sets the resistance itself, through the number of squares. For the DC side of the same trace, current capacity and temperature rise, see the trace width calculator.

Where the model stops being valid

Neighbouring conductors: the proximity effect. Everything above is an isolated conductor with its return far away. SLUP197: "When another conductor is brought into close proximity to the first, their fields add vectorially. Field intensity is no longer uniform around the conductor surfaces, so high frequency current flow will not be uniform." With the return current close by, "high frequency current flow is concentrated on the wire surfaces facing each other". Rosa and Grover draw the same line for a pair of wires: unless they "are so near together, relatively to their radius of cross section, that their mutual inductance is appreciably affected by changes in the distribution of the current within the wires, each wire may be treated by the formulas given for a straight, cylindrical wire." The isolated-wire ratio leaves the proximity effect out entirely. SLUP197 notes that most winding design uses "a sinusoidal approach based on work done by Dowell in 1966"; that is a different calculation.

Skin depth comparable to the radius. Equation 5 itself is exact for any wire: it is the depth scale of the solution. What fails is the step from δ to a resistance. SLUP197's shell is "accurate for a flat conductor surface, or when the radius of curvature is much greater than the penetration depth". The calculator uses Kelvin's exact solution for a round wire at every x, so the curvature is taken care of; the shell and r/2δ figures are shown only for comparison.

Skin depth comparable to the foil. The trace model has no exact solution behind it. ADI's formula is invalid once δ exceeds the thickness, and the calculator then reports the DC resistance, which is a floor, and says so. Near the crossover the true value lies between the two, and the model's corner is a corner in the model, not in the copper. The trace edges are left out too: the model treats the width as much larger than the thickness.

Surface roughness. At high frequency the current lives in the outermost micrometres, where a real foil is not flat. None of the sources on this page quantifies roughness, and the calculator's figures are for a smooth surface.

Non-sinusoidal currents. SLUP197: "Although the current waveforms encountered in most switching power supplies are not sinusoidal, most papers dealing with the design of high frequency transformer windings use a sinusoidal approach". and "Some authors use Fourier analysis to extend the sinusoidal method to non-sinusoidal waveforms." Each harmonic sees its own skin depth, and its loss is its own RMS current squared times the resistance at its frequency. For an inductor's ripple current, as in the buck inductor calculator, run the calculator at the fundamental and at the harmonics that carry the current.

Temperature and permeability. Resistivity rises with temperature. SLUP197 works its winding example at 100 °C, "ρ = 2.3·10⁻⁶ Ω-cm", which makes δ 1.17 times its 20 °C value and the DC resistance 1.37 times; enter it as a custom resistivity. For a magnetic conductor the calculator takes μ_ras a constant, as Rosa and Grover did in assuming 100 for iron.

Common skin depth mistakes

Putting the diameter where the radius goes. Kelvin's argument, and the paper's "ρ = radius of conductor", both use the radius. Enter the 2.5 mm wire as 5 mm and the ratio at 500 kHz comes out 13.8109 instead of 7.0356.
Mixing up pure copper and the IACS standard. The 1913 standard, 1.7241 µΩ·cm at 20 °C, is a reference for measuring conductivity, not the resistivity of pure copper: NBS Handbook 100 reports annealed 99.999 % wire "slightly over 103 percent", and Matula's 1.678 µΩ·cm is 2.7 % better than the standard. It moves δ by 1.4 % and the DC resistance by 2.7 %. For hard-drawn wire the handbook goes the other way: "when a conductivity must be assumed for hard-drawn copper wire, it should be taken as 97.5 percent of that of standard annealed copper." Use the figure the conductor's datasheet gives, as a custom resistivity.
Using the shell or r/2δ at low frequency. At 1 kHz the 2.5 mm copper wire has δ = 2.06 mm, more than its radius: r/2δ gives 0.30, less than the DC resistance, and a shell one skin depth deep is larger than the wire. The exact ratio is 1.00281.
Trusting a printed constant over the equation it came from. SLUP197's Equation 6, "D_PEN = 7.5/(f)^1/2 cm" for copper at 100 °C, is 1.7 % below its own Equation 5 with its own ρ = 2.3 µΩ·cm, which gives 7.63/√f cm. The calculator always computes δ from ρ.
Counting both faces of a microstrip. Over a plane the current uses the face towards it, and the resistance is twice what a two-sided calculation gives.
Applying skin depth to a pulse edge as if it were a frequency. SLUP197: the concept "has no meaning in the time domain". Decompose the waveform into its harmonics first.