The resistance that grows with frequency

The winding resistance in every figure of this field so far has been a constant — half an ohm, the same at every frequency. That is a model, and like every model here it has a range.

Above some frequency the current in a conductor stops filling it. The changing magnetic field inside the conductor induces circulating currents that oppose the flow at the centre and reinforce it at the surface, so the current crowds into a layer near the rim and the resistance rises. The usual statement of where that begins is the frequency at which the skin depth equals the radius.

Measured exactly, the effect has already arrived there.

The two expressions #

The skin depth is

$$\delta = \sqrt{\frac{2\rho}{\omega\mu}}$$

which for copper is 2.09 mm at 1 kHz, 209 µm at 100 kHz and 66 µm at 1 MHz. It is a length, it falls as the inverse square root of frequency, and it is the quantity everything about this effect is quoted in.

The asymptote treats the current as flowing uniformly in an annulus one skin depth thick, so the resistance ratio is the whole area over the annulus’s:

$$\frac{R(f)}{R_0} \approx \frac{r^2}{2r\delta - \delta^2}$$

which is a good approximation when δ ≪ r and is nonsense when it is not — at δ > r the annulus is larger than the wire.

The exact answer comes from solving the diffusion of the field into a cylindrical conductor, which gives Bessel functions of complex argument — conventionally written as the Kelvin functions ber and bei — and the ratio is

$$\frac{R(f)}{R_0} = \frac{q}{2}\cdot\frac{\mathrm{ber}(q)\,\mathrm{bei}'(q) - \mathrm{bei}(q)\,\mathrm{ber}'(q)}{\mathrm{ber}'(q)^2 + \mathrm{bei}'(q)^2}$$

with q = r√2/δ. That is computed here from the series definitions rather than looked up, which is the site’s habit and which turns out to have a boundary of its own.

What the measurement says #

Read the third row. At the frequency the rule of thumb names as the start of the effect, the asymptote’s own arithmetic returns exactly one — the annulus of thickness δ is the whole wire, so it predicts no increase at all — and the exact answer is 2.05% up.

That is the same shape as a result the limits field already carries. The conventional tenth-of-a-wavelength criterion for when a circuit becomes electrically large marks a point at which the lumped model is already 30% wrong. In both cases a convenient rule names a frequency, and in both cases the effect the rule is about has comfortably begun by the time the frequency arrives.

Read the last two rows for the other half. At 1 MHz the two agree to 0.14% and at 10 MHz to 0.01%. The asymptote is not wrong; it is asymptotic, and its error is entirely in the region where a designer would like an answer.

Where the series stops working, and how that was found #

The exact expression is a pair of alternating series whose terms grow like $(q/2)^{4m}$, and the functions themselves grow like $e^{q/\sqrt2}$. In double precision that means the sum loses digits to cancellation, and above about q = 15 it loses all of them.

The failure is silent. It does not throw and it does not return an obvious sentinel; it returns a smaller number, so the computed resistance starts falling with frequency.

It was caught by the figure’s own monotonicity assertion, on a 2 mm conductor where q reaches 135 at the top of the axis: the ratio went 22.78 at one frequency and 15.32 at the next, which is a resistance decreasing as the frequency rises and is not a thing that happens.

The repair is a handover. The series is used below q = 12 and the asymptote above it, and the two are required to agree at the join — they do, to 0.199%, which is the size of the asymptote’s own error at that argument. The gate holds the handover, holds the agreement across it, and holds monotonicity at three conductor radii over five decades, which is what caught the problem in the first place.

That is the third time this collection has had a series or an iteration fail by returning a plausible number. polyRoots needed Durand–Kerner rather than deflation because deflation contaminates the last pole recovered. enbw integrated from a lower limit that was not zero and came out 0.064% short of π/2. And a bilinear recursion in the noise field carried its past output with the wrong sign and produced a stable, plausible sequence for two phases.

What the effect does to the site’s other results #

Three figures elsewhere in the collection contain a winding resistance or a conductor, and it is worth saying what this curve does to each rather than leaving the correction implicit.

The noise floor a resistor sets is 4kTR, and if R rises with frequency then so does the density. The noise field’s figures use resistors of a kilohm at audio frequencies, where a wire-wound part’s skin effect is nothing — but an inductor’s winding used as a noise source at a megahertz is noisier than its direct-current resistance says, in proportion to this curve.

The equivalent series resistance of a capacitor, which the real-capacitor essay treats as a constant, has the same frequency dependence from the same mechanism in its leads and plates. That essay’s departure at 4.69 MHz is dominated by the lead inductance rather than by the resistance, so the correction does not move its headline number — but it does mean the loss at that frequency is understated.

A transmission line’s conductor loss goes as the square root of frequency for the same reason, which is why a cable’s attenuation is quoted per unit length at a frequency. The lines field treats its ladder as lossless throughout, which is the right model for what it is measuring — the convergence of a lumped model to a wave one — and would be the wrong one for a length of coaxial cable.

None of the three is repaired here. What the essay contributes is the number that says how large the correction is: a factor of 1.45 at 100 kHz for half-millimetre wire, 4.05 at a megahertz, and 1.0001 at a kilohertz — so below about ten kilohertz the constant-resistance model in the rest of the site is right to a part in a thousand, and above a hundred kilohertz it is not right at all.

What it costs a winding #

The numbers above are for one conductor. A winding is many, and the effect is worse than this essay measures for a reason worth stating rather than leaving as a caveat.

A conductor in a winding sits in the magnetic field of every other conductor as well as its own, and those fields drive their own circulating currents — the proximity effect, which in a multi-layer winding can dominate the skin effect entirely. A single isolated conductor is the easy case and the one with a closed form; a winding is a field problem.

So this essay’s curve is a lower bound on what a winding does, and it is stated as one. What it establishes is the frequency at which the effect can no longer be ignored for the simplest possible geometry, which is a floor on the answer for any real one.

Two consequences follow for the rest of the field.

The transformer figures’ constant winding resistance is a low-frequency model. At the upper band edge of the k = 0.99 design — 81.3 kHz — this curve is already at 1.336, so the half-ohm used throughout is really two thirds of an ohm there. That moves the upper corner by a similar fraction, in the direction of a wider band, which is the comfortable direction and is still an error.

The remedy is geometry rather than material. Copper’s resistivity is what it is; what changes the skin depth is frequency, and what changes the ratio at a given frequency is the conductor’s shape. Splitting a conductor into many thin insulated strands — litz wire — keeps every strand below its own skin depth, at the cost of a worse packing factor and therefore more direct-current resistance. That trade is the same shape as every other one in this field: two boundaries pulling in opposite directions with the design between them.

The shape of the curve between the two limits #

Between the two regimes the exact curve does something worth naming, because it is the reason a single expression cannot cover the range.

Below the corner it rises as the fourth power of frequency. The ratio is $1 + q^4/192$ for small $q$, so the departure from unity quadruples for every doubling of frequency below the corner — 1.0001 at 1 kHz, 1.0017 at 5 kHz, 1.0205 at 17.4 kHz. A quantity rising as the fourth power is invisible and then suddenly is not, which is exactly why the effect surprises people.

Above the corner it rises as the square root. The asymptote is $r/2\delta$, and $\delta$ goes as $f^{-1/2}$, so the ratio goes as $f^{1/2}$ — 4.05 at 1 MHz and 12.23 at 10 MHz is a factor of 3.02 for a factor of ten in frequency, against √10 = 3.16.

So the curve turns from a fourth power into a half power over about a decade around the corner, and neither exponent describes the turn. That is what the exact expression is for, and it is the region a designer works in — nobody operates a winding two decades above its skin corner if they can help it.

The site has drawn this shape before in a different subject. The lumped ladder’s convergence to the wave limit is a power law with a fitted exponent, and the interesting part is the residual rather than the exponent; the trapezoidal rule’s error falls by exactly four per doubling because it is second order, and the exactness of the four is the finding. Here there are two exponents and the transition between them, and the transition is the answer.

What the radius slider shows #

The slider moves the conductor’s radius, and it moves the whole curve left as the wire gets thicker — because the frequency at which the skin depth equals the radius goes as the inverse square of the radius.

Two readings of that table.

Thick wire is worse than thin wire at high frequency, by more than the ratio of their areas. A 2 mm conductor has sixteen times the cross-section of a 0.5 mm one and 3.80 times its resistance ratio at 1 MHz, so its advantage there is a factor of four rather than of sixteen. Above the corner the useful measure of a conductor is its perimeter rather than its area, which is what “the current flows in the skin” means read as an economic statement.

Which is the argument for many thin strands. Sixteen conductors of 0.125 mm have the same total area as one of 0.5 mm and sixteen times its perimeter, and each stays below its own skin depth to a much higher frequency. That is litz wire, and it is the reason a high-frequency winding looks nothing like a low-frequency one carrying the same current.

The table also makes the essay’s opening point visible in a second way. Every row’s “δ = radius” column is a frequency at which that conductor’s ratio is 1.0205 — the same 2%, at every radius, because the ratio at q = √2 is a pure number with no geometry left in it. The rule of thumb is therefore not merely approximate; it names a fixed and knowable error, and the error is two per cent for every round conductor there is.

The one number worth carrying #

If the essay reduces to a single frequency it is not the skin-depth-equals-radius one, and that is the point.

For a conductor of radius r the useful frequency is where the ratio reaches some stated per cent — a design decision like every other on this site — and for a half-millimetre wire and one per cent that is about 12 kHz, well below the 17.4 kHz the rule names. For ten per cent it is about 39.8 kHz. The rule’s frequency sits between them, at 2%, which is not a bad place for a rule of thumb to be as long as nobody reads it as the beginning.

The general instruction is the one the collection keeps arriving at from different directions: a criterion that names a frequency has an error at that frequency, and the useful question is what it is. Kirchhoff’s tenth of a wavelength is 30%. The skin depth equal to the radius is 2%. The quadrature rule for rise times is 2.18× against a real chain of poles’ 2.79× to 5.62×. None of the three is wrong; all three are answers to a slightly different question from the one they are used for, and the number that makes them usable is the one nobody prints.

Why this is a magnetics essay and not a materials one #

A closing note on where the boundary of the field is, because the mechanism here is arguably not about magnetics at all.

The skin effect is a consequence of Faraday’s law inside a conductor: a changing current makes a changing field, the field induces an opposing electric field, and the current redistributes. That is electromagnetism, which illustrated-physics.com owns and which this site does not derive.

What is owned here is what it does to a circuit element. The half-ohm in every transformer figure in this field is a number in a netlist, and this essay establishes the frequency above which that number is a function of frequency rather than a constant — which is a statement about a model’s range and is the site’s entire business.

The same division applies to the two neighbouring essays. The flux integral is Faraday’s law read forwards; what is owned is that a core’s limit is a volt-second product and what that does to a winding’s voltage rating. The gap’s share of the energy is a field calculation done as a circuit; what is owned is the reluctance divider and the design rule it produces.

In each case the physics is taken as given, the circuit consequence is measured, and the boundary between the two is the one the whole collection has kept since its first field: this site says what a model of a component stops being true at, and takes the mechanism from wherever mechanisms come from.

Where 2.05 per cent turns into a factor of fourteen #

A rule of thumb 2 per cent optimistic at the frequency it names is a mild finding on one conductor, and this ladder is where it stops being mild.

The copper that makes it worse is the first step: stack the conductors and the quantity changes character, because each layer sits in the field of the ones below it and the loss that field drives has no upper bound in the thickness. The product turns over, so a portion of four layers has a best foil thickness — the fourth root of three over the square root of the layer count, in skin depths — and above it more copper is more resistance, which no single-conductor picture predicts.

The assumption that is a geometry is where the whole one-dimensional account is tested against a solved field, and its answer is the factor: a portion of foils filling its window returns the closed form to 0.155 per cent, and the same copper filling a quarter of it returns 9.00 against the expression’s 16.38, dissipating 0.528 watts a metre against 0.232.

And the turns nearest the gap is where the field is not merely two-dimensional but points the wrong way entirely, with a gapped inductor’s worst turn at 37.5 times its direct-current loss.

Which is the ladder’s shape. Two per cent for one wire, a turnover for a stack, a factor of two for a window that is not full, and a factor of fourteen for a gap — each of them a mechanism the previous picture had no term for rather than a correction to it.