Wire Self Inductance Calculator turns a straight wire’s length, diameter, and drive frequency into precise inductance, reactance, skin depth, AC resistance, and Q-factor value.
The Logarithmic Formula Behind a Straight Wire’s Self-Inductance
Any current-carrying wire generates a magnetic field around itself, and that field stores energy — which is exactly what inductance measures. For an isolated straight round wire, that relationship has a closed-form solution known as Rosa’s formula, developed at what’s now NIST in the early 1900s and still the standard reference for straight-wire self-inductance. It isn’t an approximation pulled from a datasheet; it comes directly from integrating the wire’s magnetic field energy.
With length $l$ and radius $r$ in centimeters, low-frequency inductance in nanohenries is:
$$L_{DC} = 2l\left[\ln\left(\frac{2l}{r}\right) – 0.75\right]$$
That 0.75 constant assumes current is spread evenly across the wire’s cross-section, which is only true at DC and low frequency. Push frequency up and skin effect crowds current toward the surface, changing the internal field contribution and shifting the constant to 1.0:
$$L_{ext} = 2l\left[\ln\left(\frac{2l}{r}\right) – 1.0\right]$$
The gap between those two values is the wire’s internal inductance, and it collapses to something surprisingly simple:
$$L_{int} = L_{DC} – L_{ext} = 0.5 \, l$$
The logarithmic term cancels out entirely. Internal inductance of a straight non-magnetic wire depends only on length, never on diameter — a 10cm wire has 5nH of internal inductance whether it’s hair-thin or as thick as a pencil. That works out to a constant 50nH per meter, which is $\mu_0/8\pi$, a fixed property of free space rather than of the wire’s geometry.
Finding the Inductance of a 10cm, 1mm-Diameter Wire at 10 MHz
Take a 10cm length of 1mm-diameter copper wire — roughly 18 AWG — driven at 10 MHz. Radius is 0.05cm, so $2l/r = 20/0.05 = 400$, and $\ln(400) = 5.99$.
DC inductance is $2(10)(5.99 – 0.75) = 104.83$nH. The high-frequency external limit is $2(10)(5.99 – 1.0) = 99.83$nH, leaving exactly 5.00nH of internal inductance — matching the 0.5nH-per-centimeter rule regardless of the 1mm diameter. At 10 MHz, that DC inductance value produces a reactance of $2\pi(10{,}000{,}000)(104.83 \times 10^{-9}) = 6.59\Omega$.
| Quantity | Value |
|---|---|
| DC resistance | 2.14 mΩ |
| Skin depth at 10 MHz | 20.63 µm |
| AC resistance | 25.92 mΩ |
| Q factor | 254.1 |
| Length-to-diameter ratio | 100.0 |
DC resistance comes from the plain $R = \rho l / A$ relationship using copper’s resistivity, 2.14mΩ for this wire. But skin depth at 10 MHz works out to just 20.63 micrometers — far thinner than the wire’s 500-micrometer radius — so current can’t use the full cross-section anymore. AC resistance jumps to 25.92mΩ, roughly 12 times the DC value, because effectively only a thin annular ring near the surface is carrying current. Dividing reactance by that AC resistance gives a Q factor of 254.1.
Why the Length-to-Diameter Ratio Decides How Much to Trust the Number
Rosa’s formula is a thin-wire approximation, and its accuracy depends entirely on how thin “thin” is. The calculator flags any wire with a length-to-diameter ratio above 1 as valid, but only labels it optimal once that ratio clears 10. In the example above, 10cm over 1mm gives a ratio of exactly 100 — well past the point where the formula’s underlying assumptions hold cleanly.
That distinction matters because the formula treats the wire as infinitely thin relative to its length, ignoring end effects at the wire’s two tips. A ratio just above 1 — a wire barely longer than it is thick — sits in territory where those end effects stop being negligible, and the calculated inductance will diverge from a real physical measurement. Nothing stops the calculator from returning a number for a low ratio; it just won’t be as trustworthy as one from a properly slender wire.
Skin Effect: Why AC Resistance Can Dwarf DC Resistance Long Before Inductance Notices
Frequency barely touches the inductance number directly — it only enters through reactance and the skin-depth-driven resistance jump, not through $L_{DC}$ itself. Resistance is a different story entirely. Skin depth shrinks as the inverse square root of frequency, so a tenfold jump in frequency only cuts skin depth by about a third, but that’s often enough to flip a wire from carrying current across its full cross-section to squeezing it into a thin surface shell.
Diameter cuts both ways. A thicker wire lowers DC resistance, since resistance falls with cross-sectional area, but it also makes skin effect kick in at a lower frequency, since the radius has farther to go before it exceeds the skin depth. Doubling the wire’s diameter doesn’t meaningfully change inductance — diameter only appears inside a logarithm there — but it can noticeably change where the crossover into strong skin effect happens.
One conditional worth knowing about directly: the calculator only applies the skin-effect resistance penalty once the wire’s radius exceeds its skin depth. Below that crossover, AC resistance is reported as equal to DC resistance rather than scaled down further — this model doesn’t attempt to capture the gradual transition zone right around the point where radius and skin depth are close, only the two limiting regimes on either side of it.
Wire Self-Inductance and Skin Effect Questions
What is the self-inductance of a straight wire?
It’s the inductance a straight, isolated conductor has purely from its own magnetic field, without any nearby loop or return path involved. It depends primarily on the wire’s length, with a much weaker logarithmic dependence on its diameter.
What is Rosa’s formula for wire inductance?
It’s the closed-form expression $L = 2l[\ln(2l/r) – k]$, where $k$ is 0.75 for a uniform DC current distribution and 1.0 for the high-frequency limit where current sits at the surface. It remains the standard reference calculation for a single straight round conductor’s self-inductance.
Why does frequency affect a wire’s inductance?
Frequency doesn’t change the wire’s geometry, but it changes how current distributes across it. As skin effect pushes current toward the surface, the magnetic field inside the conductor weakens, which slightly lowers total inductance from its DC value down toward the external-only limit.
What is skin depth and why does it matter for wire resistance?
Skin depth is how far current penetrates into a conductor before its density drops off substantially, and it shrinks as frequency rises. Once skin depth falls below the wire’s radius, current is effectively confined to a thin outer shell, shrinking the usable conducting area and raising AC resistance well above the DC value.
How does wire diameter affect inductance?
Only weakly — diameter sits inside a logarithm in Rosa’s formula, so even a large change in diameter shifts inductance by a relatively small amount. Wire length has a far bigger effect, since it multiplies the whole expression directly.
What length-to-diameter ratio gives an accurate inductance calculation?
A ratio above 10 is generally treated as the point where Rosa’s thin-wire assumption holds well, since end effects at the wire’s tips become negligible relative to its length. Ratios close to 1 are still calculable but drift further from what a real measurement would show.
Why is AC resistance higher than DC resistance in a wire?
Because skin effect confines high-frequency current to a thin layer near the surface instead of letting it use the wire’s entire cross-section. Less usable conducting area for the same current means more resistance, and that effect grows stronger as frequency increases and skin depth keeps shrinking.
What’s the difference between internal and external inductance?
External inductance comes from the magnetic field outside the conductor; internal inductance comes from the field inside it, which only exists when current flows through the interior rather than just along the surface. For a straight wire, internal inductance depends solely on length, not diameter, and disappears entirely once skin effect pushes all the current to the surface.
This models a single isolated straight wire in free space, with no return conductor, loop, or ground plane nearby — real circuits always have a current return path, and its proximity changes the effective loop inductance well beyond what a single wire’s self-inductance alone predicts.