Determine exact transmission line characteristics instantly through the Coax Inductance Calculator. Compute total capacitance, velocity factor, and precision high-frequency limits.
The Inductance Formula Behind a Coaxial Cable’s Center Conductor
A coaxial cable’s inductance comes from the magnetic field trapped in the ring-shaped space between the center conductor and the outer shield. Unlike two parallel wires side by side, coax confines that field entirely inside the shield, which is why coax radiates far less than open two-wire line at the same frequency. The whole calculation reduces to one ratio: outer shield diameter over center conductor diameter.
$$L_{ext} = \frac{\mu_0 \mu_r}{2\pi}\, l \, \ln\!\left(\frac{D}{d}\right)$$
$d$ is the center conductor’s diameter, $D$ is the inner diameter of the shield, $l$ is cable length, $\mu_r$ is the relative permeability of the dielectric between them, and $\mu_0 = 4\pi \times 10^{-7}$ H/m. This is external inductance — the field living in the dielectric space — and it’s what governs the cable’s behavior at radio frequency, where skin effect pushes current to each conductor’s surface.
At DC and low frequency, current isn’t confined to the surface, and the center conductor picks up an additional internal inductance term that doesn’t depend on the D/d ratio at all:
$$L_{int} = \frac{\mu_0 \mu_r}{8\pi}\, l$$
Add the two together and you get total DC inductance. Capacitance follows the same logarithmic ratio in reverse — it shrinks as $\ln(D/d)$ grows, rather than growing with it:
$$C = \frac{2\pi \varepsilon_0 \varepsilon_r}{\ln(D/d)}\, l$$
where $\varepsilon_r$ is the dielectric constant of the insulation separating the two conductors. Characteristic impedance, propagation velocity, and delay all fall out of $L_{ext}$ and $C$ together — $Z_0 = \sqrt{L_{ext}/C}$, and velocity is $1/\sqrt{L_{ext} \cdot C}$ per unit length.
A 1 mm Center Conductor Inside a 3 mm Shield
Take a 1 mm diameter center conductor, a 3 mm inner shield diameter, a 1 meter cable length, air-equivalent permeability ($\mu_r = 1$), and a dielectric constant of 2.1 — close to the foam polyethylene used in a lot of RF coax. $D/d = 3$, and $\ln(3) \approx 1.0986$.
External inductance per meter works out to $$L_{ext}/l = \frac{4\pi \times 10^{-7}}{2\pi} \times 1.0986 \approx 219.72$$ nH/m, which for a 1 meter cable is 219.72 nH outright. Capacitance comes out to 106.34 pF, and dividing the two gives a characteristic impedance of roughly 45.46 Ω.
| Quantity | Result |
|---|---|
| External inductance | 219.72 nH |
| Characteristic impedance | 45.46 Ω |
| Total capacitance | 106.34 pF |
| Propagation delay | 4.83 ns |
| Velocity factor | 69.01% |
| Propagation velocity | 206.89 Mm/s |
| Inductance per meter | 219.72 nH/m |
| Inductance per foot | 66.97 nH/ft |
| Total DC inductance | 269.72 nH |
| Internal contribution | 50.00 nH (22.76% of external) |
Notice the internal inductance term adds nearly 23% on top of the external value here. That gap matters at low frequency and audio applications, and shrinks toward irrelevant at RF once skin effect takes over and current stops flowing through the conductor’s interior at all.
Judging a 45 Ω Result Against Real Coax
There’s no code-mandated pass/fail line built into this calculation — it’s transmission-line electromagnetics, not an installation standard. What’s useful instead is a comparison point: commercial coax overwhelmingly clusters around 50 Ω for general RF and test equipment use, or 75 Ω for video and broadcast applications. A 45.46 Ω result sits close enough to the 50 Ω family that the input geometry resembles a real manufactured cable rather than an arbitrary one.
Velocity factor is worth checking the same way. Solid polyethylene dielectric coax typically runs a velocity factor near 66%, foam dielectric pushes it up toward 78-84%, and air-spaced designs can approach 90% or higher. A 69.01% result lands squarely in foam-to-solid-polyethylene territory, consistent with the 2.1 dielectric constant used in the example.
Why the D/d Ratio Moves the Number More Than Either Diameter Alone
Every output above traces back to $\ln(D/d)$, and logarithms compress large changes into small ones. Push $D/d$ from 3 to 6 — doubling the shield diameter relative to the center conductor — and $\ln(6) \approx 1.79$ against $\ln(3) \approx 1.10$: inductance and impedance rise about 63%, not the 100% a linear relationship would suggest. Capacitance moves the opposite direction, since it sits in the denominator.
Dielectric constant works differently, and pulls harder. Raising $\varepsilon_r$ increases capacitance directly and proportionally, which drops impedance and slows propagation velocity by a factor of $1/\sqrt{\varepsilon_r}$.
Go from air ($\varepsilon_r \approx 1$) to solid PTFE ($\varepsilon_r \approx 2.1$) and velocity factor falls from close to 100% down toward 69% — exactly the value in the worked example above, because that’s the dielectric constant used.
Permeability barely ever needs to move from 1. It describes the dielectric medium, not the conductor metal, and almost every coax dielectric — polyethylene, PTFE, foam, air — is non-magnetic.
The one place internal inductance actually matters is low-frequency or DC work, where current still fills the center conductor’s cross-section; at RF, skin effect confines current to the surface and the external term alone governs behavior, which is why it’s the hero number and the internal term is reported separately.
Frequently Asked Questions
Why does coax have less inductance than two parallel wires of the same spacing?
Coax confines its magnetic field entirely inside the shield, so the field never extends outward the way it does around two open parallel wires. That contained geometry, plus the shield acting as a return path directly around the center conductor, keeps inductance per unit length lower for a comparable size.
What’s the difference between external and internal inductance in a coax cable?
External inductance comes from the magnetic field in the dielectric between the conductors and dominates at RF once skin effect pushes current to each conductor’s surface. Internal inductance comes from current still flowing through the interior of the center conductor, which only matters meaningfully at DC or low frequency.
Does the dielectric material change the cable’s inductance?
Barely, unless it’s magnetic, which almost none are. Dielectric constant instead controls capacitance directly, which is why swapping dielectrics changes impedance and velocity factor far more than it changes inductance.
Why do most coax cables target 50 Ω or 75 Ω specifically?
50 Ω balances power-handling capacity and loss for RF transmission and test equipment, while 75 Ω minimizes signal loss for video and broadcast applications where power handling matters less. Both values became industry standards because connectors, amplifiers, and equipment converged around them, not because other impedances don’t work.
Why does the calculator reject a shield diameter smaller than the center conductor?
Because the geometry stops being physically possible at that point — the center conductor can’t be larger than the space it sits inside. The formula is only defined once $D$ exceeds $d$.
What does velocity factor actually tell you about a cable?
It’s the fraction of the speed of light a signal travels through that particular cable, set almost entirely by the dielectric constant. A lower velocity factor means more propagation delay over the same physical length, which matters for time-critical RF and digital signaling applications.
Can I use this formula for a cable with a braided shield instead of a solid tube?
The formula treats the shield as a continuous conducting boundary at diameter $D$, which is a reasonable approximation for braided shields at lower frequencies. At higher frequencies, gaps in the braid weave introduce leakage and coupling effects this idealized model doesn’t capture.
These results model an idealized, uniform coaxial geometry with a lossless dielectric — they don’t account for conductor resistance and its frequency-dependent skin-effect losses, dielectric loss tangent, connector or manufacturing tolerances, or the reduced velocity factor and impedance shifts real braided or foil shields introduce compared to an ideal cylindrical boundary.