Coax Impedance Calculator
Coax Impedance Calculator
Characteristic impedance of a coaxial cable from the inner conductor diameter, the dielectric diameter and its permittivity — built from the capacitance and inductance per metre rather than quoted from a table — with the velocity factor, the delay per metre, the diameter ratio that gives 50 Ω or 75 Ω, why those two numbers and not others, and the frequency at which the cable stops carrying one mode.
coaxial impedance
a cable with a 2.74 mm inner conductor inside 7.24 mm of foam polyethylene, εᵣ = 1.42 — the published dimensions of Times Microwave LMR-400
Where 138 comes from
Z₀ = √(L/C) = (η₀/2π) · ln(D/d) / √εᵣ = 59.9585 ln(D/d) / √εᵣ
= 138.0595 log₁₀(D/d) / √εᵣ since 59.9585 × ln 10 = 138.0595
√(LC) = √εᵣ / c (the logarithm cancels — delay does not depend on shape)
fc,TE11 ≈ 2c / (π√εᵣ (D + d))
- d, D
- diameter over the inner conductor and over the dielectric. Only their ratio sets the impedance
- eps r
- relative permittivity of the dielectric. It is the only thing the velocity factor depends on: VF = 1/√εᵣ
- eta 0
- impedance of free space, µ₀c = 376.7303 Ω. Divided by 2π it is 59.9585 — the whole of the coax constant
- f c
- the frequency at which a second mode can propagate. Below it the cable carries one TEM wave and this page applies
Worked example
a cable with a 2.74 mm inner conductor inside 7.24 mm of foam polyethylene, εᵣ = 1.42 — the published dimensions of Times Microwave LMR-400
D ÷ d = 7.24 ÷ 2.74 = 2.6423, so ln(D/d) = 0.97166
C = 2πε₀εᵣ ÷ ln(D/d) = 81.3 pF and L = (µ₀/2π) ln(D/d) = 194.3 nH
Z₀ = √(L/C) = 48.89 Ω — and 59.9585 × 0.97166 ÷ √1.42 gives the same number, because that is where the constant came from. The datasheet calls this cable 50 Ω; the 2.2% difference is rounded dimensions and a foam whose permittivity is not quoted
Velocity factor = 1/√εᵣ = 0.8392, so the delay is √εᵣ ÷ c = 3.9749 ns per metre. The datasheet says 84%
The first higher-order mode arrives at about 2c ÷ (π√εᵣ(D+d)) = 16.05 GHz; below that the cable carries one mode and everything above is true
At 1 GHz the smooth-metal loss floor is 12.82 dB/100 m, or 3.91 dB/100 ft. The datasheet's measured figure interpolates to about 4.1 dB/100 ft, so the floor is 4.8% under it — surface roughness, the braid and a copper-clad-aluminium centre
The two impedances that are not arbitrary
| Criterion | D ÷ d | Impedance × √εᵣ | In air | In solid PE (εᵣ 2.25) |
|---|---|---|---|---|
| Minimum attenuation | 3.5911 | 76.65 Ω | 76.7 Ω | 51.1 Ω |
| Maximum voltage before breakdown | 2.7183 | 59.96 Ω | 60.0 Ω | 40.0 Ω |
| Maximum power before breakdown | 1.6487 | 29.98 Ω | 30.0 Ω | 20.0 Ω |
| Geometric mean of the last two rows | — | 47.94 Ω | 47.9 Ω | 32.0 Ω |
Real cables, checked against their own datasheets
| Cable | d (mm) | D (mm) | εᵣ used | This page | Datasheet |
|---|---|---|---|---|---|
| Times Microwave LMR-400 | 2.74 | 7.24 | 1.417 | 48.9 Ω | 50 Ω, VF 84% |
| Belden 8240 (RG-58, solid inner) | 0.838 | 2.946 | 2.25 | 50.2 Ω | 51.5 ± 1.5 Ω, VF 66% |
| Belden 8259 (RG-58C/U, stranded inner) | 0.889 | 2.946 | 2.25 | 47.9 Ω | 50 Ω, VF 66% |
| Semi-rigid 0.141 in, solid PTFE | 0.92 | 2.98 | 2.10 | 48.6 Ω | 50 Ω, VF 69% |
Two cylinders, one ratio, and two numbers that are not arbitrary
A coaxial line is the easiest transmission line in the world to analyse, because the field between the conductors is a textbook problem. Gauss’s law around the inner conductor gives the capacitance per metre, 2πε₀εᵣ/ln(D/d). Ampère’s law around the same cylinder gives the inductance per metre, (µ₀/2π)ln(D/d). A lossless line’s characteristic impedance is √(L/C), and the logarithms leave a single factor: Z₀ = (η₀/2π)·ln(D/d)/√εᵣ. η₀ is the impedance of free space, 376.7303 Ω, so the constant is 59.9585 — or 138.0595 if you prefer base-ten logarithms, which is the 138 everybody quotes. It was never measured. It is the impedance of free space divided by 2π.
Only the ratio matters. Scale a cable up by ten and D/d is unchanged, so the impedance, the velocity factor and the delay per metre are all unchanged. What does change is the loss, which falls roughly in proportion to the size, and the frequency at which a second mode appears, which falls too. That is the whole trade in cable selection: big cable, low loss, low frequency ceiling.
Why the delay does not depend on the shape. Multiply L and C rather than dividing them and the logarithm cancels: √(LC) = √(µ₀ε₀εᵣ) = √εᵣ/c. Every coaxial cable with the same dielectric has the same delay per metre whatever its impedance or its size, and the velocity factor is 1/√εᵣ and nothing else. A foam dielectric is mostly air, which is why foam cables run at 80–88% of the speed of light and solid polyethylene ones at 66%.
Why 50 Ω and why 75 Ω. Two different optima, and they are not in the same place. Conductor loss per metre goes as (1/a + 1/b)/ln(b/a); differentiate and the minimum sits where x ln x = 1 + x, which is D/d = 3.591 and an impedance of 76.7 Ω in air. That is where 75 Ω comes from, and it is why 75 Ω is the video and broadcast impedance: those links are long and loss is what hurts. Power handling is a different question. The field is strongest at the surface of the inner conductor, V/(a·ln(b/a)); for a fixed breakdown field the transmitted power peaks at D/d = √e, which is 30.0 Ω, and the voltage peaks at D/d = e, which is 60.0 Ω. So minimum loss and maximum power want ratios a factor of two apart in impedance, and 50 Ω is very close to the geometric mean of the two, 47.9 Ω. There is a second and better reason: fill the same line with polyethylene and the minimum-loss impedance becomes 76.65/√2.25 = 51.1 Ω. For the solid cables that actually got built, 50 Ω is the minimum-loss impedance. The historical record is messier than either story — 50 Ω was settled in US military practice in the 1930s and 1940s and the reasons given at the time were partly mechanical — so treat the arithmetic above as the reason it stuck rather than the reason it was chosen.
Where this page stops being true. Everything here is the TEM mode: the electric field purely radial, the magnetic field purely circumferential, no cutoff, valid down to DC. Above a certain frequency the cable can also carry a TE₁₁ mode, whose cutoff wavelength is about the mean circumference, π(a+b). Past that the cable has two ways to carry energy, they travel at different speeds, and the impedance above stops meaning anything. The rule of thumb is checked on this page against the exact coaxial eigenvalue — the root of J′₁(ka)Y′₁(kb) = J′₁(kb)Y′₁(ka) — and comes out about 2.4% low, so it errs safe. The loss figure is also a floor and not a specification: it assumes smooth solid conductors of the chosen metal, so a braided shield, a copper-clad-aluminium centre and surface roughness all push a real cable above it. Measured against two published attenuation tables it comes out 4.8% low for LMR-400 at 1 GHz and 9.6% low for braided RG-58 at 100 MHz — close enough to be useful for comparing candidate geometries, not close enough to design a link budget with. Use the manufacturer’s measured attenuation when you have it.
What the number is for: the VSWR and return loss converter turns a mismatch between this impedance and a load into standing-wave ratio and reflected power, the microstrip calculator does the same job for a trace on a board, and the L-network calculator matches something that is not this impedance to something that is.
Frequently asked questions
What is the formula for coaxial cable impedance?
Z₀ = 138.0595 × log₁₀(D/d) ÷ √εᵣ, with D the inside diameter of the outer conductor and d the outside diameter of the inner one. In natural logarithms the constant is 59.9585, which is the impedance of free space divided by 2π. The formula is √(L/C) with the capacitance and inductance of two concentric cylinders substituted in, which is how this page derives it.
Why is coax 50 ohms?
Because two different optima sit on either side of it. An air-filled line has its minimum attenuation at 76.7 Ω and its maximum power handling at 30 Ω, and 50 Ω is near the geometric mean of those two, 47.9 Ω. Better still: fill the line with polyethylene, as real cables are, and the minimum-loss impedance falls to 51.1 Ω. So for a solid-dielectric cable 50 Ω essentially is the minimum-loss impedance.
Why is video cable 75 ohms?
75 Ω is close to the minimum-attenuation impedance of an air-spaced line, 76.7 Ω. Video and broadcast distribution runs are long and carry very little power, so attenuation is the thing worth optimising and breakdown is not. It also happens to sit near the feedpoint impedance of a half-wave dipole in free space, around 73 Ω.
What is the velocity factor of coax and what sets it?
1/√εᵣ, and nothing else — not the impedance, not the diameters. Solid polyethylene gives 0.66, foamed polyethylene 0.80 to 0.88, PTFE 0.69 and an air-spaced line about 0.98. Multiply by 3.336 ns to get the delay per metre.
What is the highest frequency a coaxial cable can be used at?
The frequency at which the TE₁₁ mode starts to propagate, whose cutoff wavelength is about the mean circumference of the dielectric, π(a+b). Smaller cable, higher ceiling: that is the reason 3.5 mm and 2.4 mm connectors exist. The page prints the figure for your dimensions; it is roughly 2.4% conservative against the exact answer.
Why does my cable’s measured loss not match this page?
The figure here is the floor for smooth solid conductors of the metal you chose, with only the dielectric loss tangent you entered. A real cable has a braided or foil shield, often a copper-clad-aluminium centre, a rough surface at the skin depths involved, and connectors. On the two cables checked here the floor came out about 5% low for LMR-400 at 1 GHz and 10% low for braided RG-58 at 100 MHz, and the manufacturer’s measured table is still the number to design a link budget with.
Related calculators
References
- Ellingson SW. Electromagnetics, Volume 2 (Virginia Tech Publishing, open access), §7.5 Why 50 Ohms? — gives 60–77 Ω as the low-attenuation band for air-filled coax, b/a ≈ 1.65 (about 30 Ω) for maximum power handling, and 40–50 Ω as the equivalent low-attenuation band once the line is filled with a plastic spacer of εᵣ ≈ 2.25.
- Times Microwave Systems, LMR-400 Flexible Low Loss Communications Coax datasheet: 0.108 in (2.74 mm) solid copper-clad-aluminium centre, 0.285 in (7.24 mm) foam-PE dielectric, 50 Ω, velocity of propagation 84%, 23.9 pF/ft (78.4 pF/m), attenuation 3.9 dB/100 ft at 900 MHz and 5.1 dB/100 ft at 1,500 MHz. Used above as the page’s worked example and as the check on its loss estimate.
- Belden technical data sheets 8240 (RG-58 type, 20 AWG solid bare copper centre, 0.116 in polyethylene, 51.5 ± 1.5 Ω, VP 66%, 28.5 pF/ft) and 8259 (RG-58A/U type, 20 AWG stranded, 0.889 mm centre over 2.9464 mm PE, 50 Ω, VP 66%, 101.055 pF/m, 5.05274 ns/m). The stranded and solid pair is what lets the stranding correction in the table above be quantified.
- Pozar DM. Microwave Engineering, 4th ed., Wiley, 2011 — Chapter 2 for the coaxial line’s L, C and TEM fields, and Chapter 3 for the higher-order modes of a coaxial guide and the π(a+b) cutoff approximation. The exact eigenvalue equation J′₁(ka)Y′₁(kb) = J′₁(kb)Y′₁(ka) used to check that approximation is solved numerically in this page’s build script rather than taken on trust.
- IEC 61196-1:2005, Coaxial communication cables — Part 1: Generic specification — General, definitions and requirements: the standard that defines how characteristic impedance, velocity of propagation, capacitance and attenuation are to be measured and declared, which is why datasheet columns look the same from one manufacturer to the next. Referenced for the measurement definitions, not for any number on this page.