A coaxial cable, or coax, is the transmission line you meet most often on an RF bench. Its whole electrical behaviour follows from three numbers: the inner diameter of the outer conductor, the diameter of the center conductor, and the dielectric constant of the material between them. This page starts from those three numbers and derives the capacitance, the inductance, the characteristic impedance and the highest usable frequency of the cable, and then compares the result with common connector sizes.

• What defines a coaxial cable ?
• Capacitance and Inductance
• Characteristic Impedence
• Cutoff Frequency

What defines a coaxial cable ?

Before any formula, let's fix the names. Every formula on this page uses the same two diameters and the same dielectric constant, so it helps to see where each one sits in the cross section.

Cross section of a coaxial cable. The two diameters and the dielectric constant are the only inputs the formulas below need.

• D is the diameter of the outer conductor, measured on its inner surface, and d is the diameter of the center conductor.
• The grey ring between them is the dielectric, marked ε_R. The note asks what material is used and what its dielectric constant is.
• The notes on the conductors ask for their radius. The formulas below use the ratio D/d, so diameters and radii give the same result.

The signal travels in the dielectric between the two conductors, as a wave whose electric field points from the center conductor to the outer one. This field pattern is called the TEM mode, and it has no lower cutoff frequency. So a coax works from DC upward, which a waveguide cannot do. The dielectric also sets the speed. The wave travels at c/√ε_r, so the velocity factor is 1/√ε_r. For solid polyethylene, ε_r = 2.25 and the velocity factor is 0.667.

• Three numbers define a coax : the outer diameter D, the inner diameter d and the dielectric constant ε_r.
• The coax carries a TEM wave : the TEM mode has no lower cutoff, so a coax passes everything from DC up to its upper cutoff.
• The dielectric slows the wave : the velocity factor is 1/√ε_r, which is 0.667 for solid polyethylene.

Capacitance and Inductance

A transmission line is modelled by its capacitance and inductance per meter. For a coax both values come from the geometry alone, so let's look at the two formulas and then put numbers into them.

Capacitance per meter of a coax. It grows with the dielectric constant and falls as D/d grows.

• First line: C = 2πε₀ε_r/ln(D/d) in farads per meter, with ε₀ = 8.854 x 10^-12 F/m.
• Second line: the same formula in pF/m with the constant 24.13. The constant 2πε₀ is 55.63 pF/m, and 55.63/ln(10) = 24.16. So the second line holds with log₁₀(D/d) in the denominator, not ln(D/d) as printed.

Inductance per meter of a coax. It grows with ln(D/d) and does not depend on the dielectric.

• First line: L = (μ₀μ_r/2π) ln(D/d) in henries per meter, with μ₀ = 4π x 10^-7 H/m.
• Second line: 460.6 x ln(D/d) in nH/m. The constant μ₀/2π is 200 nH/m, and 200 x ln(10) = 460.5. So this line also holds with log₁₀(D/d), not ln(D/d) as printed. With ln, the constant is 200.

Let's check the formulas with a real case. A 50 ohm cable with a solid polyethylene dielectric, ε_r = 2.25, needs D/d = 3.49, as the next section shows. Then ln(D/d) = 1.25. The capacitance is 55.63 x 2.25/1.25 = 100 pF/m, and the inductance is 200 x 1.25 = 250 nH/m. These are the same values used on the Characteristic Impedance page.

• C and L depend only on geometry and material : C = 2πε₀ε_r/ln(D/d) and L = (μ₀/2π) ln(D/d) for a non-magnetic dielectric.
• Watch the logarithm : the constants 24.13 pF/m and 460.6 nH/m in the pictures belong to log₁₀, and the constants 55.63 pF/m and 200 nH/m belong to ln.
• A 50 ohm polyethylene cable has about 100 pF/m and 250 nH/m : these two numbers give both its impedance and its speed.

Characteristic Impedence

Now we can combine the two values. Because L grows with ln(D/d) and C falls with it, their ratio depends strongly on D/d, while the dielectric only scales the result.

Characteristic impedance of a coax. It depends on the diameter ratio and on the dielectric constant.

• First line: Z₀ = √(L/C) = (1/2π) √(μ₀μ_r/ε₀ε_r) ln(D/d). The picture leaves out the ln in front of D/d.
• Second line: Z₀ = (138/√ε_r) log(D/d), where log is log₁₀. The same formula with ln is (60/√ε_r) ln(D/d), because 60 x ln(10) = 138.2.

For a target impedance you can turn the formula around: D/d = exp(Z₀ x √ε_r/60). For 50 ohm with air, D/d = 2.30. For 50 ohm with solid polyethylene, D/d = 3.49. For 75 ohm, the ratio is larger, so a 75 ohm cable has a thinner center conductor than a 50 ohm cable with the same outer size and dielectric.

Why is 50 ohm the common value? For an air line with a fixed outer diameter, three goals give three different ratios. The conductor loss is lowest at D/d = 3.59, which is 76.7 ohm. The power handling is highest at D/d = 1.65, which is 30 ohm. The breakdown voltage is highest at D/d = 2.72, which is 60 ohm. 50 ohm sits between these values and is a compromise. 75 ohm, close to the low loss value, is used where the power is small and the loss matters, for example in video and cable TV. Note also that with a polyethylene dielectric, the low loss ratio 3.59 gives 76.7/1.5 = 51 ohm.

• Z₀ depends on the ratio D/d : Z₀ = (60/√ε_r) ln(D/d), so scaling the whole cable keeps its impedance.
• 50 ohm is a compromise : an air coax has its lowest loss at 76.7 ohm, its highest power at 30 ohm and its highest breakdown voltage at 60 ohm.
• The dielectric changes the geometry : for 50 ohm, air needs D/d = 2.30 and solid polyethylene needs D/d = 3.49.

Cutoff Frequency

A coax has no lower cutoff, but it has an upper one. Above a certain frequency, the space between the conductors is large enough for a second field pattern, the TE11 mode, to propagate. The two modes travel at different speeds, so the signal is distorted. The formula below gives that frequency.

Upper cutoff of a coax. A smaller cable and a lower dielectric constant give a higher cutoff frequency.

• Top: the cutoff wavelength is approximately the mean circumference of the dielectric, π(D + d)/2, multiplied by √(μ_Rε_R).
• Middle: the cutoff frequency is c divided by that wavelength.
• Bottom: with D and d in millimeters, f_cutoff = 190.85/((D + d)√ε_R) in GHz. The constant is 2c/π = 190.85 GHz x mm, which checks.

For example, take a polyethylene 50 ohm cable with D = 3 mm. Then d = 3/3.49 = 0.86 mm, and the cutoff is 190.85/(3.86 x 1.5) = 33 GHz. A cable twice as thick has half this cutoff. So the thin cables and connectors in the table below are the ones that reach the highest frequencies.

Tables for Max Frequency for Typical Connectors

A precision connector is named after the inner diameter of its outer conductor, the D of the formula. The table below gives the frequency up to which each size is normally used.

These connectors use air as the dielectric, so ε_r = 1 and D/d = 2.30 for 50 ohm. The cutoff formula then gives the values in the table below. Each recommended maximum in the table above sits below the computed cutoff. The difference is a safety margin, and the support beads and the mating interface of a real connector also lower the limit.

• A coax has an upper cutoff : above it the TE11 mode propagates too, and the signal is distorted.
• Smaller means higher frequency : the cutoff is inversely proportional to D + d, so a higher frequency needs a thinner cable or connector.
• The connector name is its outer diameter : the 3.5 mm to 1 mm connectors follow the same cutoff rule, with a margin below the computed value.