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Coaxial Cable Impedance: Diameter Ratio, Dielectric and Delay

Coax impedance comes from a diameter ratio, not a particular cable diameter. Making both conductors twice as large keeps an ideal coax at the same characteristic impedance. It changes loss and the frequency at which higher-order modes can propagate, so impedance alone does not identify a usable cable. [1, 2]

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The concentric-coax formula

Let d be the outside diameter of the centre conductor and D the inside diameter of the shield. For a uniform dielectric with relative permittivity εr, non-magnetic conductors and a TEM wave, the characteristic impedance is [1, 2]:

Z0 = (η0 / 2π√εr) ln(D/d) ≈ (59.96 Ω / √εr) ln(D/d)

Use the natural logarithm, ln. If the logarithm on a calculator is base 10, the equivalent coefficient is about 138 Ω. The dielectric and the ratio D/d both matter: a higher εr lowers Z0, while a larger space between the conductors raises it.

A worked 50 Ω example

Take d = 1.00 mm, D = 3.50 mm and εr = 2.25. The formula gives Z0 = 59.96/1.5 × ln(3.5) = 50.08 Ω. The velocity factor is 1/√2.25 = 0.667, so the delay is √2.25/c = 5.00 ns/m. At 1 GHz a quarter wavelength in this dielectric is about 50.0 mm. These are ideal uniform-line values, not a guarantee for a finished cable. [2]

Check the diameter ratio and delay in the coax calculator →

Choosing a diameter or a cable

For a target impedance, rearrange the formula: D/d = exp(2πZ0√εr/η0). This solves an ideal geometry, but a cable data sheet may specify a velocity factor rather than a single bulk εr. It may also use a foamed dielectric, a plated or stranded conductor and a braided shield. Enter its documented dimensions or velocity factor; do not treat the ideal smooth-conductor loss estimate as a cable data-sheet rating. [2, 3]

The coax tool also estimates the first higher-order TE11 cutoff. TEM operation does not end at that exact number in a real connector assembly: connectors, bends and an off-centre conductor can disturb the field, and loss usually matters well before the cutoff. Use the cable and connector data sheets for operating limits. [2, 4]

Tools used in this guide

  • Coaxial Line Impedance Calculator – Characteristic impedance of a coaxial line from D, d and εr, or the diameter for a target Z0, with offset centre conductor, delay, C and L per metre, TE11 cutoff and loss.
  • Quarter-Wave Transformer & Stub Calculator – Single and multisection binomial or Chebyshev λ/4 transformers with exact response and bandwidth, open and short stubs, and single-stub matching, with physical lengths and trace widths.

Sources

  1. Analog Devices, Transmission Lines and Standing Waves laboratory guide: characteristic impedance from distributed L and C and the coaxial diameter-ratio formula.
  2. D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012, §2.2/Table 2.1 and §2.7: coaxial TEM line parameters and conductor loss.
  3. Analog Devices, Impact of Cable Losses: skin-effect and dielectric-loss contributions to cable attenuation.
  4. Rohde & Schwarz, Cable Impedance Measurement: measuring a cable impedance with a VNA and a quarter-wave section.

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