A transmission line's characteristic impedance Z₀ depends only on its cross-section geometry and the dielectric between the conductors — not on length. For coaxial cable Z₀ = (138/√εr) · log10(D/d), where D is the outer-shield ID and d is the center conductor OD. For parallel two-wire (ladder line) Z₀ = (276/√εr) · log10(2s/d). 50 Ω is the practical compromise between minimum loss (~77 Ω) and maximum power handling (~30 Ω); 75 Ω is the minimum-loss point for air-dielectric coax and the broadcast/CATV standard.

What this calculator does

This calculator returns the characteristic impedance Z₀ of either a coaxial cable (given the outer-shield inner diameter D, center-conductor outer diameter d, and dielectric constant εr) or a parallel two-wire transmission line (given conductor diameter d, center-to-center spacing s, and εr). Both formulas are textbook quasi-static derivations from the L and C per unit length and are accurate to <1% for any standard cable. The dimension defaults match RG-58-class coax (D≈7.25 mm, d≈2 mm, polyethylene); other defaults can be entered for whatever physical cable you are characterizing.

How it works — the formula

Coax:           Z₀ = (138 / √εr) · log10(D / d)
Parallel wire:  Z₀ = (276 / √εr) · log10(2s / d)

Both forms derive from Z₀ = √(L/C), with L the inductance per unit length and C the capacitance per unit length of the geometry. The √εr in the denominator captures the dielectric's effect on capacitance. The 138 and 276 constants encode the natural-log-to-base-10 conversion and the free-space wave impedance (η₀ ≈ 376.7 Ω). These formulas are quasi-static and apply when conductor spacing is much less than wavelength — true for any practical coax through tens of GHz.

Sources: Pozar — Microwave Engineering, 4th ed., §2.5 (coaxial line) and §2.6 (parallel-wire line) · ITU-R Handbook — Transmission line dimensions and characteristic impedance · IEEE Std 145-2013 — antenna and transmission line terminology

Worked examples

Example 1

RG-58 coax (PE dielectric)

Inputs:: type=coax, D=2.95 mm, d=0.81 mm, εr=2.3
Output:: Z₀ ≈ 51 Ω

Matches the nominal 50 Ω rating.

Example 2

300 Ω twin-lead (TV ribbon)

Inputs:: type=parallel, d=0.81 mm, s=7.5 mm, εr=1.5
Output:: Z₀ ≈ 301 Ω

Classic FM antenna feed-line.

Example 3

450 Ω ladder line

Inputs:: type=parallel, d=1.5 mm, s=25 mm, εr=1.0
Output:: Z₀ ≈ 462 Ω

Open-wire ladder line for HF antennas — very low loss but more environmentally sensitive than coax.

When to use this vs other tools

Use this to verify cable characteristic impedance from physical dimensions, or to design a custom transmission line. For ready-made cables, the impedance is already on the spec sheet.

Coax Loss
Use to estimate per-100-ft attenuation for the same cable once you have confirmed the impedance.
SWR
Use to check whether the line and load impedances are matched well enough at the operating frequency.
Antenna Length
Use to size the antenna that the transmission line will feed.
Audio Impedance
Use for low-frequency audio applications where lumped-element analysis suffices.

Authority note

ITU-R Handbook & IEEE Std 145-2013

International Telecommunication Union Radiocommunication Sector — transmission line formulas; companion IEEE 145-2013 terminology

The coax and parallel-wire Z₀ formulas appear in every microwave-engineering reference; the ITU and IEEE codify them for international consistency in radiocommunication and antenna engineering.

Limitations

Quasi-static — assumes line geometry is much smaller than the operating wavelength.
Lossless model — does not capture frequency-dependent loss or dispersion in the dielectric.
Foamed-dielectric cables have an effective εr different from the solid-dielectric value of the same polymer; check the cable spec.
Mechanical tolerances of ±5% on D, d, or s typically yield ±2-3% on Z₀ — measure the physical cable for production-critical work.

For lab-quality precision, characterize the actual line with a vector network analyzer rather than relying on nominal geometry.

Frequently asked

Why are common cables 50 Ω and 75 Ω?+

50 Ω is the geometric mean of the loss minimum (~77 Ω) and the power-handling maximum (~30 Ω) for typical coax dielectrics. 75 Ω is the loss minimum and is used where loss matters more than power (TV/satellite).

What is εr (relative permittivity)?+

A number telling you how much the dielectric slows electromagnetic waves vs vacuum. Air ≈ 1.0, solid polyethylene ≈ 2.3, foamed polyethylene ≈ 1.5, PTFE (Teflon) ≈ 2.1, FR-4 PCB ≈ 4.4.

Do these formulas account for loss?+

No — they give the lossless characteristic impedance. Real cables have a small resistive component that you can usually ignore for matching purposes.

Why is parallel-line Z₀ usually higher?+

Less capacitance per length, more inductance per length — both push Z₀ up. 300-600 Ω ladder line is common in HF antennas; 50-100 Ω coax is more common otherwise.

PCB microstrip / stripline?+

Different formulas (Wheeler / Hammerstad). They depend on the substrate height and have many more correction terms. This calculator covers coax and parallel-wire only.