Transmission Line Impedance Calculator
Coax or parallel-wire dimensions + dielectric → characteristic impedance Z₀.
Result
- GeometryCoax (concentric)
- Outer ID D7.25 mm
- Inner OD d2 mm
- D/d ratio3.625
- Dielectric εr2.3
- Characteristic Z₀50.89 Ω
Step-by-step
- Z₀ = (138 / √εr) · log10(D / d)
- = (138 / √2.3) · log10(7.25/2)
- = 90.995 · log10(3.6250)
- = 50.894 Ω.
How to use this calculator
- Pick line geometry (coax or parallel wire).
- Enter conductor and spacing dimensions in mm.
- Enter dielectric constant εr (air ≈ 1.0, polyethylene ≈ 2.3, PTFE ≈ 2.1).
- Read Z₀.
About this calculator
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.
Worked examples
- Inputs:
- type=coax, D=2.95 mm, d=0.81 mm, εr=2.3
- Output:
- Z₀ ≈ 51 Ω
Matches the nominal 50 Ω rating.
- Inputs:
- type=parallel, d=0.81 mm, s=7.5 mm, εr=1.5
- Output:
- Z₀ ≈ 301 Ω
Classic FM antenna feed-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
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.