Coax Impedance Calculator computes characteristic impedance per transmission-line theory from inner conductor diameter, outer shield diameter, and relative dielectric constant.
Calculate Coaxial Cable Characteristic Impedance from Conductor Diameters
This calculator finds the characteristic impedance of a coaxial cable from the inner conductor diameter, the outer conductor (shield) diameter, and the dielectric constant between them. RF and microwave engineers, antenna and broadcast technicians, and test-equipment designers use it to verify impedance matching and predict propagation delay, distributed inductance and capacitance, and the onset of higher-order-mode interference.
How to Use
Enter inner conductor diameter (d1) and outer shield diameter (d2) in mm, cm, in, or mil, and the relative dielectric constant (εr). The tool returns characteristic impedance (Z₀) in ohms, propagation delay, velocity factor, distributed inductance and capacitance per unit length, and the TE₁₁ higher-order-mode cutoff frequency.
Formula
The formulas below follow classical transmission-line theory as presented in Pozar’s Microwave Engineering, the standard graduate-level reference for coaxial and waveguide analysis.
Step 1 — distributed inductance and capacitance per unit length:
$$L’ = \frac{\mu_0}{2\pi}\ln\left(\frac{d_2}{d_1}\right), \qquad C’ = \frac{2\pi \varepsilon_0 \varepsilon_r}{\ln(d_2/d_1)}$$
Step 2 — characteristic impedance from $Z_0 = \sqrt{L’/C’}$, which reduces to:
$$Z_0 = \frac{60}{\sqrt{\varepsilon_r}}\ln\left(\frac{d_2}{d_1}\right) = \frac{138}{\sqrt{\varepsilon_r}}\log_{10}\left(\frac{d_2}{d_1}\right)$$
Step 3 — velocity factor and propagation delay, from $v_p = c/\sqrt{\varepsilon_r}$:
$$\frac{v}{c} = \frac{1}{\sqrt{\varepsilon_r}}$$
Step 4 — the TE₁₁ higher-order-mode cutoff frequency, using the standard closed-form approximation for the coaxial TE₁₁ mode (exact solution requires a transcendental Bessel-function equation with no closed form, so this approximation is the one used in practice):
$$f_c \approx \frac{2c}{\pi\sqrt{\varepsilon_r}\,(d_1 + d_2)}$$
Non-obvious nuance: d2 must be the outer conductor’s inside diameter — the surface actually touching the dielectric — not the physical outside diameter of the shield braid or cable jacket. Two cables with identical outer jacket diameters can have noticeably different Z₀ if their shield wall thickness differs, because d2 in this formula is defined by where the dielectric ends, not where the cable ends.
Common input mistake: using the catalog (1 kHz or DC) dielectric constant for foam or air-spaced dielectrics, where εr is frequency-dependent and can differ from the nominal datasheet value at RF, shifting both Z₀ and velocity factor.
FAQ
Why does the same coax jacket diameter give different impedance readings?
Impedance depends on d2, the shield’s inside diameter against the dielectric, not the outer jacket diameter. Thicker braid or foil shielding reduces the effective d2 without changing the cable’s outside size.
Why are 50 Ω and 75 Ω the standard coax impedances?
For air-dielectric coax, minimum attenuation occurs near 77 Ω and maximum power handling near 30 Ω; 50 Ω was adopted as an engineering compromise between the two, per the SCTE’s Broadband Library technical history.
What happens if I operate above the TE₁₁ cutoff frequency?
Above cutoff, energy can couple into the TE₁₁ mode alongside the intended TEM mode, causing signal distortion and increased loss. Most designs stay below roughly 90% of fc for margin.
Does cable length affect characteristic impedance?
No. Z₀ is a property of the cross-section and dielectric only, independent of length, as long as the cable is uniform along its run per transmission-line theory.
Can this calculator be used for semi-rigid or corrugated hardline cable?
Semi-rigid cable follows these equations directly. Corrugated hardline has a non-circular effective cross-section and should be checked against the manufacturer’s published impedance instead.