Coaxial Line Impedance Calculator

Method, formulas and references

Concentric line

For a centre conductor of diameter d inside a shield of inner diameter D, filled with a dielectric of relative permittivity ε_r (μ_r = 1), the line constants per metre are (Pozar §2.2, Table 2.1):

with η₀ = 376.730 Ω. The wave travels at c/√ε_r, so the velocity factor is 1/√ε_r and the delay is √ε_r/c per metre. L′ is the external inductance; at radio frequencies the field inside the conductors is confined to the skin depth and adds little. Cable datasheets give the velocity of propagation rather than ε_r; the velocity factor input uses ε_r = 1/VF².

To solve for a target impedance, (2) is inverted: D = d e^G or d = D e^−G with G = 2π√ε_r Z₀/η₀.

Offset (eccentric) centre conductor

If the centre of the inner conductor is displaced by s from the axis of the shield, the capacitance of the two eccentric cylinders (McDonald, eq. 14, from the image method) gives

This is the familiar handbook form (60/√ε_r) acosh(…) with the exact η₀/2π = 59.958 Ω instead of 60. At s = 0 the argument equals cosh(ln(D/d)), so (3) reduces to (2). The offset always lowers Z₀, slowly at first (by about (η₀/2π√ε_r) · 4s²/(D² − d²) for a small offset) and towards zero as the conductors touch at s = (D − d)/2. For a target impedance the quadratic in D or d is solved directly. The tests compare (3) with an independent image-charge solution of the same geometry.

Highest usable frequency

The first higher-order mode of a coaxial line is TE11. Pozar (§3.5) gives the approximate cutoff wavenumber k_c ≈ 2/(a + b) for radii a and b, so

Above this frequency the line can carry TE11 as well as TEM, and connectors and bends convert energy between them. The value shown with an offset is the concentric approximation.

Attenuation

For smooth conductors of conductivity σ, Pozar (§2.7, attenuation of a coaxial line) gives the conductor loss, and the dielectric loss of any TEM line filled with one dielectric is (§3.1):

One neper is 20 log₁₀ e = 8.686 dB. α_c rises with √f and α_d with f. Equation (5) needs the skin depth δ = 1/√(πfμ₀σ) to be small compared with the centre conductor; the tool warns when δ exceeds a tenth of its radius. It also assumes solid, smooth conductors with the current spread evenly around each, so it is not used with an offset, and braided shields, plating and roughness make real cable lossier. The default σ = 58 MS/m is annealed copper (1/58 Ω·mm²/m at 20 °C).

No cable presets

Cable datasheets list conductor and dielectric diameters and the velocity of propagation, but seldom ε_r itself, and nominal dimensions carry tolerances that move Z₀ by an ohm or more. Rather than ship unverified preset values, enter the dimensions from the datasheet of the cable you use, with its velocity factor.

References

1. D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012: §2.2 and Table 2.1 (coaxial line parameters), §2.7 (attenuation of a coaxial line), §3.1 (dielectric loss of TEM lines) and §3.5 (coaxial line, higher-order modes).

Data sources

1. K. T. McDonald, "An Off-Center 'Coaxial' Cable", Joseph Henry Laboratories, Princeton University, 1999
   Eq. (14): capacitance of eccentric cylinders, C = 1/(2 cosh⁻¹((a² + b² − δ²)/2ab)) in Gaussian units, i.e. 2πε / acosh(…) in SI; eq. (12) small-offset expansion, used as a test case.
2. NBS Handbook 100, Copper Wire Tables (1966)
   International Annealed Copper Standard: 1/58 Ω·mm²/m at 20 °C, the default conductivity of 58 MS/m (the same value as IEC 60028).
3. NIST CODATA: characteristic impedance of vacuum
   η0 = 376.730313668 Ω (CODATA 2018), with c = 299 792 458 m/s; μ0 = η0/c and ε0 = 1/(η0 c) follow.