The usual explanation for 50 Ω and 75 Ω coax: three optima, using electrostatic FEM and formulas - DEV Community

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The usual explanation for 50 Ω and 75 Ω coax: three optima, using electrostatic FEM and formulas

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Power and voltage at a fixed peak field, and inverse conductor loss, against characteristic impedance for an air-filled coax of fixed outer diameter. Studio points sit on the formula curves; the optima are at 30, 60 and 77 ohms

50 Ω and 75 Ω are common coaxial-cable impedances. The usual explanation is that there is no single best impedance. For an air-filled coax with a fixed outer diameter, three different goals have three different optima:

  • most power before the dielectric breaks down: 30 Ω
  • most voltage before the dielectric breaks down: 60 Ω
  • lowest conductor loss: 77 Ω

50 Ω is commonly described as a compromise between the first and the last, and 75 Ω as sitting close to the low-loss optimum.

For the voltage and power optima, I computed capacitance per metre (which sets Z0) and peak electric field with NovaSolver Studio's electrostatic solver for six inner radii and compared them with formulas. The conductor-attenuation optimum comes from a separate formula.

Setup

Outer conductor radius b = 3.5 mm, air (εr = 1), 1000 V on the inner conductor, 0 V on the outer, 10 mm long with zero normal electric flux at both end faces, representing a uniform section of a long coax. The inner radius is a = b/q for six ratios q = b/a. The formulas are:

C′=2πε0ln⁡q,Z0=1c C′≈60ln⁡q Ω,Emax⁡=Valn⁡q C' = \frac{2\pi\varepsilon_0}{\ln q}, \qquad Z_0 = \frac{1}{c\,C'} \approx 60\ln q\ \Omega, \qquad E_{\max} = \frac{V}{a\ln q}

Results

b/a Z0 from Studio C, Studio C, formula Peak field, Studio Peak field, formula
1.65 (√e) 30.0 Ω 1.1127 pF 1.1127 pF 0.943 kV/mm 0.942
2 41.6 Ω 0.8026 pF 0.8026 pF 0.826 kV/mm 0.824
2.72 (e) 60.0 Ω 0.5563 pF 0.5563 pF 0.779 kV/mm 0.777
3 65.9 Ω 0.5064 pF 0.5064 pF 0.783 kV/mm 0.780
3.59 76.7 Ω 0.4352 pF 0.4352 pF 0.803 kV/mm 0.803
5 96.5 Ω 0.3457 pF 0.3457 pF 0.892 kV/mm 0.888

The capacitance matched the formula to within 0.02% in all six cases, and Studio's own check of the capacitance against the coax formula (tolerance 1%) passed each time. The peak field is the largest value anywhere in the mesh and came out 0.1–0.5% above the formula.

The peak field is lowest at b/a = e. That is the 60 Ω optimum. For a fixed breakdown field, you can apply the most voltage there.

From two numbers to three optima

If the dielectric breaks down at a field E_bd, the largest voltage is V = E_bd · a · ln q. Studio's peak field per 1000 V gives this directly. For a matched line, V is the sinusoidal peak voltage and P = V² / (2 Z0) is the average transmitted power. At a fixed frequency, with equal surface resistance on both conductors and strong skin effect in a low-loss line, conductor attenuation αc is proportional to (1/a + 1/b) / ln q; this comes from a formula, because an electrostatic solve does not give it.

b/a Z0 Max voltage Max power Conductor loss
1.65 30 Ω 0.83 1.00 1.48
2.72 60 Ω 1.00 0.73 1.04
3.59 77 Ω 0.97 0.54 1.00

Each column is relative to its own best value. Voltage and power come from Studio's peak field and Z0, and loss from the formula. From the formula curves, a 50 Ω line gets 98% of the best voltage and 86% of the best power, with 10% more loss than the minimum. A 75 Ω line has essentially the minimum loss but carries only 56% of the best power.

Two notes on the numbers:

  • These optima assume air. A solid dielectric scales Z0 by 1/√εr. With polyethylene (εr ≈ 2.25), the low-loss geometry gives about 51 Ω instead of 77 Ω.
  • The power optimum assumes the limit is the peak field. In real cables, heating often limits continuous power first.

How it ran

The cases are Studio's coax template with two parameters changed. The main MCP calls for one case are excerpted below; ID extraction, input checks and completion waits via job_status are omitted:

case = create_case(template="coax")
update_case(case_id=case, expected_revision=1, changes={
    "geometry.params.a": "1.28757 mm",     # b/a = e
    "materials.solid.eps_r": 1,
})
check_convergence(case_id=case)
verify_case(case_id=case)   # -> computed 0.55634 pF, reference 0.55632 pF, error +0.00% (tolerance 1.0%) -> PASS
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These runs used a local development build after v0.7.6 (2026-11-11).

Takeaways

  • For an air-filled coax with b/a from 1.65 to 5, Studio's capacitance matched the formula to within 0.02%, and its peak field to within 0.5%
  • For an air-filled line with a fixed outer diameter, the best power is at 30 Ω, the best voltage at 60 Ω and the lowest conductor loss at 77 Ω
  • In air, 50 Ω gives up little voltage and some power to stay near low loss, and 75 Ω sits near the low-loss optimum but gives up power. With a polyethylene dielectric the low-loss optimum moves to about 51 Ω

NovaSolver Studio is free and open source (GPL-3.0). It solves structural, thermal, flow and electrostatic problems in one app and can be driven from MCP clients such as Claude Desktop.

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