Visual calculator 02 · Transmission lines
Coaxial line
The field lives between two conductors. Its shape sets the impedance, the loss, the power limit and the first higher mode.
Preset
Why 50 Ω: the ratio trade-off
Fixed outer diameter. Each curve peaks at its own optimum, labeled with its Z0 in this dielectric (30, 60 and 77 Ω in air).
Loss vs frequency
Smooth conductors, skin-effect limit. Shaded: TE11 can propagate.
Check
Method
Z0 = (η0 / 2π√εr) · ln(b/a) a = d/2, b = D/2
Rs = √(ωµ0 / 2σ)
αc = Rs (1/a + 1/b) / (4π Z0) [Np/m]
αd = (π f √εr / c) · tan δ [Np/m]
E(r) = V / (r ln(b/a)), max at r = a
Pmax = (Ebd · a · ln(b/a))² / (2 Z0) (peak, matched)
TE11: J1′(x) Y1′(cx) − J1′(cx) Y1′(x) = 0,
x = kc·a, c = b/a (smallest root)
approx. kc ≈ 2/(a+b)
Conductor loss uses a smooth surface (no roughness) and the same metal for both conductors. Power is the peak-field limit only, not the thermal (average-power) limit. The default breakdown field, 3 kV/mm, is air at sea level; for filled cables treat it as a conservative air-gap figure.
Reference values
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012, Sec. 2.7 and 3.5 (coax line constants, loss, TE11).
- N. Marcuvitz, Waveguide Handbook, MIT Rad. Lab. Series vol. 10, 1951, Sec. 2.4 (coaxial modes).
- M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 1964, §9.1–9.2 (Bessel functions).
- Cable data: Belden 8240 (RG-58/U); MIL-DTL-17 M17/130-RG402; Times Microwave LMR-400; IEC 61169-16 (N), IEC 61169-4 (7/16); IEEE Std 287 (3.5 mm, 2.4 mm air lines).